Abstract group 1536.201107024: $S_3\times C_2^5:D_4$

• Label: 1536.201107024
• Order: \( 2^{9} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \cdot 3 \)
• Perm deg.: not computed
• Trans deg.: $96$
• Rank: $5$

Group information

Description:	$S_3\times C_2^5:D_4$
Order:	\(1536\)\(\medspace = 2^{9} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2\times C_2^7.A_4.C_2^5\times S_3$, of order \(1179648\)\(\medspace = 2^{17} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, metabelian, and rational.

Group statistics

Order	1	2	3	4	6	12
Elements	1	383	2	640	190	320	1536
Conjugacy classes	1	57	1	40	28	20	147
Divisions	1	57	1	40	28	20	147
Autjugacy classes	1	14	1	6	8	3	33

Dimension	1	2	4	8	16
Irr. complex chars.	32	72	36	6	1	147
Irr. rational chars.	32	72	36	6	1	147

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$96$
Rank:	$5$
Inequivalent generating 5-tuples:	$2133196800$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{4}=e^{2}=f^{4}=g^{2}=[a,c]= \!\cdots\! \rangle}$
Direct product:	$S_3$ $\, \times\, $ $(C_2^5:D_4)$
Semidirect product:	$C_2^5$ $\,\rtimes\,$ $(S_3\times D_4)$	$(D_6.D_4^2)$ $\,\rtimes\,$ $C_2$	$(D_6\times C_2^4)$ $\,\rtimes\,$ $D_4$	$(D_6:C_2^4)$ $\,\rtimes\,$ $D_4$	all 109
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(D_4\times D_6)$ (3)	$D_6$ . $(C_2^4:D_4)$	$D_6$ . $(C_2^4:D_4)$	$C_6$ . $(C_2^5:D_4)$	all 49

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{13}$
Commutator length:	$1$

Subgroups

There are 198282 subgroups in 39956 conjugacy classes, 1147 normal (43 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^5:D_6$
Commutator:	$G' \simeq$ $C_2^3\times C_6$	$G/G' \simeq$ $C_2^5$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2^3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_2^5:D_4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $S_3\times C_2^5:D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4:D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times C_2^4.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 1536: $S_3\times C_2^5:D_4$

Order 768: $(C_2^2\times D_{12}):D_4$ x 4, $(C_2^2\times D_{12}):D_4$ x 4, $S_3\times C_2^4.D_4$ x 4, $C_2\wr C_2^2\times D_6$ x 4, $D_6.D_4^2$ x 3, $C_2^5:D_{12}$ x 3, $(C_2^3\times C_{12}):D_4$ x 3, $C_2^5:C_4\times S_3$ x 3, $(C_2^3\times C_{12}):D_4$, $C_3\times C_2^5:D_4$, $D_{12}:C_2^5$

Order 512: $C_2\times C_2^4.C_2^4$

Order 384: $D_{12}:C_2^4$ x 52, $C_2\wr C_2^2\times S_3$ x 32, $(C_2^2\times D_{12}):C_4$ x 12, $C_2^4.D_{12}$ x 12, $(C_2^3\times C_4):D_6$ x 12, $C_2^5:D_6$ x 12, $C_2^3:C_4\times D_6$ x 12, $C_2^4:D_{12}$ x 12, $C_6.D_4^2$ x 9, $(C_4\times D_6):D_4$ x 9, $C_2^5:D_6$ x 9, $C_2^5:D_6$ x 9, $C_{12}:C_2^5$ x 7, $C_6.D_4^2$ x 6, $S_3\times C_2^3:D_4$ x 6, $C_2^4.D_{12}$ x 6, $D_{12}:C_2^4$ x 6, $C_2^5:D_6$ x 6, $(C_2^3\times C_{12}):C_2^2$ x 4, $(C_6\times D_4).D_4$ x 4, $C_2\wr C_2^2\times C_6$ x 4, $(C_6\times D_4):D_4$ x 4, $C_2^4:D_{12}$ x 3, $C_2^5.D_6$ x 3, $C_2^5:C_{12}$ x 3, $C_2^5.D_6$ x 3, $D_{12}:C_2^4$ x 3, $D_{12}:C_2^4$ x 3, $C_6.D_4^2$ x 3, $C_{12}.C_2^5$

Order 256: $C_2^5:D_4$ x 16, $C_2^5:D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^5:D_4$ x 3, $C_2^6:C_4$ x 3, $D_4:C_2^5$

Order 192: $D_{12}:C_2^3$ x 528, $C_{12}:C_2^4$ x 281, $D_{12}:C_2^3$ x 222, $D_{12}:C_2^3$ x 144, $D_{12}:C_2^3$ x 144, $C_2^4:D_6$ x 72, $C_2^4:D_6$ x 48, $C_2^3:C_4\times S_3$ x 48, $C_2^3:D_{12}$ x 48, $C_{12}.C_2^4$ x 26, $C_2^3.D_{12}$ x 24, $C_2^4.D_6$ x 24, $C_2^4:D_6$ x 21, $C_2^3:D_{12}$ x 21, $C_2^4.D_6$ x 18, $C_2^3:D_{12}$ x 18, $C_3\times C_2\wr C_2^2$ x 16, $(C_2\times C_{12}):D_4$ x 16, $D_6:C_2^4$ x 14, $C_2^4:C_{12}$ x 12, $C_2^4.D_6$ x 12, $C_2^4.D_6$ x 12, $C_2^3.D_{12}$ x 12, $(C_2\times C_4):D_{12}$ x 12, $(C_2\times C_{12}):D_4$ x 9, $C_2^4.D_6$ x 9, $C_2^3.D_{12}$ x 9, $C_2^3.D_{12}$ x 9, $C_{12}.C_2^4$ x 9, $D_{12}:C_2^3$ x 9, $C_2^5:C_6$ x 9, $C_2^4:D_6$ x 9, $C_{12}:C_2^4$ x 7, $C_2^3\times D_{12}$ x 7, $C_{12}:C_2^4$ x 7, $C_3\times C_2^3:D_4$ x 6, $(C_2\times C_4):D_{12}$ x 6, $D_6.C_4^2$ x 6, $D_6\times C_2^4$ x 6, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $C_{12}.C_2^4$ x 3, $D_{12}:C_2^3$ x 3, $C_{12}.C_2^4$

Order 128: $C_2^4:D_4$ x 64, $D_4:C_2^4$ x 62, $C_2^4:D_4$ x 36, $C_2^4.D_4$ x 32, $C_2^5:C_4$ x 24, $C_2^5:C_4$ x 12, $C_2^4:D_4$ x 9, $C_2^4.D_4$ x 9, $D_4\times C_2^4$ x 7, $C_2^4:D_4$ x 6, $D_4:C_2^4$ x 3, $C_2^5:C_4$ x 3

Order 96: $D_4\times D_6$ x 2008, $D_4:D_6$ x 1176, $D_4:D_6$ x 768, $D_4:D_6$ x 768, $C_2^3:D_6$ x 278, $D_4:D_6$ x 216, $D_{12}:C_2^2$ x 216, $C_2^3\times D_6$ x 153, $C_{12}:C_2^3$ x 142, $C_{12}:C_2^3$ x 139, $C_2^2\times D_{12}$ x 139, $C_6.C_2^4$ x 136, $D_6.D_4$ x 90, $C_2^2:D_{12}$ x 84, $D_6:D_4$ x 84, $C_6.C_2^4$ x 72, $Q_8:D_6$ x 72, $C_2^2.D_{12}$ x 48, $C_2^2.D_{12}$ x 42, $C_2^4:C_6$ x 36, $C_2^4:S_3$ x 36, $C_2^3:C_{12}$ x 24, $C_2^3.D_6$ x 24, $Q_8\times D_6$ x 24, $C_2^3:C_{12}$ x 24, $C_2^3.D_6$ x 24, $C_6.C_4^2$ x 18, $C_2^3\times C_{12}$ x 7, $C_6.C_2^4$ x 7, $C_6.C_4^2$ x 6, $C_2^4\times C_6$ x 6, $C_6.C_2^4$ x 3, $C_6.C_2^4$

Order 64: $D_4:C_2^3$ x 920, $D_4\times C_2^3$ x 309, $D_4:C_2^3$ x 168, $C_2^3:D_4$ x 132, $C_2\wr C_2^2$ x 128, $C_2^4:C_4$ x 96, $C_2^3.D_4$ x 60, $C_2^3:D_4$ x 48, $C_2^4:C_4$ x 24, $C_2^4:C_4$ x 18, $C_2^4\times C_4$ x 7, $C_2^2.C_4^2$ x 6, $C_2^6$ x 6, $Q_8\times C_2^3$

Order 48: $S_3\times D_4$ x 1952, $C_6:D_4$ x 944, $C_2^2\times D_6$ x 912, $D_4:S_3$ x 576, $D_{12}:C_2$ x 576, $C_4\times D_6$ x 514, $C_6\times D_4$ x 472, $C_2\times D_{12}$ x 472, $D_4:C_6$ x 192, $D_{12}:C_2$ x 192, $D_6:C_4$ x 78, $C_2^3\times C_6$ x 73, $C_2^2\times C_{12}$ x 71, $C_6.C_2^3$ x 71, $S_3\times Q_8$ x 64, $C_2^2:C_{12}$ x 45, $C_6.D_4$ x 45, $C_6:Q_8$ x 36, $C_6\times Q_8$ x 12

Order 32: $C_2^2\times D_4$ x 2564, $D_4:C_2^2$ x 2080, $D_4:C_2^2$ x 1344, $C_2^2\wr C_2$ x 240, $C_2^3:C_4$ x 180, $C_2^5$ x 159, $C_2^3\times C_4$ x 156, $C_2^3:C_4$ x 96, $C_2^2\times Q_8$ x 28, $C_2.C_4^2$ x 24

Order 24: $C_2\times D_6$ x 1361, $C_3:D_4$ x 368, $C_4\times S_3$ x 232, $C_2^2\times C_6$ x 220, $D_{12}$ x 184, $C_3\times D_4$ x 184, $C_2\times C_{12}$ x 124, $C_6:C_4$ x 124, $C_3:Q_8$ x 48, $C_3\times Q_8$ x 16

Order 16: $C_2\times D_4$ x 3840, $D_4:C_2$ x 1536, $C_2^4$ x 985, $C_2^2\times C_4$ x 656, $C_2^2:C_4$ x 168, $C_2\times Q_8$ x 112

Order 12: $D_6$ x 465, $C_2\times C_6$ x 164, $C_{12}$ x 20, $C_3:C_4$ x 20

Order 8: $C_2^3$ x 1581, $D_4$ x 736, $C_2\times C_4$ x 480, $Q_8$ x 64

Order 6: $S_3$ x 29, $C_6$ x 28

Order 4: $C_2^2$ x 629, $C_4$ x 40

Order 3: $C_3$

Order 2: $C_2$ x 57

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1536: $S_3\times C_2^5:D_4$

Order 768: $D_6.D_4^2$, $C_2^5:D_{12}$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2^2\times D_{12}):D_4$, $C_3\times C_2^5:D_4$, $S_3\times C_2^4.D_4$, $C_2^5:C_4\times S_3$, $D_{12}:C_2^5$, $C_2\wr C_2^2\times D_6$

Order 512: $C_2\times C_2^4.C_2^4$

Order 384: $D_{12}:C_2^4$ x 5, $C_{12}:C_2^5$ x 3, $C_6.D_4^2$ x 2, $(C_4\times D_6):D_4$ x 2, $D_{12}:C_2^4$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^5:D_6$ x 2, $C_6.D_4^2$, $S_3\times C_2^3:D_4$, $C_2^4:D_{12}$, $C_2^5.D_6$, $(C_2^3\times C_{12}):C_2^2$, $C_2^5:C_{12}$, $(C_6\times D_4).D_4$, $C_2^5.D_6$, $(C_2^2\times D_{12}):C_4$, $C_2^4.D_{12}$, $(C_2^3\times C_4):D_6$, $C_2^4.D_{12}$, $C_{12}.C_2^5$, $D_{12}:C_2^4$, $D_{12}:C_2^4$, $C_2\wr C_2^2\times C_6$, $C_6.D_4^2$, $(C_6\times D_4):D_4$, $C_2^5:D_6$, $C_2\wr C_2^2\times S_3$, $C_2^3:C_4\times D_6$, $C_2^4:D_{12}$

Order 256: $C_2^5:D_4$ x 6, $D_4:C_2^5$, $C_2^5:D_4$, $C_2^5:D_4$, $C_2^5.D_4$, $C_2^6:C_4$

Order 192: $C_{12}:C_2^4$ x 23, $D_{12}:C_2^3$ x 8, $D_{12}:C_2^3$ x 8, $D_6:C_2^4$ x 6, $D_{12}:C_2^3$ x 5, $C_2^4.D_6$ x 5, $D_6\times C_2^4$ x 4, $D_{12}:C_2^3$ x 4, $C_2^4:D_6$ x 4, $C_2^3:D_{12}$ x 4, $C_2^4.D_6$ x 3, $C_2^3:D_{12}$ x 3, $C_2^3.D_{12}$ x 3, $C_{12}.C_2^4$ x 3, $C_{12}:C_2^4$ x 3, $C_2^3\times D_{12}$ x 3, $C_{12}:C_2^4$ x 3, $(C_2\times C_{12}):D_4$ x 2, $C_2^4.D_6$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3.D_{12}$ x 2, $C_2^4.D_6$ x 2, $(C_2\times C_4):D_{12}$ x 2, $C_{12}.C_2^4$ x 2, $D_{12}:C_2^3$ x 2, $C_2^5:C_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4:D_6$ x 2, $C_3\times C_2\wr C_2^2$, $C_2^4:C_{12}$, $C_3\times C_2^3:D_4$, $C_2^4:C_{12}$, $C_2^4.D_6$, $(C_2\times C_{12}):D_4$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^3.D_{12}$, $C_2^3:C_4\times S_3$, $C_2^3:D_{12}$, $(C_2\times C_4):D_{12}$, $D_6.C_4^2$, $C_{12}.C_2^4$, $D_{12}:C_2^3$, $C_{12}.C_2^4$

Order 128: $D_4:C_2^4$ x 9, $C_2^4:D_4$ x 9, $C_2^4:D_4$ x 5, $C_2^4.D_4$ x 4, $C_2^5:C_4$ x 4, $D_4\times C_2^4$ x 3, $C_2^4:D_4$ x 2, $C_2^4.D_4$ x 2, $D_4:C_2^4$, $C_2^5:C_4$, $C_2^4:D_4$, $C_2^5:C_4$

Order 96: $D_4\times D_6$ x 51, $C_2^3\times D_6$ x 31, $C_2^3:D_6$ x 25, $C_{12}:C_2^3$ x 15, $C_{12}:C_2^3$ x 13, $C_2^2\times D_{12}$ x 13, $D_4:D_6$ x 12, $C_2^2.D_{12}$ x 7, $D_4:D_6$ x 6, $D_4:D_6$ x 6, $D_{12}:C_2^2$ x 6, $D_6.D_4$ x 5, $D_4:D_6$ x 5, $C_2^3:C_{12}$ x 5, $C_2^3.D_6$ x 5, $C_2^2:D_{12}$ x 4, $C_2^4\times C_6$ x 4, $C_6.C_2^4$ x 4, $D_6:D_4$ x 4, $C_6.C_4^2$ x 3, $C_6.C_2^4$ x 3, $C_2^3\times C_{12}$ x 3, $C_6.C_2^4$ x 3, $Q_8:D_6$ x 3, $Q_8\times D_6$ x 3, $C_2^4:C_6$ x 2, $C_2^4:S_3$ x 2, $C_2^3:C_{12}$, $C_6.C_4^2$, $C_2^3.D_6$, $C_6.C_2^4$, $C_6.C_2^4$, $C_2^2.D_{12}$

Order 64: $D_4\times C_2^3$ x 35, $D_4:C_2^3$ x 23, $C_2^3:D_4$ x 14, $C_2^3.D_4$ x 12, $D_4:C_2^3$ x 11, $C_2^3:D_4$ x 8, $C_2^4:C_4$ x 5, $C_2^4:C_4$ x 5, $C_2^4:C_4$ x 4, $C_2^6$ x 4, $C_2\wr C_2^2$ x 4, $C_2^4\times C_4$ x 3, $C_2^2.C_4^2$, $Q_8\times C_2^3$

Order 48: $C_2^2\times D_6$ x 85, $C_6:D_4$ x 41, $C_4\times D_6$ x 25, $C_6\times D_4$ x 22, $S_3\times D_4$ x 21, $C_2\times D_{12}$ x 21, $C_2^3\times C_6$ x 20, $C_2^2\times C_{12}$ x 9, $C_6.C_2^3$ x 9, $D_6:C_4$ x 7, $D_4:S_3$ x 5, $D_{12}:C_2$ x 5, $C_2^2:C_{12}$ x 5, $C_6.D_4$ x 5, $D_4:C_6$ x 3, $C_6\times Q_8$ x 2, $D_{12}:C_2$ x 2, $S_3\times Q_8$ x 2, $C_6:Q_8$ x 2

Order 32: $C_2^2\times D_4$ x 102, $C_2^5$ x 35, $D_4:C_2^2$ x 24, $C_2^3:C_4$ x 22, $D_4:C_2^2$ x 21, $C_2^3\times C_4$ x 21, $C_2^2\wr C_2$ x 12, $C_2^2\times Q_8$ x 5, $C_2.C_4^2$ x 4, $C_2^3:C_4$ x 3

Order 24: $C_2\times D_6$ x 91, $C_2^2\times C_6$ x 35, $C_2\times C_{12}$ x 12, $C_6:C_4$ x 12, $C_3:D_4$ x 11, $C_4\times S_3$ x 8, $D_{12}$ x 6, $C_3\times D_4$ x 6, $C_3:Q_8$, $C_3\times Q_8$

Order 16: $C_2^4$ x 105, $C_2\times D_4$ x 105, $C_2^2\times C_4$ x 43, $C_2^2:C_4$ x 17, $D_4:C_2$ x 15, $C_2\times Q_8$ x 6

Order 12: $D_6$ x 40, $C_2\times C_6$ x 26, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 8: $C_2^3$ x 126, $C_2\times C_4$ x 32, $D_4$ x 23, $Q_8$ x 2

Order 6: $C_6$ x 8, $S_3$ x 6

Order 4: $C_2^2$ x 66, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 14

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1536: $S_3\times C_2^5:D_4$ ($C_1$)

Order 768: $(C_2^2\times D_{12}):D_4$ ($C_2$) x 4, $(C_2^2\times D_{12}):D_4$ ($C_2$) x 4, $S_3\times C_2^4.D_4$ ($C_2$) x 4, $C_2\wr C_2^2\times D_6$ ($C_2$) x 4, $D_6.D_4^2$ ($C_2$) x 3, $C_2^5:D_{12}$ ($C_2$) x 3, $(C_2^3\times C_{12}):D_4$ ($C_2$) x 3, $C_2^5:C_4\times S_3$ ($C_2$) x 3, $(C_2^3\times C_{12}):D_4$ ($C_2$), $C_3\times C_2^5:D_4$ ($C_2$), $D_{12}:C_2^5$ ($C_2$)

Order 384: $(C_2^2\times D_{12}):C_4$ ($C_2^2$) x 12, $C_2^4.D_{12}$ ($C_2^2$) x 12, $(C_2^3\times C_4):D_6$ ($C_2^2$) x 12, $C_2^5:D_6$ ($C_2^2$) x 12, $C_2^3:C_4\times D_6$ ($C_2^2$) x 12, $C_2^4:D_{12}$ ($C_2^2$) x 12, $D_{12}:C_2^4$ ($C_2^2$) x 8, $C_6.D_4^2$ ($C_2^2$) x 6, $(C_4\times D_6):D_4$ ($C_2^2$) x 6, $C_2^4.D_{12}$ ($C_2^2$) x 6, $C_2^5:D_6$ ($C_2^2$) x 6, $C_2^5:D_6$ ($C_2^2$) x 6, $C_2^5:D_6$ ($C_2^2$) x 6, $(C_2^3\times C_{12}):C_2^2$ ($C_2^2$) x 4, $(C_6\times D_4).D_4$ ($C_2^2$) x 4, $C_{12}:C_2^5$ ($C_2^2$) x 4, $C_2\wr C_2^2\times C_6$ ($C_2^2$) x 4, $(C_6\times D_4):D_4$ ($C_2^2$) x 4, $C_2^4:D_{12}$ ($C_2^2$) x 3, $C_2^5.D_6$ ($C_2^2$) x 3, $C_2^5:C_{12}$ ($C_2^2$) x 3, $C_2^5.D_6$ ($C_2^2$) x 3, $D_{12}:C_2^4$ ($C_2^2$) x 3, $C_6.D_4^2$ ($C_2^2$) x 3, $C_{12}.C_2^5$ ($C_2^2$)

Order 256: $C_2^5:D_4$ ($S_3$)

Order 192: $D_{12}:C_2^3$ ($D_4$) x 32, $C_2^3.D_{12}$ ($C_2^3$) x 24, $C_2^4:C_{12}$ ($C_2^3$) x 12, $C_2^4.D_6$ ($C_2^3$) x 12, $D_{12}:C_2^3$ ($C_2^3$) x 12, $C_{12}:C_2^4$ ($C_2^3$) x 12, $C_{12}:C_2^4$ ($D_4$) x 12, $(C_2\times C_{12}):D_4$ ($C_2^3$) x 6, $C_2^4.D_6$ ($C_2^3$) x 6, $C_2^3.D_{12}$ ($C_2^3$) x 6, $C_2^3.D_{12}$ ($C_2^3$) x 6, $C_2^3.D_{12}$ ($C_2^3$) x 6, $C_2^5:C_6$ ($C_2^3$) x 6, $C_2^4:D_6$ ($C_2^3$) x 6, $C_2^4:D_6$ ($C_2^3$) x 6, $C_2^3:D_{12}$ ($C_2^3$) x 6, $C_2^4.D_6$ ($C_2^3$) x 6, $D_6:C_2^4$ ($C_2^3$) x 5, $C_{12}.C_2^4$ ($C_2^3$) x 4, $C_{12}:C_2^4$ ($C_2^3$) x 4, $C_2^4:C_{12}$ ($C_2^3$) x 3, $C_2^4.D_6$ ($C_2^3$) x 3, $D_6\times C_2^4$ ($D_4$) x 3, $D_6:C_2^4$ ($D_4$) x 3, $C_2^3\times D_{12}$ ($D_4$) x 3, $C_{12}:C_2^4$ ($D_4$) x 3, $D_6\times C_2^4$ ($C_2^3$) x 2, $C_2^3\times D_{12}$ ($C_2^3$), $C_{12}:C_2^4$ ($C_2^3$)

Order 128: $C_2^4.D_4$ ($D_6$) x 4, $C_2^4:D_4$ ($D_6$) x 4, $C_2^5:C_4$ ($D_6$) x 3, $C_2^4:D_4$ ($D_6$) x 3, $D_4:C_2^4$ ($D_6$)

Order 96: $D_4:D_6$ ($C_2\times D_4$) x 48, $D_4\times D_6$ ($C_2\times D_4$) x 48, $C_2^3\times D_6$ ($C_2\times D_4$) x 27, $C_2^3:D_6$ ($C_2\times D_4$) x 18, $C_6.C_2^4$ ($C_2\times D_4$) x 16, $C_2^2\times D_{12}$ ($C_2\times D_4$) x 12, $C_{12}:C_2^3$ ($C_2\times D_4$) x 12, $C_{12}:C_2^3$ ($C_2^4$) x 6, $C_{12}:C_2^3$ ($C_2\times D_4$) x 6, $C_2^3:D_6$ ($C_2^4$) x 6, $C_2^3:C_{12}$ ($C_2^4$) x 6, $C_2^3.D_6$ ($C_2^4$) x 6, $C_2^4\times C_6$ ($C_2\times D_4$) x 3, $C_2^3\times D_6$ ($C_2^4$) x 3, $C_2^3\times C_{12}$ ($C_2\times D_4$) x 3, $C_6.C_2^4$ ($C_2\times D_4$) x 3, $C_2^4\times C_6$ ($C_2^4$) x 2, $C_2^3\times C_{12}$ ($C_2^4$), $C_6.C_2^4$ ($C_2^4$)

Order 64: $C_2^4:C_4$ ($C_2\times D_6$) x 12, $C_2^3.D_4$ ($C_2\times D_6$) x 6, $C_2^3:D_4$ ($C_2\times D_6$) x 6, $D_4:C_2^3$ ($C_2\times D_6$) x 4, $D_4\times C_2^3$ ($C_2\times D_6$) x 4, $C_2^4:C_4$ ($C_2\times D_6$) x 3

Order 48: $C_2^2\times D_6$ ($C_2^2\wr C_2$) x 40, $C_2^2\times D_6$ ($C_2^2\times D_4$) x 26, $C_6\times D_4$ ($C_2^2\times D_4$) x 24, $C_6:D_4$ ($C_2^2\times D_4$) x 24, $C_6:D_4$ ($C_2^2\wr C_2$) x 24, $C_2\times D_{12}$ ($C_2^2\wr C_2$) x 24, $C_4\times D_6$ ($C_2^2\wr C_2$) x 24, $C_2^3\times C_6$ ($C_2^2\times D_4$) x 12, $C_2^2\times C_{12}$ ($C_2^2\times D_4$) x 6, $C_6.C_2^3$ ($C_2^2\times D_4$) x 6, $C_2^3\times C_6$ ($C_2^5$)

Order 32: $D_4:C_2^2$ ($S_3\times D_4$) x 16, $C_2^2\times D_4$ ($C_2^2\times D_6$) x 6, $C_2^2\times D_4$ ($S_3\times D_4$) x 6, $C_2^3:C_4$ ($C_2^2\times D_6$) x 6, $C_2^5$ ($S_3\times D_4$) x 3, $C_2^3\times C_4$ ($S_3\times D_4$) x 3, $C_2^5$ ($C_2^2\times D_6$) x 2, $C_2^3\times C_4$ ($C_2^2\times D_6$)

Order 24: $C_2\times D_6$ ($C_2^3:D_4$) x 44, $C_2^2\times C_6$ ($C_2^3:D_4$) x 16, $C_2\times C_{12}$ ($C_2^3:D_4$) x 12, $C_6:C_4$ ($C_2^3:D_4$) x 12, $C_2^2\times C_6$ ($D_4\times C_2^3$) x 7, $C_2\times D_6$ ($C_2\wr C_2^2$) x 4

Order 16: $C_2\times D_4$ ($D_4\times D_6$) x 24, $C_2^4$ ($D_4\times D_6$) x 12, $C_2^2\times C_4$ ($D_4\times D_6$) x 6, $C_2^4$ ($C_2^3\times D_6$)

Order 12: $D_6$ ($C_2^3\wr C_2$) x 8, $C_2\times C_6$ ($C_2^4:D_4$) x 7, $D_6$ ($C_2^4:D_4$) x 4, $C_2\times C_6$ ($C_2^4:D_4$) x 2, $D_6$ ($C_2^4:D_4$) x 2

Order 8: $C_2^3$ ($C_2^4:D_6$) x 16, $C_2\times C_4$ ($C_2^4:D_6$) x 12, $C_2^3$ ($C_{12}:C_2^4$) x 7

Order 6: $S_3$ ($C_2^5:D_4$) x 4, $C_6$ ($C_2^5:D_4$), $C_6$ ($C_2^5:D_4$), $C_6$ ($C_2^5:D_4$)

Order 4: $C_2^2$ ($C_2^5:D_6$) x 7, $C_2^2$ ($C_2\wr C_2^2\times S_3$) x 2

Order 3: $C_3$ ($C_2\times C_2^4.C_2^4$)

Order 2: $C_2$ ($C_2^6:D_6$), $C_2$ ($C_2^4.C_2^3\times S_3$), $C_2$ ($C_2\wr C_2^2\times D_6$)

Order 1: $C_1$ ($S_3\times C_2^5:D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1536: $S_3\times C_2^5:D_4$ ($C_1$)

Order 768: $D_6.D_4^2$ ($C_2$), $C_2^5:D_{12}$ ($C_2$), $(C_2^2\times D_{12}):D_4$ ($C_2$), $(C_2^3\times C_{12}):D_4$ ($C_2$), $(C_2^3\times C_{12}):D_4$ ($C_2$), $(C_2^2\times D_{12}):D_4$ ($C_2$), $C_3\times C_2^5:D_4$ ($C_2$), $S_3\times C_2^4.D_4$ ($C_2$), $C_2^5:C_4\times S_3$ ($C_2$), $D_{12}:C_2^5$ ($C_2$), $C_2\wr C_2^2\times D_6$ ($C_2$)

Order 384: $D_{12}:C_2^4$ ($C_2^2$) x 2, $C_{12}:C_2^5$ ($C_2^2$) x 2, $C_2^5:D_6$ ($C_2^2$) x 2, $C_6.D_4^2$ ($C_2^2$), $(C_4\times D_6):D_4$ ($C_2^2$), $C_2^4:D_{12}$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $(C_2^3\times C_{12}):C_2^2$ ($C_2^2$), $C_2^5:C_{12}$ ($C_2^2$), $(C_6\times D_4).D_4$ ($C_2^2$), $C_2^5.D_6$ ($C_2^2$), $(C_2^2\times D_{12}):C_4$ ($C_2^2$), $C_2^4.D_{12}$ ($C_2^2$), $(C_2^3\times C_4):D_6$ ($C_2^2$), $C_2^4.D_{12}$ ($C_2^2$), $C_{12}.C_2^5$ ($C_2^2$), $D_{12}:C_2^4$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2\wr C_2^2\times C_6$ ($C_2^2$), $C_6.D_4^2$ ($C_2^2$), $(C_6\times D_4):D_4$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2^5:D_6$ ($C_2^2$), $C_2^3:C_4\times D_6$ ($C_2^2$), $C_2^4:D_{12}$ ($C_2^2$)

Order 256: $C_2^5:D_4$ ($S_3$)

Order 192: $D_6:C_2^4$ ($C_2^3$) x 3, $D_6\times C_2^4$ ($C_2^3$) x 2, $C_{12}:C_2^4$ ($C_2^3$) x 2, $C_{12}:C_2^4$ ($C_2^3$) x 2, $C_{12}:C_2^4$ ($D_4$) x 2, $C_2^4:C_{12}$ ($C_2^3$), $(C_2\times C_{12}):D_4$ ($C_2^3$), $C_2^4:C_{12}$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2^3$), $C_2^3.D_{12}$ ($C_2^3$), $D_6\times C_2^4$ ($D_4$), $C_{12}.C_2^4$ ($C_2^3$), $D_6:C_2^4$ ($D_4$), $D_{12}:C_2^3$ ($D_4$), $D_{12}:C_2^3$ ($C_2^3$), $C_2^3\times D_{12}$ ($C_2^3$), $C_2^3\times D_{12}$ ($D_4$), $C_{12}:C_2^4$ ($C_2^3$), $C_{12}:C_2^4$ ($D_4$), $C_2^5:C_6$ ($C_2^3$), $C_2^4:D_6$ ($C_2^3$), $C_2^4:D_6$ ($C_2^3$), $C_2^3:D_{12}$ ($C_2^3$), $C_2^4.D_6$ ($C_2^3$)

Order 128: $C_2^4.D_4$ ($D_6$), $C_2^5:C_4$ ($D_6$), $D_4:C_2^4$ ($D_6$), $C_2^4:D_4$ ($D_6$), $C_2^4:D_4$ ($D_6$)

Order 96: $C_2^3\times D_6$ ($C_2\times D_4$) x 7, $C_2^3\times D_6$ ($C_2^4$) x 3, $C_2^3:D_6$ ($C_2\times D_4$) x 3, $C_2^4\times C_6$ ($C_2^4$) x 2, $C_2^2\times D_{12}$ ($C_2\times D_4$) x 2, $C_{12}:C_2^3$ ($C_2\times D_4$) x 2, $C_2^4\times C_6$ ($C_2\times D_4$), $C_6.C_2^4$ ($C_2\times D_4$), $C_{12}:C_2^3$ ($C_2^4$), $C_{12}:C_2^3$ ($C_2\times D_4$), $C_2^3\times C_{12}$ ($C_2^4$), $C_2^3\times C_{12}$ ($C_2\times D_4$), $C_2^3:D_6$ ($C_2^4$), $C_6.C_2^4$ ($C_2^4$), $C_6.C_2^4$ ($C_2\times D_4$), $D_4:D_6$ ($C_2\times D_4$), $D_4\times D_6$ ($C_2\times D_4$), $C_2^3:C_{12}$ ($C_2^4$), $C_2^3.D_6$ ($C_2^4$)

Order 64: $D_4\times C_2^3$ ($C_2\times D_6$) x 2, $C_2^4:C_4$ ($C_2\times D_6$), $C_2^3.D_4$ ($C_2\times D_6$), $C_2^4:C_4$ ($C_2\times D_6$), $D_4:C_2^3$ ($C_2\times D_6$), $C_2^3:D_4$ ($C_2\times D_6$)

Order 48: $C_2^2\times D_6$ ($C_2^2\times D_4$) x 6, $C_2^2\times D_6$ ($C_2^2\wr C_2$) x 6, $C_2^3\times C_6$ ($C_2^2\times D_4$) x 3, $C_2^3\times C_6$ ($C_2^5$), $C_6\times D_4$ ($C_2^2\times D_4$), $C_2^2\times C_{12}$ ($C_2^2\times D_4$), $C_6:D_4$ ($C_2^2\times D_4$), $C_6:D_4$ ($C_2^2\wr C_2$), $C_6.C_2^3$ ($C_2^2\times D_4$), $C_2\times D_{12}$ ($C_2^2\wr C_2$), $C_4\times D_6$ ($C_2^2\wr C_2$)

Order 32: $C_2^5$ ($C_2^2\times D_6$) x 2, $C_2^5$ ($S_3\times D_4$), $D_4:C_2^2$ ($S_3\times D_4$), $C_2^2\times D_4$ ($C_2^2\times D_6$), $C_2^2\times D_4$ ($S_3\times D_4$), $C_2^3\times C_4$ ($C_2^2\times D_6$), $C_2^3\times C_4$ ($S_3\times D_4$), $C_2^3:C_4$ ($C_2^2\times D_6$)

Order 24: $C_2\times D_6$ ($C_2^3:D_4$) x 7, $C_2^2\times C_6$ ($C_2^3:D_4$) x 3, $C_2^2\times C_6$ ($D_4\times C_2^3$) x 2, $C_2\times C_{12}$ ($C_2^3:D_4$), $C_6:C_4$ ($C_2^3:D_4$), $C_2\times D_6$ ($C_2\wr C_2^2$)

Order 16: $C_2^4$ ($D_4\times D_6$) x 3, $C_2^4$ ($C_2^3\times D_6$), $C_2\times D_4$ ($D_4\times D_6$), $C_2^2\times C_4$ ($D_4\times D_6$)

Order 12: $C_2\times C_6$ ($C_2^4:D_4$) x 2, $D_6$ ($C_2^4:D_4$) x 2, $D_6$ ($C_2^3\wr C_2$) x 2, $C_2\times C_6$ ($C_2^4:D_4$), $D_6$ ($C_2^4:D_4$)

Order 8: $C_2^3$ ($C_2^4:D_6$) x 3, $C_2^3$ ($C_{12}:C_2^4$) x 2, $C_2\times C_4$ ($C_2^4:D_6$)

Order 6: $C_6$ ($C_2^5:D_4$), $C_6$ ($C_2^5:D_4$), $C_6$ ($C_2^5:D_4$), $S_3$ ($C_2^5:D_4$)

Order 4: $C_2^2$ ($C_2^5:D_6$) x 2, $C_2^2$ ($C_2\wr C_2^2\times S_3$)

Order 3: $C_3$ ($C_2\times C_2^4.C_2^4$)

Order 2: $C_2$ ($C_2^6:D_6$), $C_2$ ($C_2^4.C_2^3\times S_3$), $C_2$ ($C_2\wr C_2^2\times D_6$)

Order 1: $C_1$ ($S_3\times C_2^5:D_4$)

Series

Derived series

$S_3\times C_2^5:D_4$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_1$

Chief series

$S_3\times C_2^5:D_4$

$\rhd$

$S_3\times C_2^4.D_4$

$\rhd$

$(C_2^3\times C_{12}):C_2^2$

$\rhd$

$C_2^4.D_4$

$\rhd$

$D_4:C_2^3$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2\times D_4$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$S_3\times C_2^5:D_4$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^5$

$\lhd$

$C_2^5:D_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $147 \times 147$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.