Abstract group 12288.bmm: $(C_2^8\times S_3).D_4$

• Label: 12288.bmm
• Order: \( 2^{12} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{26} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{16} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $6$

Group information

Description:	$(C_2^8\times S_3).D_4$
Order:	\(12288\)\(\medspace = 2^{12} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(201326592\)\(\medspace = 2^{26} \cdot 3 \)
Composition factors:	$C_2$ x 12, $C_3$
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	1919	2	5248	958	1024	2624	512	12288
Conjugacy classes	1	177	1	178	88	8	89	4	546
Divisions	1	177	1	178	88	8	89	4	546

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$6$
Inequivalent generating 6-tuples:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=c^{4}=d^{2}=e^{2}=f^{4}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $28$ $\langle(3,6,10,13)(4,8,12,14)(9,19)(11,21)(15,25)(16,20,23,26)(17,27)(18,22,24,28) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	$C_3$ $\,\rtimes\,$ $(C_2^9.D_4)$				more information
Trans. wreath product:	not computed
Possibly split product:	$(C_2^8.D_4)$ . $S_3$	$(C_2^8\times S_3)$ . $D_4$	$S_3$ . $(C_2^8.D_4)$ (4)	$(C_2^7:D_4)$ . $D_6$ (8)	all 211
Aut. group:	$\Aut(C_4^2.D_6)$	$\Aut(C_4^2.D_6)$	$\Aut(C_4^2.D_6)$	$\Aut(C_4^2.D_6)$	all 6

Elements of the group are displayed as permutations of degree 28.

Homology

Abelianization:	$C_{2}^{6} $
Schur multiplier:	not computed
Commutator length:	$1$

Subgroups

There are 18515 normal subgroups (106 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to $C_{12}:C_2^4$
Frattini:	a subgroup isomorphic to $D_4\times C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 3 (order at least 4096) are shown, as well as normal subgroups of index up to 12 or of order up to 16.

Order 12288: $(C_2^8\times S_3).D_4$

Order 6144: $C_3:(C_2.C_2^6.C_2^4)$ x 23, $C_3:(C_2^4.C_2^4.C_2^3)$ x 11, $C_3:(C_2^3.C_2^5.C_2^3)$ x 9, $D_4^2.(D_4\times D_6)$ x 7, $C_3:(C_2^3.C_2^6.C_2^2)$ x 5, $C_3:(C_2^4.C_2^5.C_2^2)$ x 3, $C_3:(C_2^9.C_2^2)$, $C_3:(C_2^6.C_2^5)$, $C_3:(C_2^5.C_2^6)$, $C_3:(C_2^4.C_2^6.C_2)$, $C_2^8.(C_4\times S_3)$

Order 4096: $C_2^9.D_4$

Order 3072: $C_3:(C_2^3.C_2^4.C_2^3)$ x 150, $C_3:(C_2^3.C_2^5.C_2^2)$ x 117, $C_3:(C_2.C_2^6.C_2^3)$ x 70, $C_3:(C_2^4.C_2^5.C_2)$ x 60, $C_3:(C_2^5.C_2^5)$ x 50, $C_3:(C_2^2.C_2^5.C_2^3)$ x 39, $C_3:(C_2^2.C_2^6.C_2^2)$ x 24, $C_3:(C_2^3.C_2^6.C_2)$ x 23, $C_3:(C_2^8.C_2^2)$ x 22, $C_3:(C_2^4.C_2^3.C_2^3)$ x 21, $C_3:(C_2^5.C_2^2.C_2^3)$ x 17, $C_3:(C_2^4.C_2^4.C_2^2)$ x 15, $C_3\times C_2^7:D_4$ x 8, $C_3:((C_2^3\times D_4).C_2^4)$ x 7, $C_3:(C_2^6.C_2^4)$ x 6, $S_3\times C_2^5.C_2^4$ x 6, $C_3:((C_2\times D_4).C_2^6)$ x 5, $C_3:(C_2^5.C_2^3.C_2^2)$ x 4, $C_3:(C_2^7.C_2^3)$ x 2, $C_2^7.C_2^2\times S_3$ x 2, $C_3:(C_2^4.C_2^6)$, $(D_6\times C_2^6).C_4$, $C_2^2.(D_6\times C_2^6)$

Order 2048: $C_2^8.D_4$

Order 1536: $S_3\times C_4^2:C_2^4$ x 128, $C_2\times C_2^5.C_2^2\times S_3$ x 64, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 48, $C_2\times D_4^2:D_6$ x 35, $C_2\times C_3:(C_2^6.C_2^2)$ x 32, $C_6\times C_2^4.C_2^3.C_2$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 32, $C_6.(D_4\times C_2^5)$ x 26, $C_3:(C_2^7.C_2^2)$ x 24, $C_6\times D_4^2.C_2^2$ x 16, $C_2\times C_3:(C_2^6.C_2^2)$ x 16, $C_3\times C_2^4.C_2^4.C_2$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 16, $C_3:((C_2\times D_4).C_2^5)$ x 16, $C_2\times C_2^4:C_2^2.C_2\times S_3$ x 16, $C_2^5:C_4\times D_6$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_3:(C_2^4.C_2^4.C_2)$ x 16, $C_2^7:C_{12}$ x 12, $C_2\times C_3:(C_2^5.C_2^3)$ x 12, $C_2\times C_3:(C_2^5.C_2^3)$ x 12, $C_2\times C_2^4.C_2^3\times S_3$ x 12, $C_2\times C_2^5.C_2^2\times S_3$ x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 12, $C_3:(C_2^7.C_2^2)$ x 12, $C_2\times C_3:(C_2^6.C_2^2)$ x 8, $C_6\times C_2^4.C_2^3.C_2$ x 8, $C_6\times C_2^4.C_2^3.C_2$ x 8, $C_2\times C_3:(C_2^4.C_2^4)$ x 8, $C_2\times C_3:(C_2^4.C_2^4)$ x 8, $C_2\times C_3:(C_2^4.C_2^4)$ x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 8, $C_2\times C_2^4.C_2^3\times S_3$ x 8, $C_2\times C_4.C_2^2:D_4\times S_3$ x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ x 8, $C_3:(C_2^4.C_2^4.C_2)$ x 8, $C_3:(C_2^3.C_2^5.C_2)$ x 8, $C_3:(C_2^3.C_2^5.C_2)$ x 8, $C_3:(C_2^5.C_2^4)$ x 8, $C_3:(C_2^5.C_2^4)$ x 8, $C_3:(C_2^3.C_2^5.C_2)$ x 8, $C_3:(C_2^6.C_2^3)$ x 8, $C_6\times C_2^6.C_2^2$ x 8, $C_6\times C_2^5:D_4$ x 6, $C_3\times C_2^7.C_2^2$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_3:(C_2^5.C_2^3)$ x 6, $C_2\times C_4.C_2^2\wr C_2\times S_3$ x 6, $C_3:(C_4.C_2^6.C_2)$ x 6, $C_3:(C_2^7.C_2^2)$ x 6, $C_3:(C_2^7.C_2^2)$ x 6, $C_3:(C_2^7.C_2^2)$ x 6, $C_6\times C_2^4.C_2^4$ x 4, $C_6\times C_2^3.C_2^4.C_2$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ x 4, $C_2^2\times C_2^2\wr C_2:C_2\times S_3$ x 4, $C_2^2\times (C_2\times C_4).C_2^3\times S_3$ x 4, $C_2^2\times C_2^2\wr C_2.C_2\times S_3$ x 4, $C_2\times D_6\times D_4^2$ x 4, $C_2\times C_2^4.C_2^3\times S_3$ x 4, $C_2^4:D_4\times D_6$ x 4, $C_2\times (C_2\times C_4).C_2^4\times S_3$ x 4, $C_3\times C_2^4.C_2^4.C_2$ x 4, $C_3\times C_2^4.C_2^4.C_2$ x 4, $C_3\times C_2^3.C_2^5.C_2$ x 4, $C_3\times C_2^5.C_2^4$ x 4, $C_2\times C_3:(C_2^4.C_2^4)$ x 4, $C_3:(C_2^6.C_2^3)$ x 4, $C_3:(C_2^6.C_2^3)$ x 4, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ x 4, $C_3:(C_2^3.C_2^5.C_2)$ x 4, $C_3:(C_2^3.C_2^5.C_2)$ x 4, $C_2^5.C_2^3\times S_3$ x 4, $(C_2^3\times D_4).C_2^2\times S_3$ x 4, $C_3:(C_2^4.C_2^4.C_2)$ x 4, $C_3:(C_2^4.C_2^4.C_2)$ x 4, $C_3:(C_2^4.C_2^4.C_2)$ x 4, $C_3:(C_2^4.C_2^4.C_2)$ x 4, $C_3:(C_2^3.C_2^5.C_2)$ x 4, $C_3:(C_2^3.C_2^5.C_2)$ x 4, $C_3:(C_2^3.C_2^5.C_2)$ x 4, $C_3:(C_2^5.C_2^4)$ x 4, $C_3:(C_2^5.C_2^4)$ x 4, $C_3:(C_2^5.C_2^4)$ x 4, $C_2\times C_6\times C_2^5.C_2^2$ x 4, $C_3:(C_2^5.C_2^4)$ x 3, $C_3:(C_2^5.C_2^4)$ x 3, $C_2^5.C_2^3\times S_3$ x 3, $C_3\times C_2^6.C_2^3$ x 2, $C_3\times C_2^3.C_2^5.C_2$ x 2, $C_2^3\times Q_8:C_2^2\times S_3$ x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^4)$ x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ x 2, $C_3:(C_2^5.C_2^4)$ x 2, $C_2^2\times (C_2^2\times D_4):C_2\times S_3$ x 2, $C_2^2\times C_2^4:C_2^2\times S_3$ x 2, $C_3:(C_2^3.C_2^6)$ x 2, $C_2^2\times C_2^3:(C_2\times C_4)\times S_3$ x 2, $C_3\times C_2^7.C_2^2$ x 2, $C_3\times C_2^3.C_2^5.C_2$ x 2, $C_3\times C_2^6.C_2^3$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^7.C_2^2)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^3.C_2^5.C_2)$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_3:(C_2^6.C_2^3)$ x 2, $C_6\times C_2^5.C_2^3$ x 2, $C_3\times C_2^5.C_2^4$, $C_2^8\times S_3$, $C_2^5\times S_3\times D_4$, $C_2^4\times C_2^2:D_{12}$, $C_2^4\times C_2^4:S_3$, $C_2^4\times C_2^3:D_6$, $C_2^4\times C_2^2:C_4\times S_3$, $C_3\times C_4.C_2^6.C_2$, $C_3:(C_2^5.C_2^4)$, $C_3:(C_4.C_2^6.C_2)$, $C_6\times C_2^4.C_2^4$

Order 1024: $C_2.C_2^6.C_2^3$ x 10, $C_2^7:D_4$ x 8, $C_2^4.C_2^5.C_2$ x 5, $C_2^4.C_2^4.C_2^2$ x 4, $C_2^8.C_2^2$, $C_2^6.C_2^4$, $C_2^8:C_4$, $C_2^7:D_4$

Order 16: $C_2^4$ x 67, $C_2^2:C_4$ x 32, $C_2\times D_4$ x 24, $C_2^2\times C_4$ x 20, $C_4^2$ x 16, $C_2\times Q_8$ x 8

Order 12: $D_6$ x 16, $C_2\times C_6$ x 11

Order 8: $C_2^3$ x 27, $C_2\times C_4$ x 4

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 11

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 12 or order up to 16 are shown.

Order 12288: $(C_2^8\times S_3).D_4$ ($C_1$)

Order 6144: $C_3:(C_2.C_2^6.C_2^4)$ ($C_2$) x 23, $C_3:(C_2^4.C_2^4.C_2^3)$ ($C_2$) x 11, $C_3:(C_2^3.C_2^5.C_2^3)$ ($C_2$) x 9, $D_4^2.(D_4\times D_6)$ ($C_2$) x 7, $C_3:(C_2^3.C_2^6.C_2^2)$ ($C_2$) x 5, $C_3:(C_2^4.C_2^5.C_2^2)$ ($C_2$) x 3, $C_3:(C_2^9.C_2^2)$ ($C_2$), $C_3:(C_2^6.C_2^5)$ ($C_2$), $C_3:(C_2^5.C_2^6)$ ($C_2$), $C_3:(C_2^4.C_2^6.C_2)$ ($C_2$), $C_2^8.(C_4\times S_3)$ ($C_2$)

Order 3072: $C_3:(C_2^3.C_2^4.C_2^3)$ ($C_2^2$) x 150, $C_3:(C_2^3.C_2^5.C_2^2)$ ($C_2^2$) x 117, $C_3:(C_2.C_2^6.C_2^3)$ ($C_2^2$) x 70, $C_3:(C_2^4.C_2^5.C_2)$ ($C_2^2$) x 60, $C_3:(C_2^5.C_2^5)$ ($C_2^2$) x 50, $C_3:(C_2^2.C_2^5.C_2^3)$ ($C_2^2$) x 39, $C_3:(C_2^2.C_2^6.C_2^2)$ ($C_2^2$) x 24, $C_3:(C_2^3.C_2^6.C_2)$ ($C_2^2$) x 23, $C_3:(C_2^8.C_2^2)$ ($C_2^2$) x 22, $C_3:(C_2^4.C_2^3.C_2^3)$ ($C_2^2$) x 21, $C_3:(C_2^5.C_2^2.C_2^3)$ ($C_2^2$) x 17, $C_3:(C_2^4.C_2^4.C_2^2)$ ($C_2^2$) x 15, $C_3\times C_2^7:D_4$ ($C_2^2$) x 8, $C_3:((C_2^3\times D_4).C_2^4)$ ($C_2^2$) x 7, $C_3:(C_2^6.C_2^4)$ ($C_2^2$) x 6, $S_3\times C_2^5.C_2^4$ ($C_2^2$) x 6, $C_3:((C_2\times D_4).C_2^6)$ ($C_2^2$) x 5, $C_3:(C_2^5.C_2^3.C_2^2)$ ($C_2^2$) x 4, $C_3:(C_2^7.C_2^3)$ ($C_2^2$) x 2, $C_2^7.C_2^2\times S_3$ ($C_2^2$) x 2, $C_3:(C_2^4.C_2^6)$ ($C_2^2$), $(D_6\times C_2^6).C_4$ ($C_2^2$), $C_2^2.(D_6\times C_2^6)$ ($C_2^2$)

Order 2048: $C_2^8.D_4$ ($S_3$)

Order 1536: $S_3\times C_4^2:C_2^4$ ($D_4$) x 128, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 48, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 32, $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^3$) x 32, $C_2\times C_2^5.C_2^2\times S_3$ ($D_4$) x 32, $C_6\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 32, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 24, $C_2\times D_4^2:D_6$ ($C_2^3$) x 19, $C_6\times D_4^2.C_2^2$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 16, $C_2\times D_4^2:D_6$ ($D_4$) x 16, $C_3\times C_2^4.C_2^4.C_2$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:((C_2\times D_4).C_2^5)$ ($C_2^3$) x 16, $C_2\times C_2^4:C_2^2.C_2\times S_3$ ($C_2^3$) x 16, $C_2^5:C_4\times D_6$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 16, $C_6.(D_4\times C_2^5)$ ($D_4$) x 14, $C_6.(D_4\times C_2^5)$ ($C_2^3$) x 12, $C_2^7:C_{12}$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 12, $C_2\times C_2^4.C_2^3\times S_3$ ($C_2^3$) x 12, $C_2\times C_2^5.C_2^2\times S_3$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 12, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 12, $C_2\times C_3:(C_2^6.C_2^2)$ ($C_2^3$) x 8, $C_6\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_6\times C_2^4.C_2^3.C_2$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 8, $C_2\times C_2^4.C_2^3\times S_3$ ($C_2^3$) x 8, $C_2\times C_4.C_2^2:D_4\times S_3$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 8, $C_2\times C_3:(C_2^4.C_2^3.C_2)$ ($C_2^3$) x 8, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 8, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 8, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 8, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 8, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 8, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 8, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 8, $C_6\times C_2^6.C_2^2$ ($C_2^3$) x 8, $C_6\times C_2^5:D_4$ ($C_2^3$) x 6, $C_3\times C_2^7.C_2^2$ ($C_2^3$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 6, $C_2\times C_3:(C_2^5.C_2^3)$ ($C_2^3$) x 6, $C_2\times C_4.C_2^2\wr C_2\times S_3$ ($C_2^3$) x 6, $C_3:(C_4.C_2^6.C_2)$ ($C_2^3$) x 6, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 6, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 6, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 6, $C_6\times C_2^4.C_2^4$ ($C_2^3$) x 4, $C_6\times C_2^3.C_2^4.C_2$ ($C_2^3$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($D_4$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($D_4$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($C_2^3$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($D_4$) x 4, $C_2^2\times C_3:(C_2^5.C_2^2)$ ($D_4$) x 4, $C_2^2\times C_2^2\wr C_2:C_2\times S_3$ ($D_4$) x 4, $C_2^2\times (C_2\times C_4).C_2^3\times S_3$ ($D_4$) x 4, $C_2^2\times C_2^2\wr C_2.C_2\times S_3$ ($D_4$) x 4, $C_2\times D_6\times D_4^2$ ($D_4$) x 4, $C_2\times C_2^4.C_2^3\times S_3$ ($C_2^3$) x 4, $C_2^4:D_4\times D_6$ ($C_2^3$) x 4, $C_2\times (C_2\times C_4).C_2^4\times S_3$ ($C_2^3$) x 4, $C_3\times C_2^4.C_2^4.C_2$ ($C_2^3$) x 4, $C_3\times C_2^4.C_2^4.C_2$ ($C_2^3$) x 4, $C_3\times C_2^3.C_2^5.C_2$ ($C_2^3$) x 4, $C_3\times C_2^5.C_2^4$ ($C_2^3$) x 4, $C_2\times C_3:(C_2^4.C_2^4)$ ($C_2^3$) x 4, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 4, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 4, $C_2\times C_3:(C_2^3.C_2^4.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 4, $C_2^5.C_2^3\times S_3$ ($C_2^3$) x 4, $(C_2^3\times D_4).C_2^2\times S_3$ ($C_2^3$) x 4, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^4.C_2^4.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 4, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 4, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 4, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 4, $C_2\times C_6\times C_2^5.C_2^2$ ($C_2^3$) x 4, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 3, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 3, $C_2^5.C_2^3\times S_3$ ($C_2^3$) x 3, $C_3\times C_2^6.C_2^3$ ($C_2^3$) x 2, $C_3\times C_2^3.C_2^5.C_2$ ($C_2^3$) x 2, $C_2^3\times Q_8:C_2^2\times S_3$ ($D_4$) x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 2, $C_3:(C_2^5.C_2^4)$ ($C_2^3$) x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ ($C_2^3$) x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ ($C_2^3$) x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 2, $C_2^2\times C_3:(C_2^4.C_2^3)$ ($D_4$) x 2, $C_3:(C_2^5.C_2^4)$ ($D_4$) x 2, $C_2^2\times (C_2^2\times D_4):C_2\times S_3$ ($C_2^3$) x 2, $C_2^2\times C_2^4:C_2^2\times S_3$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^6)$ ($D_4$) x 2, $C_2^2\times C_2^3:(C_2\times C_4)\times S_3$ ($D_4$) x 2, $C_3\times C_2^7.C_2^2$ ($C_2^3$) x 2, $C_3\times C_2^3.C_2^5.C_2$ ($C_2^3$) x 2, $C_3\times C_2^6.C_2^3$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^7.C_2^2)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^3.C_2^5.C_2)$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_3:(C_2^6.C_2^3)$ ($C_2^3$) x 2, $C_6\times C_2^5.C_2^3$ ($C_2^3$) x 2, $C_3\times C_2^5.C_2^4$ ($C_2^3$), $C_2^8\times S_3$ ($D_4$), $C_2^5\times S_3\times D_4$ ($C_2^3$), $C_2^4\times C_2^2:D_{12}$ ($D_4$), $C_2^4\times C_2^4:S_3$ ($D_4$), $C_2^4\times C_2^3:D_6$ ($C_2^3$), $C_2^4\times C_2^2:C_4\times S_3$ ($D_4$), $C_3\times C_4.C_2^6.C_2$ ($C_2^3$), $C_3:(C_2^5.C_2^4)$ ($C_2^3$), $C_3:(C_4.C_2^6.C_2)$ ($C_2^3$), $C_6\times C_2^4.C_2^4$ ($C_2^3$)

Order 1024: $C_2.C_2^6.C_2^3$ ($D_6$) x 10, $C_2^7:D_4$ ($D_6$) x 8, $C_2^4.C_2^5.C_2$ ($D_6$) x 5, $C_2^4.C_2^4.C_2^2$ ($D_6$) x 4, $C_2^8.C_2^2$ ($D_6$), $C_2^6.C_2^4$ ($D_6$), $C_2^8:C_4$ ($D_6$), $C_2^7:D_4$ ($D_6$)

Order 768: ? (unidentified group of order 16) x 2451

Order 512: ? (unidentified group of order 24) x 155

Order 384: ? (unidentified group of order 32) x 4403

Order 256: ? (unidentified group of order 48) x 275

Order 192: ? (unidentified group of order 64) x 4379

Order 128: ? (unidentified group of order 96) x 451

Order 96: ? (unidentified group of order 128) x 1995

Order 64: ? (unidentified group of order 192) x 771

Order 48: ? (unidentified group of order 256) x 475

Order 32: ? (unidentified group of order 384) x 439

Order 24: ? (unidentified group of order 512) x 91

Order 16: $C_2^4$ ($C_2^6:D_6$) x 42, $C_2^2:C_4$ ($C_2^6:D_6$) x 32, $C_4^2$ ($C_2^6:D_6$) x 16, $C_2\times D_4$ ($C_2^6:D_6$) x 16, $C_2^2\times C_4$ ($C_2^6:D_6$) x 14, $C_2^4$ ($C_2^6:D_6$) x 13, $C_2\times Q_8$ ($C_2^6:D_6$) x 8, $C_2\times D_4$ ($C_2^6:D_6$) x 8, $C_2^4$ ($C_2^4.C_2^3\times S_3$) x 6, $C_2^4$ ($C_2\wr C_2^2\times D_6$) x 6, $C_2^2\times C_4$ ($C_2^6:D_6$) x 6

Order 12: $D_6$ ($C_2^7:D_4$) x 8, $D_6$ ($C_2^7.C_2^3$) x 6, $C_2\times C_6$ ($C_2^7:D_4$) x 4, $C_2\times C_6$ ($C_2^7.C_2^3$) x 3, $C_2\times C_6$ ($C_2^5.C_2^5$) x 2, $D_6$ ($C_2^5.C_2^5$) x 2, $C_2\times C_6$ ($C_2^4.C_2^6$), $C_2\times C_6$ ($C_2^9.C_2$)

Order 8: $C_2^3$ ($S_3\times C_2^5:D_4$) x 12, $C_2^3$ ($C_2^7:D_6$) x 11, $C_2\times C_4$ ($C_2^7:D_6$) x 4, $C_2^3$ ($C_2\times C_2^4.C_2^3\times S_3$) x 3, $C_2^3$ ($C_2^4:D_4\times D_6$)

Order 6: $S_3$ ($C_2^8.D_4$) x 4, $C_6$ ($C_2^8.D_4$) x 2, $C_6$ ($C_2^8.C_2^3$)

Order 4: $C_2^2$ ($(S_3\times D_4^2).D_4$) x 4, $C_2^2$ ($C_3:(C_2^5.C_2^5)$) x 3, $C_2^2$ ($C_3:(C_2^4.C_2^6)$) x 2, $C_2^2$ ($C_3:(C_2^9.C_2)$), $C_2^2$ ($C_3:(C_2^6.C_2^4)$)

Order 3: $C_3$ ($C_2^9.D_4$)

Order 2: $C_2$ ($C_3:(C_2^7.C_2^4)$), $C_2$ ($C_3:(C_2.C_2^6.C_2^4)$), $C_2$ ($D_4^2.(D_4\times D_6)$)

Order 1: $C_1$ ($(C_2^8\times S_3).D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 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

The $546 \times 546$ rational character table is not available for this group.