Abstract group 384.6840: $C_4^2.(C_4\times S_3)$

• Label: 384.6840
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: $23$
• Trans deg.: $384$
• Rank: $4$

Group information

Description:	$C_4^2.(C_4\times S_3)$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:D_4\times F_5$, of order \(49152\)\(\medspace = 2^{14} \cdot 3 \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	7	2	248	14	112	384
Conjugacy classes	1	7	1	42	7	20	78
Divisions	1	7	1	26	7	10	52
Autjugacy classes	1	5	1	11	5	5	28

Dimension	1	2	4	8
Irr. complex chars.	32	32	14	0	78
Irr. rational chars.	16	16	14	6	52

Minimal presentations

Permutation degree:	$23$
Transitive degree:	$384$
Rank:	$4$
Inequivalent generating quadruples:	$131040$

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:	$\langle a, b, c, d \mid b^{4}=c^{4}=d^{12}=[a,b]=[a,c]=[c,d]=1, a^{2}=c^{2}d^{6}, d^{a}=c^{2}d^{11}, c^{b}=cd^{6}, d^{b}=d^{5} \rangle$
Permutation group:	Degree $23$ $\langle(1,2,5,3)(4,7,8,6)(9,10,12,13)(11,14,15,16)(20,21,22,23), (1,3,5,2)(4,7,8,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{12}:Q_8)$ $\,\rtimes\,$ $C_4$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_4^2:C_4)$ . $S_3$	$C_4^2$ . $(C_4\times S_3)$	$C_2^2$ . $(D_4:D_6)$ (3)	$C_2^2$ . $(D_4.D_6)$	all 53

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

Homology

Abelianization:	$C_{2}^{3} \times C_{4} $
Schur multiplier:	$C_{2}^{3} \times C_{4}$
Commutator length:	$1$

Subgroups

There are 1118 subgroups in 464 conjugacy classes, 185 normal (49 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^3$	$G/Z \simeq$ $C_2^2\times D_6$
Commutator:	$G' \simeq$ $C_2\times C_6$	$G/G' \simeq$ $C_2^3\times C_4$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $C_2^2\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_4^2:C_{12}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_4^2.(C_4\times S_3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^4$
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 384: $C_4^2.(C_4\times S_3)$

Order 192: $C_6.(C_4\times D_4)$ x 3, $C_6.(C_4\times D_4)$ x 2, $C_6.(C_4\times D_4)$ x 2, $(C_2\times C_{12}):Q_8$ x 2, $C_4^2:C_{12}$, $C_{12}.C_4^2$, $C_2^2.(C_4\times D_6)$, $C_2^2.(C_4\times D_6)$, $C_6.(C_2\times C_4^2)$, $C_4^2.D_6$

Order 128: $C_2^3.C_2^4$

Order 96: $C_6.C_4^2$ x 12, $C_{12}:Q_8$ x 8, $C_6:C_4^2$ x 8, $C_6.C_4^2$ x 6, $C_2^3.D_6$ x 6, $C_6.C_2^4$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$

Order 64: $C_2^3.C_2^3$ x 4, $C_4:C_4^2$ x 3, $C_2^3.C_2^3$ x 2, $C_4^2:C_4$ x 2, $C_4^2:C_4$ x 2, $C_4^2:C_4$, $C_4^2.C_2^2$

Order 48: $C_{12}:C_4$ x 20, $C_6.D_4$ x 16, $C_6.C_2^3$ x 8, $C_6:Q_8$ x 8, $C_2^2\times C_{12}$ x 7, $C_4\times C_{12}$ x 4, $C_{12}:C_4$ x 4

Order 32: $C_2.C_4^2$ x 18, $C_2\times C_4^2$ x 9, $C_4\times Q_8$ x 8, $C_2^2.D_4$ x 7, $C_2^2\times Q_8$

Order 24: $C_6:C_4$ x 32, $C_2\times C_{12}$ x 24, $C_3:Q_8$ x 8, $C_2^2\times C_6$

Order 16: $C_4^2$ x 24, $C_4:C_4$ x 20, $C_2^2\times C_4$ x 15, $C_2\times Q_8$ x 8

Order 12: $C_3:C_4$ x 16, $C_{12}$ x 10, $C_2\times C_6$ x 7

Order 8: $C_2\times C_4$ x 56, $Q_8$ x 8, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_4$ x 26, $C_2^2$ x 7

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_4^2.(C_4\times S_3)$

Order 192: $C_6.(C_4\times D_4)$ x 2, $C_4^2:C_{12}$, $C_{12}.C_4^2$, $C_6.(C_4\times D_4)$, $C_6.(C_4\times D_4)$, $C_2^2.(C_4\times D_6)$, $C_2^2.(C_4\times D_6)$, $C_6.(C_2\times C_4^2)$, $(C_2\times C_{12}):Q_8$, $C_4^2.D_6$

Order 128: $C_2^3.C_2^4$

Order 96: $C_6.C_4^2$ x 7, $C_6:C_4^2$ x 5, $C_6.C_4^2$ x 4, $C_2^3.D_6$ x 4, $C_{12}:Q_8$, $C_6.C_2^4$, $C_2\times C_4\times C_{12}$, $C_2^2.D_{12}$

Order 64: $C_2^3.C_2^3$ x 2, $C_4^2:C_4$ x 2, $C_4:C_4^2$ x 2, $C_4^2:C_4$ x 2, $C_2^3.C_2^3$, $C_4^2:C_4$, $C_4^2.C_2^2$

Order 48: $C_6.C_2^3$ x 6, $C_2^2\times C_{12}$ x 5, $C_{12}:C_4$ x 5, $C_6.D_4$ x 4, $C_6:Q_8$ x 3, $C_4\times C_{12}$, $C_{12}:C_4$

Order 32: $C_2.C_4^2$ x 11, $C_2\times C_4^2$ x 6, $C_2^2.D_4$ x 5, $C_2^2\times Q_8$, $C_4\times Q_8$

Order 24: $C_6:C_4$ x 16, $C_2\times C_{12}$ x 13, $C_3:Q_8$, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 11, $C_4^2$ x 6, $C_4:C_4$ x 5, $C_2\times Q_8$ x 3

Order 12: $C_3:C_4$ x 6, $C_2\times C_6$ x 5, $C_{12}$ x 5

Order 8: $C_2\times C_4$ x 29, $C_2^3$, $Q_8$

Order 6: $C_6$ x 5

Order 4: $C_4$ x 11, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_4^2.(C_4\times S_3)$ ($C_1$)

Order 192: $C_6.(C_4\times D_4)$ ($C_2$) x 3, $C_6.(C_4\times D_4)$ ($C_2$) x 2, $C_6.(C_4\times D_4)$ ($C_2$) x 2, $(C_2\times C_{12}):Q_8$ ($C_2$) x 2, $C_4^2:C_{12}$ ($C_2$), $C_{12}.C_4^2$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$), $C_6.(C_2\times C_4^2)$ ($C_2$), $C_4^2.D_6$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_2^2$) x 12, $C_{12}:Q_8$ ($C_4$) x 8, $C_6:C_4^2$ ($C_2^2$) x 8, $C_6.C_4^2$ ($C_2^2$) x 6, $C_2^3.D_6$ ($C_2^2$) x 6, $C_6.C_2^4$ ($C_2^2$), $C_2\times C_4\times C_{12}$ ($C_2^2$), $C_2^2.D_{12}$ ($C_2^2$)

Order 64: $C_4^2:C_4$ ($S_3$)

Order 48: $C_6.C_2^3$ ($C_2^3$) x 8, $C_6.D_4$ ($C_2\times C_4$) x 8, $C_{12}:C_4$ ($C_2\times C_4$) x 8, $C_2^2\times C_{12}$ ($C_2^3$) x 7, $C_6:Q_8$ ($C_2\times C_4$) x 4, $C_4\times C_{12}$ ($C_2\times C_4$) x 4, $C_{12}:C_4$ ($C_2\times C_4$) x 4

Order 32: $C_2.C_4^2$ ($D_6$) x 6, $C_2\times C_4^2$ ($D_6$)

Order 24: $C_6:C_4$ ($D_4:C_2$) x 8, $C_6:C_4$ ($C_2^2\times C_4$) x 8, $C_2\times C_{12}$ ($C_2^2\times C_4$) x 6, $C_2^2\times C_6$ ($C_2^4$)

Order 16: $C_2^2\times C_4$ ($C_2\times D_6$) x 7, $C_4^2$ ($C_4\times S_3$) x 4

Order 12: $C_2\times C_6$ ($D_4:C_2^2$) x 4, $C_3:C_4$ ($C_4^2:C_2$) x 4, $C_2\times C_6$ ($C_4.C_2^3$) x 2, $C_2\times C_6$ ($C_2^3\times C_4$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 6, $C_2^3$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$) x 2, $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2:C_2^2$)

Order 4: $C_2^2$ ($D_4:D_6$) x 4, $C_2^2$ ($D_4.D_6$) x 2, $C_2^2$ ($C_{12}:C_2^3$)

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

Order 2: $C_2$ ($C_4^2.D_6$) x 2, $C_2$ ($C_4^2.D_6$) x 2, $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2:D_6$), $C_2$ ($C_4^2.D_6$)

Order 1: $C_1$ ($C_4^2.(C_4\times S_3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_4^2.(C_4\times S_3)$ ($C_1$)

Order 192: $C_6.(C_4\times D_4)$ ($C_2$) x 2, $C_4^2:C_{12}$ ($C_2$), $C_{12}.C_4^2$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $C_6.(C_4\times D_4)$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$), $C_2^2.(C_4\times D_6)$ ($C_2$), $C_6.(C_2\times C_4^2)$ ($C_2$), $(C_2\times C_{12}):Q_8$ ($C_2$), $C_4^2.D_6$ ($C_2$)

Order 96: $C_6.C_4^2$ ($C_2^2$) x 7, $C_6:C_4^2$ ($C_2^2$) x 5, $C_6.C_4^2$ ($C_2^2$) x 4, $C_2^3.D_6$ ($C_2^2$) x 4, $C_{12}:Q_8$ ($C_4$), $C_6.C_2^4$ ($C_2^2$), $C_2\times C_4\times C_{12}$ ($C_2^2$), $C_2^2.D_{12}$ ($C_2^2$)

Order 64: $C_4^2:C_4$ ($S_3$)

Order 48: $C_6.C_2^3$ ($C_2^3$) x 6, $C_2^2\times C_{12}$ ($C_2^3$) x 5, $C_6.D_4$ ($C_2\times C_4$) x 2, $C_{12}:C_4$ ($C_2\times C_4$) x 2, $C_6:Q_8$ ($C_2\times C_4$), $C_4\times C_{12}$ ($C_2\times C_4$), $C_{12}:C_4$ ($C_2\times C_4$)

Order 32: $C_2.C_4^2$ ($D_6$) x 4, $C_2\times C_4^2$ ($D_6$)

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

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

Order 12: $C_2\times C_6$ ($D_4:C_2^2$) x 3, $C_2\times C_6$ ($C_4.C_2^3$), $C_2\times C_6$ ($C_2^3\times C_4$), $C_3:C_4$ ($C_4^2:C_2$)

Order 8: $C_2\times C_4$ ($C_4\times D_6$) x 3, $C_2^3$ ($C_2^2\times D_6$)

Order 6: $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2.C_2^2$), $C_6$ ($C_4^2:C_2^2$)

Order 4: $C_2^2$ ($D_4:D_6$) x 3, $C_2^2$ ($D_4.D_6$), $C_2^2$ ($C_{12}:C_2^3$)

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

Order 2: $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2.D_6$), $C_2$ ($C_4^2:D_6$), $C_2$ ($C_4^2.D_6$)

Order 1: $C_1$ ($C_4^2.(C_4\times S_3)$)

Series

Derived series

$C_4^2.(C_4\times S_3)$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_4^2.(C_4\times S_3)$

$\rhd$

$C_6.(C_4\times D_4)$

$\rhd$

$C_6:C_4^2$

$\rhd$

$C_6.C_2^3$

$\rhd$

$C_2^2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_4^2.(C_4\times S_3)$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2^3$

$\lhd$

$C_4^2:C_4$

Character theory

Complex character table

See the $78 \times 78$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $52 \times 52$ rational character table.