Abstract group 2304.wb: $C_2^6:S_3^2$

• Label: 2304.wb
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_2^6:S_3^2$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.C_6^2.C_2^3$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	207	386	816	558	336	2304
Conjugacy classes	1	14	3	13	16	7	54
Divisions	1	14	3	13	12	5	48
Autjugacy classes	1	10	3	8	8	3	33

Dimension	1	2	3	4	6	12	24
Irr. complex chars.	4	4	12	1	24	9	0	54
Irr. rational chars.	4	4	12	1	16	9	2	48

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$18$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{6}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(4,6)(5,7)(8,14)(9,11)(10,12)(13,15), (1,2,3)(5,6)(8,13,11,12)(9,10,14,15) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^6$ $\,\rtimes\,$ $S_3^2$	$(C_2^4:S_4)$ $\,\rtimes\,$ $S_3$	$(C_2^4:S_3)$ $\,\rtimes\,$ $S_4$	$C_2^4$ $\,\rtimes\,$ $(S_3\times S_4)$	all 17
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(S_3\times S_4)$	$C_2^2$ . $(C_2^4:S_3^2)$	$(C_2^3\times C_6)$ . $(C_2\times S_4)$	$(C_2\times C_6)$ . $(C_2^3:S_4)$	more information
Aut. group:	$\Aut((C_2\times C_6).D_6^2)$

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 32312 subgroups in 2206 conjugacy classes, 28 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^6:S_3^2$
Commutator:	$G' \simeq$ $C_2^6:C_3^2$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^4:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^5\times C_6$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_2^6:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

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 2304: $C_2^6:S_3^2$

Order 1152: $(C_2^3\times C_6):S_4$, $C_3\times C_2^4:S_4$, $C_2^4:(S_3\times A_4)$

Order 768: $C_2^6:D_6$, $C_2^5:S_4$

Order 576: $C_2^4:S_3^2$ x 2, $C_2^6:C_3^2$, $C_2^4:S_3^2$

Order 384: $C_2^4:S_4$ x 3, $C_2^5.D_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2^5:D_6$ x 2, $C_2^5.D_6$, $C_2^4.D_{12}$, $C_2^4.D_{12}$, $C_2^5:D_6$, $C_2^5:D_6$, $C_2^5:A_4$, $C_2^6:C_6$, $C_2^6:S_3$

Order 288: $(C_2\times C_6):S_4$ x 5, $C_3\times C_2^2:S_4$ x 5, $(C_2\times D_6):A_4$ x 3, $S_3\times C_2^2:A_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^4:D_6$ x 11, $C_2^4:D_6$ x 8, $C_2^3.D_{12}$ x 6, $C_2^4:C_{12}$ x 5, $C_2^4.D_6$ x 5, $C_2^5:C_6$ x 5, $C_2^3:S_4$ x 2, $C_2^3.D_{12}$ x 2, $C_2^3.D_{12}$ x 2, $C_2^4.D_6$ x 2, $C_2^4:A_4$ x 2, $C_2^4:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^4:D_6$ x 2, $C_2^4.D_6$ x 2, $C_2^5\times C_6$, $C_2^3:S_4$, $D_6:C_2^4$, $D_{12}:C_2^3$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$

Order 144: $C_2^4:C_3^2$ x 10, $S_3\times S_4$ x 3

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

Order 96: $C_2^4:S_3$ x 35, $C_2^3:D_6$ x 20, $C_2^3.D_6$ x 20, $C_2^4:C_6$ x 18, $C_2^2:S_4$ x 17, $C_2^3:C_{12}$ x 13, $C_2^4\times C_6$ x 10, $C_2^2.D_{12}$ x 10, $D_4:D_6$ x 6, $C_2^3.D_6$ x 6, $C_{12}:C_2^3$ x 5, $D_6:D_4$ x 5, $D_6:D_4$ x 4, $C_2^2:D_{12}$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 4, $D_4\times D_6$ x 4, $D_6.D_4$ x 3, $C_2^3:A_4$ x 3, $C_2^2.D_{12}$ x 2, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_2^3.D_6$ x 2, $D_4:D_6$ x 2, $C_{12}:D_4$ x 2, $C_{12}:D_4$ x 2, $C_2^3.D_6$ x 2, $C_{12}:D_4$ x 2, $C_{12}:D_4$ x 2, $C_2^3.D_6$, $C_2^3\times D_6$, $C_2^3:A_4$, $C_6.C_2^4$, $D_{12}:C_2^2$, $C_2^3.D_6$, $C_2^3.D_6$

Order 72: $C_3:S_4$ x 5, $C_3\times S_4$ x 5, $S_3\times A_4$ x 4

Order 64: $C_2^3:D_4$ x 17, $C_2^4:C_4$ x 10, $C_2^4:C_4$ x 8, $C_2\wr C_2^2$ x 8, $C_2^2.C_4^2$ x 4, $D_4:D_4$ x 4, $D_4^2$ x 4, $C_2^3:D_4$ x 3, $C_2^6$, $D_4:C_2^3$, $D_4\times C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^4:C_4$

Order 48: $C_2^3\times C_6$ x 82, $C_6:D_4$ x 74, $C_6.D_4$ x 38, $C_6\times D_4$ x 26, $C_2^2:C_{12}$ x 26, $C_2^2:A_4$ x 24, $C_6.C_2^3$ x 13, $S_3\times D_4$ x 12, $D_4:S_3$ x 8, $D_6:C_4$ x 8, $C_2^2\times D_6$ x 7, $C_2^2\times C_{12}$ x 5, $D_{12}:C_2$ x 4, $C_4\times D_6$ x 4, $C_6.D_4$ x 4, $C_2\times S_4$ x 3, $C_2\times D_{12}$ x 3, $C_{12}:C_4$ x 2, $C_{12}:C_4$ x 2, $C_6:Q_8$

Order 36: $C_3\times A_4$ x 10, $S_3^2$

Order 32: $C_2^2\wr C_2$ x 62, $C_2^3:C_4$ x 36, $C_2^2\times D_4$ x 29, $C_2^3:C_4$ x 14, $C_4:D_4$ x 14, $C_2^5$ x 11, $C_4\times D_4$ x 8, $D_4:C_2^2$ x 6, $C_2^2.D_4$ x 6, $D_4:C_2^2$ x 3, $C_4:D_4$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_2^3\times C_4$, $C_4^2:C_2$

Order 24: $C_2^2\times C_6$ x 151, $C_3:D_4$ x 44, $C_6:C_4$ x 26, $C_2\times D_6$ x 25, $C_3\times D_4$ x 18, $S_4$ x 17, $C_2\times C_{12}$ x 13, $C_4\times S_3$ x 6, $D_{12}$ x 4, $C_2\times A_4$ x 4, $C_3:Q_8$ x 2

Order 18: $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$ x 115, $C_2^4$ x 89, $C_2^2:C_4$ x 72, $C_2^2\times C_4$ x 22, $D_4:C_2$ x 12, $C_4:C_4$ x 6, $C_4^2$ x 2, $C_2\times Q_8$

Order 12: $C_2\times C_6$ x 86, $A_4$ x 24, $D_6$ x 17, $C_3:C_4$ x 8, $C_{12}$ x 5

Order 9: $C_3^2$

Order 8: $C_2^3$ x 176, $D_4$ x 66, $C_2\times C_4$ x 45, $Q_8$ x 2

Order 6: $C_6$ x 12, $S_3$ x 6

Order 4: $C_2^2$ x 102, $C_4$ x 13

Order 3: $C_3$ x 3

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $C_2^6:S_3^2$

Order 1152: $(C_2^3\times C_6):S_4$, $C_3\times C_2^4:S_4$, $C_2^4:(S_3\times A_4)$

Order 768: $C_2^6:D_6$, $C_2^5:S_4$

Order 576: $C_2^6:C_3^2$, $C_2^4:S_3^2$, $C_2^4:S_3^2$

Order 384: $C_2^4:S_4$ x 3, $C_2^5.D_6$, $C_2^5.D_6$, $C_2^4:D_{12}$, $C_2^5.D_6$, $C_2^4.D_{12}$, $C_2^4.D_{12}$, $C_2^5:D_6$, $C_2^5:D_6$, $C_2^5:A_4$, $C_2^6:C_6$, $C_2^6:S_3$, $C_2^5:D_6$

Order 288: $(C_2\times C_6):S_4$ x 3, $C_3\times C_2^2:S_4$ x 3, $(C_2\times D_6):A_4$ x 2, $S_3\times C_2^2:A_4$

Order 256: $C_2^5:D_4$

Order 192: $C_2^4:D_6$ x 7, $C_2^4:D_6$ x 4, $C_2^4:C_{12}$ x 3, $C_2^4.D_6$ x 3, $C_2^3.D_{12}$ x 3, $C_2^5:C_6$ x 3, $C_2^4:A_4$ x 2, $C_2^3:S_4$, $C_2^3.D_{12}$, $C_2^3.D_{12}$, $C_2^4.D_6$, $C_2^5\times C_6$, $C_2^3:S_4$, $D_6:C_2^4$, $D_{12}:C_2^3$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^4:D_6$, $C_2^4.D_6$, $C_2^3:D_{12}$, $C_2^4.D_6$, $C_2^4.D_6$

Order 144: $C_2^4:C_3^2$ x 6, $S_3\times S_4$ x 2

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

Order 96: $C_2^4:S_3$ x 20, $C_2^3:D_6$ x 13, $C_2^3.D_6$ x 10, $C_2^2:S_4$ x 9, $C_2^4:C_6$ x 9, $C_2^4\times C_6$ x 6, $C_2^3:C_{12}$ x 6, $C_2^2.D_{12}$ x 5, $C_2^2:D_{12}$ x 3, $C_{12}:C_2^3$ x 3, $D_4:D_6$ x 3, $D_4\times D_6$ x 3, $C_2^3.D_6$ x 3, $D_6:D_4$ x 3, $C_2^2.D_{12}$ x 2, $D_6:D_4$ x 2, $C_2^3.D_6$ x 2, $D_6.D_4$ x 2, $C_2^3:A_4$ x 2, $C_2^3.D_6$ x 2, $D_4:D_6$ x 2, $C_2^3.D_6$, $D_6.D_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_2^3\times D_6$, $C_2^3:A_4$, $C_6.C_2^4$, $D_{12}:C_2^2$, $C_{12}:D_4$, $C_{12}:D_4$, $C_2^3.D_6$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$

Order 72: $S_3\times A_4$ x 3, $C_3:S_4$ x 3, $C_3\times S_4$ x 3

Order 64: $C_2^3:D_4$ x 11, $C_2^4:C_4$ x 6, $C_2^4:C_4$ x 4, $C_2\wr C_2^2$ x 4, $C_2^3:D_4$ x 3, $C_2^2.C_4^2$ x 2, $D_4:D_4$ x 2, $D_4^2$ x 2, $C_2^6$, $D_4:C_2^3$, $D_4\times C_2^3$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_2^4:C_4$

Order 48: $C_2^3\times C_6$ x 37, $C_6:D_4$ x 37, $C_6.D_4$ x 19, $C_2^2:A_4$ x 12, $C_6\times D_4$ x 12, $C_2^2:C_{12}$ x 12, $C_6.C_2^3$ x 9, $S_3\times D_4$ x 8, $C_2^2\times D_6$ x 6, $D_4:S_3$ x 4, $D_6:C_4$ x 4, $C_2^2\times C_{12}$ x 3, $C_4\times D_6$ x 3, $C_2\times S_4$ x 2, $D_{12}:C_2$ x 2, $C_2\times D_{12}$ x 2, $C_6.D_4$ x 2, $C_6:Q_8$, $C_{12}:C_4$, $C_{12}:C_4$

Order 36: $C_3\times A_4$ x 6, $S_3^2$

Order 32: $C_2^2\wr C_2$ x 35, $C_2^2\times D_4$ x 19, $C_2^3:C_4$ x 18, $C_2^3:C_4$ x 7, $C_2^5$ x 7, $C_4:D_4$ x 7, $C_2^2.D_4$ x 5, $C_4\times D_4$ x 4, $D_4:C_2^2$ x 3, $D_4:C_2^2$ x 3, $C_2^3\times C_4$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4^2:C_2$

Order 24: $C_2^2\times C_6$ x 60, $C_3:D_4$ x 22, $C_2\times D_6$ x 18, $C_6:C_4$ x 14, $S_4$ x 9, $C_3\times D_4$ x 9, $C_2\times C_{12}$ x 6, $C_4\times S_3$ x 4, $D_{12}$ x 3, $C_2\times A_4$ x 3, $C_3:Q_8$

Order 18: $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$ x 59, $C_2^4$ x 43, $C_2^2:C_4$ x 35, $C_2^2\times C_4$ x 15, $D_4:C_2$ x 6, $C_4:C_4$ x 3, $C_4^2$, $C_2\times Q_8$

Order 12: $C_2\times C_6$ x 39, $D_6$ x 13, $A_4$ x 12, $C_3:C_4$ x 5, $C_{12}$ x 3

Order 9: $C_3^2$

Order 8: $C_2^3$ x 78, $D_4$ x 34, $C_2\times C_4$ x 24, $Q_8$

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

Order 4: $C_2^2$ x 51, $C_4$ x 8

Order 3: $C_3$ x 3

Order 2: $C_2$ x 10

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $C_2^6:S_3^2$ ($C_1$)

Order 1152: $(C_2^3\times C_6):S_4$ ($C_2$), $C_3\times C_2^4:S_4$ ($C_2$), $C_2^4:(S_3\times A_4)$ ($C_2$)

Order 576: $C_2^6:C_3^2$ ($C_2^2$)

Order 384: $C_2^4:S_4$ ($S_3$), $C_2^5:D_6$ ($S_3$)

Order 192: $C_2^5\times C_6$ ($D_6$), $C_2^4:A_4$ ($D_6$)

Order 96: $C_2^4:S_3$ ($S_4$) x 2, $C_2^3\times D_6$ ($S_4$)

Order 64: $C_2^6$ ($S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_2\times S_4$) x 3

Order 24: $C_2\times D_6$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($S_3\times S_4$) x 3

Order 12: $C_2\times C_6$ ($C_2^3:S_4$) x 2, $C_2\times C_6$ ($C_2^3:S_4$)

Order 4: $C_2^2$ ($C_2^4:S_3^2$) x 2, $C_2^2$ ($C_2^4:S_3^2$)

Order 3: $C_3$ ($C_2^5:S_4$)

Order 1: $C_1$ ($C_2^6:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $C_2^6:S_3^2$ ($C_1$)

Order 1152: $(C_2^3\times C_6):S_4$ ($C_2$), $C_3\times C_2^4:S_4$ ($C_2$), $C_2^4:(S_3\times A_4)$ ($C_2$)

Order 576: $C_2^6:C_3^2$ ($C_2^2$)

Order 384: $C_2^4:S_4$ ($S_3$), $C_2^5:D_6$ ($S_3$)

Order 192: $C_2^5\times C_6$ ($D_6$), $C_2^4:A_4$ ($D_6$)

Order 96: $C_2^3\times D_6$ ($S_4$), $C_2^4:S_3$ ($S_4$)

Order 64: $C_2^6$ ($S_3^2$)

Order 48: $C_2^3\times C_6$ ($C_2\times S_4$) x 2

Order 24: $C_2\times D_6$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($S_3\times S_4$) x 2

Order 12: $C_2\times C_6$ ($C_2^3:S_4$), $C_2\times C_6$ ($C_2^3:S_4$)

Order 4: $C_2^2$ ($C_2^4:S_3^2$), $C_2^2$ ($C_2^4:S_3^2$)

Order 3: $C_3$ ($C_2^5:S_4$)

Order 1: $C_1$ ($C_2^6:S_3^2$)

Series

Derived series

$C_2^6:S_3^2$

$\rhd$

$C_2^6:C_3^2$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_2^6:S_3^2$

$\rhd$

$(C_2^3\times C_6):S_4$

$\rhd$

$C_2^6:C_3^2$

$\rhd$

$C_2^4:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^6:S_3^2$

$\rhd$

$C_2^6:C_3^2$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $48 \times 48$ rational character table.