Abstract group 2304.sb: $(C_2^4\times C_6):S_4$

• Label: 2304.sb
• Order: \( 2^{8} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$(C_2^4\times C_6):S_4$
Order:	\(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6:C_{18}:C_6$, of order \(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
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	223	386	1056	542	96	2304
Conjugacy classes	1	14	4	13	18	4	54
Divisions	1	14	4	13	15	3	50
Autjugacy classes	1	8	2	5	8	2	26

Dimension	1	2	3	6	12	24
Irr. complex chars.	4	8	12	20	10	0	54
Irr. rational chars.	4	8	12	16	8	2	50

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$35280$

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 $15$ $\langle(4,6)(5,7)(8,14)(9,11)(10,12)(13,15), (1,2,3)(8,11)(9,14)(10,15)(12,13) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^5:A_4)$ $\,\rtimes\,$ $S_3$	$(C_2^6:C_6)$ $\,\rtimes\,$ $S_3$	$(C_2^4\times C_6)$ $\,\rtimes\,$ $S_4$	$(C_2^4:C_6)$ $\,\rtimes\,$ $S_4$	all 20
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(C_6:S_4)$	$(C_2^3\times C_6)$ . $(C_2\times S_4)$	$(C_2\times C_6)$ . $(C_2^3:S_4)$	$C_2^2$ . $((C_2^2\times C_6):S_4)$	more information

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{4} \times C_{6}$
Commutator length:	$2$

Subgroups

There are 35242 subgroups in 2288 conjugacy classes, 37 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_2^4\times C_6):S_4$
Commutator:	$G' \simeq$ $C_2^6:C_3^2$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $(C_2^2\times C_6):S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_6$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_2^4\times C_6):S_4$	$G/R \simeq$ $C_1$
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

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 2304: $(C_2^4\times C_6):S_4$

Order 1152: $(C_2^3\times C_6):S_4$ x 2, $C_3\times C_2^5:A_4$

Order 768: $C_2^5:S_4$ x 3, $C_2^6:D_6$

Order 576: $(C_2^2\times C_6):S_4$ x 2, $C_2^6:C_3^2$, $(C_2^2\times C_6):S_4$

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

Order 288: $(C_2\times C_6):S_4$ x 10, $C_3\times C_2^3:A_4$ x 3, $C_2^5:C_3^2$

Order 256: $C_2^5:D_4$

Order 192: $C_2^4:D_6$ x 11, $C_2^4.D_6$ x 10, $C_2^4:D_6$ x 8, $C_2^3:S_4$ x 6, $C_2^5:C_6$ x 6, $C_2^3.D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^3.D_{12}$ x 4, $C_2^4.D_6$ x 4, $C_2^4:D_6$ x 4, $C_2^4:A_4$ x 3, $C_2^3:S_4$ x 3, $C_2^4:D_6$ x 3, $C_2^5\times C_6$, $C_{12}:C_2^4$, $D_{12}:C_2^3$, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^4.D_6$, $C_2^4.D_6$

Order 144: $C_2^4:C_3^2$ x 10, $C_6:S_4$ 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 42, $C_2^2:S_4$ x 30, $C_2^3.D_6$ x 29, $C_2^4:C_6$ x 16, $C_{12}:C_2^3$ x 14, $C_2^3:D_6$ x 13, $C_2^4\times C_6$ x 11, $C_2^3:A_4$ x 9, $C_2^3.D_6$ x 8, $C_2^2.D_{12}$ x 8, $C_2^3:C_{12}$ x 7, $C_2^3.D_6$ x 6, $D_4:D_6$ x 6, $C_{12}:D_4$ x 4, $D_6:D_4$ x 4, $C_2^3.D_6$ x 4, $C_2^3.D_6$ x 4, $C_{12}:D_4$ x 4, $C_2^3:A_4$ x 3, $D_4:D_6$ x 2, $D_4\times D_6$ x 2, $C_{12}:D_4$ 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, $C_2^3\times C_{12}$, $D_{12}:C_2^2$, $C_2^3.D_6$

Order 72: $C_3:S_4$ x 10, $C_6\times A_4$ 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 86, $C_6:D_4$ x 66, $C_6.D_4$ x 58, $C_6\times D_4$ x 40, $C_2^2:A_4$ x 30, $C_6.C_2^3$ x 13, $C_2^2:C_{12}$ x 10, $C_2\times S_4$ x 9, $D_4:S_3$ x 8, $S_3\times D_4$ x 8, $C_2^2\times C_{12}$ x 7, $D_{12}:C_2$ x 4, $D_6:C_4$ x 4, $C_6.D_4$ x 4, $C_2^2\times D_6$ x 3, $C_4\times D_6$ x 2, $C_{12}:C_4$ x 2, $C_{12}:C_4$ x 2, $C_2\times D_{12}$, $C_6:Q_8$

Order 36: $C_3\times A_4$ x 10, $C_6:S_3$

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 162, $C_3:D_4$ x 44, $C_6:C_4$ x 30, $S_4$ x 30, $C_3\times D_4$ x 20, $C_2\times D_6$ x 14, $C_2\times A_4$ x 12, $C_2\times C_{12}$ x 11, $C_4\times S_3$ x 4, $D_{12}$ x 2, $C_3:Q_8$ x 2

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

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 90, $A_4$ x 30, $D_6$ x 15, $C_3:C_4$ x 10, $C_{12}$ x 3

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 15, $S_3$ x 8

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

Order 3: $C_3$ x 4

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2304: $(C_2^4\times C_6):S_4$

Order 1152: $(C_2^3\times C_6):S_4$, $C_3\times C_2^5:A_4$

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

Order 576: $C_2^6:C_3^2$, $(C_2^2\times C_6):S_4$, $(C_2^2\times C_6):S_4$

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

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

Order 256: $C_2^5:D_4$

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

Order 144: $C_2^4:C_3^2$ x 5, $C_6:S_4$ x 2

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

Order 96: $C_2^4:S_3$ x 11, $C_2^4:C_6$ x 10, $C_{12}:C_2^3$ x 8, $C_2^3.D_6$ x 8, $C_2^4\times C_6$ x 6, $C_2^3:D_6$ x 5, $C_2^3:C_{12}$ x 4, $C_2^3.D_6$ x 3, $C_2^2:S_4$ x 3, $D_4:D_6$ x 3, $C_2^3:A_4$ x 2, $C_2^3.D_6$ x 2, $D_6:D_4$ x 2, $C_2^3.D_6$ x 2, $C_2^2.D_{12}$ x 2, $C_2^3:A_4$, $C_2^3\times C_{12}$, $D_4:D_6$, $D_4\times D_6$, $D_{12}:C_2^2$, $C_{12}:D_4$, $C_{12}:D_4$, $C_{12}.D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_2^3.D_6$, $C_{12}.D_4$

Order 72: $C_6\times A_4$ x 3, $C_3:S_4$ x 3

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

Order 48: $C_2^3\times C_6$ x 30, $C_6:D_4$ x 20, $C_6\times D_4$ x 16, $C_6.D_4$ x 14, $C_2^2:C_{12}$ x 6, $C_2^2:A_4$ x 5, $C_2^2\times C_{12}$ x 5, $C_6.C_2^3$ x 5, $C_2^2\times D_6$ x 2, $C_2\times S_4$ x 2, $D_4:S_3$ x 2, $S_3\times D_4$ x 2, $D_{12}:C_2$ x 2, $C_2\times D_{12}$, $C_4\times D_6$, $C_6:Q_8$, $D_6:C_4$, $C_{12}:C_4$, $C_6.D_4$, $C_{12}:C_4$

Order 36: $C_3\times A_4$ x 5, $C_6:S_3$

Order 32: $C_2^2\wr C_2$ x 23, $C_2^2\times D_4$ x 14, $C_2^3:C_4$ x 12, $C_2^5$ x 6, $C_2^3:C_4$ x 5, $C_4:D_4$ x 4, $D_4:C_2^2$ x 3, $C_2^2.D_4$ x 3, $D_4:C_2^2$ x 2, $C_4\times D_4$ x 2, $C_2^3\times C_4$, $C_4:D_4$, $C_4^2:C_2$, $C_2^2:Q_8$, $C_4^2:C_2$

Order 24: $C_2^2\times C_6$ x 51, $C_3:D_4$ x 12, $C_6:C_4$ x 9, $C_2\times C_{12}$ x 7, $C_2\times D_6$ x 6, $C_3\times D_4$ x 6, $C_2\times A_4$ x 3, $S_4$ x 3, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$

Order 18: $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2\times D_4$ x 39, $C_2^4$ x 32, $C_2^2:C_4$ x 21, $C_2^2\times C_4$ x 11, $D_4:C_2$ x 4, $C_4:C_4$ x 2, $C_4^2$, $C_2\times Q_8$

Order 12: $C_2\times C_6$ x 33, $D_6$ x 6, $A_4$ x 5, $C_3:C_4$ x 3, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 57, $D_4$ x 19, $C_2\times C_4$ x 17, $Q_8$

Order 6: $C_6$ x 8, $S_3$ x 2

Order 4: $C_2^2$ x 38, $C_4$ x 5

Order 3: $C_3$ x 2

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2304: $(C_2^4\times C_6):S_4$ ($C_1$)

Order 1152: $(C_2^3\times C_6):S_4$ ($C_2$) x 2, $C_3\times C_2^5:A_4$ ($C_2$)

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

Order 384: $C_2^5:A_4$ ($S_3$) x 3, $C_2^6:C_6$ ($S_3$)

Order 192: $C_2^4:A_4$ ($D_6$) x 3, $C_2^5\times C_6$ ($D_6$)

Order 128: $C_2^4:D_4$ ($C_3:S_3$)

Order 96: $C_2^4:C_6$ ($S_4$) x 2, $C_2^4\times C_6$ ($S_4$)

Order 64: $C_2^6$ ($C_6:S_3$)

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

Order 32: $C_2^2\wr C_2$ ($C_3:S_4$) x 2, $C_2^5$ ($C_3:S_4$)

Order 24: $C_2^2\times C_6$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($C_6:S_4$) 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 8: $C_2^3$ ($(C_2\times C_6):S_4$)

Order 4: $C_2^2$ ($(C_2^2\times C_6):S_4$) x 2, $C_2^2$ ($(C_2^2\times C_6):S_4$)

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

Order 1: $C_1$ ($(C_2^4\times C_6):S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2304: $(C_2^4\times C_6):S_4$ ($C_1$)

Order 1152: $(C_2^3\times C_6):S_4$ ($C_2$), $C_3\times C_2^5:A_4$ ($C_2$)

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

Order 384: $C_2^6:C_6$ ($S_3$), $C_2^5:A_4$ ($S_3$)

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

Order 128: $C_2^4:D_4$ ($C_3:S_3$)

Order 96: $C_2^4\times C_6$ ($S_4$), $C_2^4:C_6$ ($S_4$)

Order 64: $C_2^6$ ($C_6:S_3$)

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

Order 32: $C_2^5$ ($C_3:S_4$), $C_2^2\wr C_2$ ($C_3:S_4$)

Order 24: $C_2^2\times C_6$ ($C_2^2:S_4$)

Order 16: $C_2^4$ ($C_6:S_4$) 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 8: $C_2^3$ ($(C_2\times C_6):S_4$)

Order 4: $C_2^2$ ($(C_2^2\times C_6):S_4$), $C_2^2$ ($(C_2^2\times C_6):S_4$)

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

Order 1: $C_1$ ($(C_2^4\times C_6):S_4$)

Series

Derived series

$(C_2^4\times C_6):S_4$

$\rhd$

$C_2^6:C_3^2$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$(C_2^4\times C_6):S_4$

$\rhd$

$C_3\times C_2^5:A_4$

$\rhd$

$C_2^6:C_3^2$

$\rhd$

$C_2^4:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_2^4\times C_6):S_4$

$\rhd$

$C_2^6:C_3^2$

Upper central series

$C_1$

Supergroups

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

This group is a maximal quotient of 1 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 $50 \times 50$ rational character table.