Abstract group 393216.ks: $C_2^{12}.(C_2^2\times S_4)$

• Label: 393216.ks
• Order: \( 2^{17} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: 8
• $\card{\Aut(G)}$: \( 2^{35} \cdot 3^{2} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{21} \cdot 3 \cdot 7 \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$C_2^{12}.(C_2^2\times S_4)$
Order:	\(393216\)\(\medspace = 2^{17} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	Group of order \(2164663517184\)\(\medspace = 2^{35} \cdot 3^{2} \cdot 7 \)
Composition factors:	$C_2$ x 17, $C_3$
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	39167	2048	222976	96256	32768	393216
Conjugacy classes	1	3383	1	1152	31	8	4576
Divisions	1	3383	1	1152	23	6	4566

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p \mid b^{6}=c^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $40$ $\langle(2,7)(4,40)(5,11)(8,14)(20,24)(22,28)(25,31)(27,34), (2,7)(4,40)(5,11)(8,14) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^{13}$ . $(S_3\times D_4)$ (4)	$(C_2^{15}.C_2)$ . $S_3$	$(C_2^{13}.C_2)$ . $S_4$ (15)	$C_2^{12}$ . $(D_4\times D_6)$ (6)	all 53
Aut. group:	$\Aut(C_2^3.C_2^5)$	$\Aut(C_2^3.C_2^5)$	$\Aut(C_2^3.C_2^5)$	$\Aut(C_2^3.C_2^5)$	all 6

Elements of the group are displayed as permutations of degree 40.

Homology

Abelianization:	$C_{2}^{5} $
Schur multiplier:	$C_{2}^{21}$
Commutator length:	$2$

Subgroups

There are 12009 normal subgroups (126 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 $C_2^3$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_2^4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^{15}.C_2^2$

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 6 (order at least 65536) are shown, as well as normal subgroups of index up to 96 or of order up to 96.

Order 393216: $C_2^{12}.(C_2^2\times S_4)$

Order 196608: $C_2^8.A_4.D_4^2$ x 16, $C_2^{10}.C_6.C_2^4.C_2$ x 8, $C_2^8.A_4.C_2^5.C_2$ x 2, $C_2^{10}.C_6.C_2^5$, $C_2^{10}.C_6.C_2.C_2^4$, $C_2^9.A_4.C_2^5$, $C_2^2\times C_2^{10}.C_6.C_2^3$, $C_2\times C_2^{10}.C_6.C_2^4$

Order 131072: $C_2^{15}.C_2^2$

Order 98304: $C_2^{10}.C_6.C_2^4$ x 31, $C_2^8.A_4.C_2^2:D_4$ x 24, $C_2^9.A_4.C_2^4$ x 18, $C_2^{10}.C_2^4:S_3$ x 16, $C_2^{10}.C_2^3:D_6$ x 16, $C_2^8.A_4.(C_4\times D_4)$ x 16, $C_2\times C_2^{10}.C_6.C_2^3$ x 16, $C_2^8.C_6.C_2^5.C_2$ x 15, $C_2\times C_2^8.A_4.D_4.C_2$ x 12, $C_2^{10}.D_6:D_4$ x 8, $C_2^8.A_4.C_4:D_4$ x 8, $C_2^8.A_4.C_2^4.C_2$ x 5, $C_2^{10}.C_6.C_2.C_2^3$ x 4, $C_2^2\times C_2^8.A_4.D_4$ x 4, $C_2^{11}.C_6.C_2^3$ x 2, $C_2^{10}.C_6.C_2^3.C_2$ x 2, $C_2^2\times C_2^8.A_4.C_2^3$ x 2, $C_2\times C_2^8.A_4.C_2^4$ x 2, $C_2^2\times C_2^{10}.C_6.C_2^2$

Order 65536: $C_2^{14}.C_2^2$ x 505, $C_2^{13}.C_2^3$ x 4, $C_2^{15}.C_2$ x 2

Order 49152: $C_2^{10}.C_6.C_2^3$ x 36, $C_2\times C_2^8.A_4.D_4$ x 28, $C_2^9.A_4.C_2^3$ x 24, $C_2^{10}.(C_6.D_4)$ x 20, $C_2\times C_2^8.A_4.C_2^3$ x 15, $C_2^{10}.D_6:C_4$ x 12, $C_2\times C_2^{10}.C_6.C_2^2$ x 5, $C_2^{10}.C_{12}:C_4$ x 4, $C_2^8.A_4.D_4.C_2$ x 4, $C_2^8.A_4.C_4^2$ x 4, $C_2^8.A_4.C_4:C_4$ x 4, $C_2\times C_2^{10}.D_6.C_2$ x 4, $C_2\times C_2^8.A_4.C_4.C_2$ x 4, $C_2^2\times C_2^{10}.C_6.C_2$ x 3, $C_2^2\times C_2^{10}.D_6$ x 2, $C_2^2\times C_2^8.A_4.C_4$ x 2

Order 32768: $C_2^{14}.C_2$ x 14, $C_2^{15}$

Order 24576: $C_2^{11}.C_6.C_2$ x 16, $C_2^8.A_4.D_4$ x 16, $C_2\times C_2^{10}.C_6.C_2$ x 16, $C_2^{10}.C_6.C_2^2$ x 14, $C_2^8.A_4.C_2^3$ x 6, $C_2\times C_2^{10}.D_6$ x 6, $C_2\times C_2^8.A_4.C_4$ x 6, $C_2^2\times C_2^{10}.C_6$ x 3, $C_2^{10}.D_6.C_2$ x 2, $C_2^8.A_4.C_4.C_2$ x 2

Order 16384: $C_2^{13}.C_2$ x 50, $C_2^{14}$ x 7, $C_2^9.C_2^5$

Order 12288: $C_2\times C_2^{10}.C_6$ x 10, $C_2^{10}.C_6.C_2$ x 8, $C_2^{10}.D_6$ x 4, $C_2^{10}.C_{12}$ x 4, $C_2^8.A_4.C_4$ x 4, $C_2^2\times C_2^8.A_4$

Order 8192: $C_2^{12}.C_2$ x 222, $C_2^{13}$ x 31, $C_2^9.C_2^4$ x 18, $C_2^{11}\times C_4$

Order 6144: $C_2^{10}.C_6$ x 6, $C_2\times C_2^8.A_4$

Order 4096: $C_2^{11}.C_2$ x 504, $C_2^{12}$ x 126, $C_2^9.C_2^3$ x 67, $C_2^{10}\times C_4$ x 6

Order 64: $C_2^6$ x 827, $C_2^4\times C_4$ x 34, $C_2^4:C_4$ x 32, $C_2^3:D_4$ x 32, $D_4\times C_2^3$ x 24

Order 32: $C_2^5$ x 535, $C_2^2\times D_4$ x 16, $C_2^3\times C_4$ x 8

Order 16: $C_2^4$ x 279, $C_2^2\times C_4$ x 4

Order 8: $C_2^3$ x 120

Order 4: $C_2^2$ x 30

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

All normal subgroups of index up to 96 or order up to 96 are shown.

Order 393216: $C_2^{12}.(C_2^2\times S_4)$ ($C_1$)

Order 196608: $C_2^8.A_4.D_4^2$ ($C_2$) x 16, $C_2^{10}.C_6.C_2^4.C_2$ ($C_2$) x 8, $C_2^8.A_4.C_2^5.C_2$ ($C_2$) x 2, $C_2^{10}.C_6.C_2^5$ ($C_2$), $C_2^{10}.C_6.C_2.C_2^4$ ($C_2$), $C_2^9.A_4.C_2^5$ ($C_2$), $C_2^2\times C_2^{10}.C_6.C_2^3$ ($C_2$), $C_2\times C_2^{10}.C_6.C_2^4$ ($C_2$)

Order 98304: $C_2^8.A_4.C_2^2:D_4$ ($C_2^2$) x 24, $C_2^{10}.C_2^4:S_3$ ($C_2^2$) x 16, $C_2^{10}.C_2^3:D_6$ ($C_2^2$) x 16, $C_2^8.A_4.(C_4\times D_4)$ ($C_2^2$) x 16, $C_2^{10}.C_6.C_2^4$ ($C_2^2$) x 15, $C_2\times C_2^{10}.C_6.C_2^3$ ($C_2^2$) x 12, $C_2^9.A_4.C_2^4$ ($C_2^2$) x 10, $C_2^{10}.D_6:D_4$ ($C_2^2$) x 8, $C_2^8.A_4.C_4:D_4$ ($C_2^2$) x 8, $C_2\times C_2^8.A_4.D_4.C_2$ ($C_2^2$) x 8, $C_2^8.A_4.C_2^4.C_2$ ($C_2^2$) x 5, $C_2^{10}.C_6.C_2.C_2^3$ ($C_2^2$) x 4, $C_2^2\times C_2^8.A_4.D_4$ ($C_2^2$) x 4, $C_2^{11}.C_6.C_2^3$ ($C_2^2$) x 2, $C_2^{10}.C_6.C_2^3.C_2$ ($C_2^2$) x 2, $C_2^2\times C_2^8.A_4.C_2^3$ ($C_2^2$) x 2, $C_2\times C_2^8.A_4.C_2^4$ ($C_2^2$) x 2, $C_2^2\times C_2^{10}.C_6.C_2^2$ ($C_2^2$)

Order 65536: $C_2^{15}.C_2$ ($S_3$)

Order 49152: $C_2^{10}.C_6.C_2^3$ ($C_2^3$) x 32, $C_2\times C_2^8.A_4.D_4$ ($C_2^3$) x 28, $C_2^{10}.(C_6.D_4)$ ($C_2^3$) x 20, $C_2^9.A_4.C_2^3$ ($C_2^3$) x 16, $C_2\times C_2^8.A_4.C_2^3$ ($C_2^3$) x 15, $C_2^{10}.D_6:C_4$ ($C_2^3$) x 12, $C_2^9.A_4.C_2^3$ ($D_4$) x 8, $C_2\times C_2^{10}.C_6.C_2^2$ ($C_2^3$) x 5, $C_2^{10}.C_{12}:C_4$ ($C_2^3$) x 4, $C_2^{10}.C_6.C_2^3$ ($D_4$) x 4, $C_2^8.A_4.D_4.C_2$ ($D_4$) x 4, $C_2^8.A_4.C_4^2$ ($C_2^3$) x 4, $C_2^8.A_4.C_4:C_4$ ($C_2^3$) x 4, $C_2\times C_2^{10}.D_6.C_2$ ($C_2^3$) x 4, $C_2\times C_2^8.A_4.C_4.C_2$ ($C_2^3$) x 4, $C_2^2\times C_2^{10}.C_6.C_2$ ($C_2^3$) x 3, $C_2^2\times C_2^{10}.D_6$ ($C_2^3$) x 2, $C_2^2\times C_2^8.A_4.C_4$ ($C_2^3$) x 2

Order 32768: $C_2^{14}.C_2$ ($D_6$) x 14, $C_2^{15}$ ($D_6$)

Order 24576: $C_2^{11}.C_6.C_2$ ($C_2\times D_4$) x 16, $C_2^8.A_4.D_4$ ($C_2\times D_4$) x 16, $C_2\times C_2^{10}.C_6.C_2$ ($C_2^4$) x 10, $C_2^{10}.C_6.C_2^2$ ($C_2^4$) x 8, $C_2^{10}.C_6.C_2^2$ ($C_2\times D_4$) x 6, $C_2^8.A_4.C_2^3$ ($C_2\times D_4$) x 6, $C_2\times C_2^{10}.D_6$ ($C_2^4$) x 6, $C_2\times C_2^{10}.C_6.C_2$ ($C_2\times D_4$) x 6, $C_2\times C_2^8.A_4.C_4$ ($C_2^4$) x 6, $C_2^{10}.D_6.C_2$ ($C_2\times D_4$) x 2, $C_2^8.A_4.C_4.C_2$ ($C_2\times D_4$) x 2, $C_2^2\times C_2^{10}.C_6$ ($C_2\times D_4$) x 2, $C_2^2\times C_2^{10}.C_6$ ($C_2^4$)

Order 16384: $C_2^{13}.C_2$ ($C_2\times D_6$) x 28, $C_2^{13}.C_2$ ($S_4$) x 14, $C_2^{13}.C_2$ ($C_3:D_4$) x 8, $C_2^{14}$ ($C_2\times D_6$) x 7, $C_2^9.C_2^5$ ($S_4$)

Order 12288: $C_2\times C_2^{10}.C_6$ ($C_2^2\times D_4$) x 9, $C_2^{10}.C_6.C_2$ ($C_2^2\times D_4$) x 6, $C_2^{10}.D_6$ ($C_2^2\times D_4$) x 4, $C_2^{10}.C_{12}$ ($C_2^2\times D_4$) x 4, $C_2^8.A_4.C_4$ ($C_2^2\times D_4$) x 4, $C_2^{10}.C_6.C_2$ ($D_4:C_2^2$) x 2, $C_2^2\times C_2^8.A_4$ ($C_2^2\times D_4$), $C_2\times C_2^{10}.C_6$ ($C_2^5$)

Order 8192: $C_2^{12}.C_2$ ($C_2\times S_4$) x 191, $C_2^{12}.C_2$ ($C_6:D_4$) x 24, $C_2^9.C_2^4$ ($C_2\times S_4$) x 17, $C_2^{13}$ ($C_2\times S_4$) x 16, $C_2^{13}$ ($C_2^2\times D_6$) x 7, $C_2^{12}.C_2$ ($C_2^2\times D_6$) x 7, $C_2^{13}$ ($C_6:D_4$) x 4, $C_2^{13}$ ($S_3\times D_4$) x 4, $C_2^{11}\times C_4$ ($C_2\times S_4$), $C_2^9.C_2^4$ ($C_2^2\times D_6$)

Order 6144: $C_2^{10}.C_6$ ($D_4^2$) x 4, $C_2^{10}.C_6$ ($D_4:C_2^3$), $C_2^{10}.C_6$ ($D_4\times C_2^3$), $C_2\times C_2^8.A_4$ ($D_4\times C_2^3$)

Order 4096: $C_2^{11}.C_2$ ($C_2^2\times S_4$) x 354, $C_2^{11}.C_2$ ($\GL(2,\mathbb{Z}/4)$) x 112, $C_2^{12}$ ($C_2^2\times S_4$) x 111, $C_2^9.C_2^3$ ($C_2^2\times S_4$) x 54, $C_2^{11}.C_2$ ($C_2^2:S_4$) x 31, $C_2^9.C_2^3$ ($\GL(2,\mathbb{Z}/4)$) x 8, $C_2^{11}.C_2$ ($C_2^3:D_6$) x 7, $C_2^{10}\times C_4$ ($C_2^2\times S_4$) x 6, $C_2^{12}$ ($C_2^3:D_6$) x 6, $C_2^{12}$ ($D_4\times D_6$) x 6, $C_2^9.C_2^3$ ($C_2^2:S_4$) x 4, $C_2^{12}$ ($D_4:D_6$) x 2, $C_2^9.C_2^3$ ($C_2^3:D_6$), $C_2^{12}$ ($C_2^3\times D_6$)

Order 3072: ? (unidentified group of order 128)

Order 2048: ? (unidentified group of order 192) x 1245

Order 1024: ? (unidentified group of order 384) x 1922

Order 512: ? (unidentified group of order 768) x 2123

Order 256: ? (unidentified group of order 1536) x 1782

Order 128: ? (unidentified group of order 3072) x 1455

Order 64: $C_2^6$ ($C_2^6.C_6.C_2^4$) x 299, $C_2^6$ ($C_2^9.D_6$) x 168, $C_2^6$ ($C_2^6.\GL(2,\mathbb{Z}/4)$) x 118, $C_2^6$ ($C_2^9.D_6$) x 56, $C_2^6$ ($C_2^8.D_6.C_2$) x 42, $C_2^6$ ($C_2^8.C_6.C_2^2$) x 42, $C_2^4:C_4$ ($C_2^6.\GL(2,\mathbb{Z}/4)$) x 32, $C_2^3:D_4$ ($C_2^6.\GL(2,\mathbb{Z}/4)$) x 32, $C_2^4\times C_4$ ($C_2^6.\GL(2,\mathbb{Z}/4)$) x 28, $C_2^6$ ($C_2^4:C_3.C_2^5.C_2^2$) x 24, $C_2^6$ ($C_2\times C_2^6.D_6.C_2^2$) x 21, $C_2^6$ ($C_2^4:C_3.C_2^4.C_2^3$) x 18, $D_4\times C_2^3$ ($C_2^8.S_4$) x 16, $C_2^6$ ($C_2\times C_2^8.D_6$) x 14, $C_2^6$ ($C_2^8.S_4$) x 13, $C_2^6$ ($C_2\times C_2^6.C_6.C_2^3$) x 8, $D_4\times C_2^3$ ($C_2^6.\GL(2,\mathbb{Z}/4)$) x 8, $C_2^4\times C_4$ ($C_2^8.S_4$) x 6, $C_2^6$ ($C_2^6.D_6.C_2^3$) x 3, $C_2^6$ ($C_2^4:C_3.D_4^2.C_2$)

Order 32: $C_2^5$ ($C_2^8.C_6.C_2^3$) x 187, $C_2^5$ ($C_2^6:C_3.D_4^2$) x 172, $C_2^5$ ($C_2^6.C_6.C_2^5$) x 92, $C_2^5$ ($C_2\times C_2^6.C_6.C_2^4$) x 34, $C_2^5$ ($C_2^8.D_6.C_2^2$) x 24, $C_2^2\times D_4$ ($C_2^7.\GL(2,\mathbb{Z}/4)$) x 16, $C_2^5$ ($C_2\times C_2^8.C_6.C_2^2$) x 8, $C_2^5$ ($C_2^7.\GL(2,\mathbb{Z}/4)$) x 8, $C_2^5$ ($C_2\times C_2^8.D_6.C_2$) x 6, $C_2^3\times C_4$ ($C_2^8.C_6.C_2^3$) x 4, $C_2^3\times C_4$ ($C_2^7.\GL(2,\mathbb{Z}/4)$) x 4, $C_2^5$ ($C_2\times C_2^6.D_6.C_2^3$) x 2, $C_2^5$ ($C_2^{10}.D_6$), $C_2^5$ ($C_2^6.D_6.C_2^4$)

Order 16: $C_2^4$ ($C_2^8.C_6.C_2^4$) x 190, $C_2^4$ ($C_2^6.C_6.C_2^5.C_2$) x 22, $C_2^4$ ($C_2^6.C_6.C_2.C_2^5$) x 18, $C_2^4$ ($C_2\times C_2^8.C_6.C_2^3$) x 17, $C_2^4$ ($C_2\times C_2^8.D_6.C_2^2$) x 12, $C_2^4$ ($C_2^8.\GL(2,\mathbb{Z}/4)$) x 10, $C_2^4$ ($C_2^8.A_4.C_2^3$) x 4, $C_2^2\times C_4$ ($C_2^8.\GL(2,\mathbb{Z}/4)$) x 4, $C_2^4$ ($C_2^6:C_3.D_4^2.C_2$) x 3, $C_2^4$ ($C_2^{10}.D_6.C_2$) x 2, $C_2^4$ ($C_2\times C_2^{10}.D_6$)

Order 8: $C_2^3$ ($C_2^{12}.D_6$) x 56, $C_2^3$ ($C_2^8.C_6.C_2^5$) x 31, $C_2^3$ ($C_2\times C_2^8.C_6.C_2^4$) x 14, $C_2^3$ ($C_2^8.A_4.C_2^4$) x 6, $C_2^3$ ($C_2^8.C_3.D_4^2$) x 4, $C_2^3$ ($C_2^8.(D_4\times S_4)$) x 4, $C_2^3$ ($C_2^{10}.C_6.C_2^3$) x 3, $C_2^3$ ($C_2^8.A_4.D_4.C_2$), $C_2^3$ ($C_2\times C_2^{10}.D_6.C_2$)

Order 4: $C_2^2$ ($C_2^{10}.C_6.C_2^4$) x 12, $C_2^2$ ($C_2^8.C_6.C_2^5.C_2$) x 8, $C_2^2$ ($C_2^8.C_6.C_2.C_2^5$) x 6, $C_2^2$ ($C_2\times C_2^8.(D_4\times S_4)$) x 2, $C_2^2$ ($C_2^8.C_3.D_4^2.C_2$), $C_2^2$ ($C_2^8.A_4.C_2^5$)

Order 2: $C_2$ ($C_2^8.A_4.D_4^2$) x 4, $C_2$ ($C_2^{10}.C_6.C_2^5$) x 2, $C_2$ ($C_2\times C_2^{10}.C_6.C_2^4$)

Order 1: $C_1$ ($C_2^{12}.(C_2^2\times S_4)$)

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

Character theory

Complex character table

The $4576 \times 4576$ character table is not available for this group.

Rational character table

The $4566 \times 4566$ rational character table is not available for this group.