Abstract group 768.1085665: $C_4\times C_2^3.S_4$

• Label: 768.1085665
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{3} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $16$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$C_4\times C_2^3.S_4$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_4^2:C_3.C_2^6.C_2$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	15	32	304	96	192	128	768
Conjugacy classes	1	7	1	48	3	16	4	80
Divisions	1	7	1	26	3	8	2	48
Autjugacy classes	1	5	1	12	2	1	1	23

Dimension	1	2	3	4	6	12
Irr. complex chars.	16	8	48	0	8	0	80
Irr. rational chars.	4	10	4	2	26	2	48

Minimal presentations

Permutation degree:	$16$
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	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{12}=c^{4}=d^{4}=[a,d]=[c,d]=1, b^{a}=b^{5}, c^{a}=c^{3}d, c^{b}=d^{3}, d^{b}=cd^{3} \rangle$
Permutation group:	Degree $16$ $\langle(1,2,3,5)(4,8,7,12)(6,11,10,9)(13,14,15,16), (13,14,15,16), (13,15)(14,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 4 \\ 2 & 2 & 4 & 1 \\ 1 & 4 & 0 & 2 \\ 2 & 0 & 3 & 3 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 3 & 1 & 3 & 3 \\ 0 & 4 & 3 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 1 & 2 \\ 0 & 0 & 4 & 3 \\ 1 & 4 & 4 & 2 \\ 2 & 1 & 3 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 0 & 0 & 0 \\ 0 & 4 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrrr} 4 & 3 & 3 & 3 \\ 0 & 1 & 2 & 2 \\ 3 & 2 & 4 & 1 \\ 3 & 4 & 4 & 2 \end{array}\right), \left(\begin{array}{rrrr} 3 & 0 & 3 & 1 \\ 3 & 1 & 2 & 4 \\ 3 & 2 & 0 & 1 \\ 3 & 0 & 4 & 0 \end{array}\right), \left(\begin{array}{rrrr} 3 & 1 & 0 & 3 \\ 1 & 3 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 2 & 3 & 0 & 3 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 4 & 3 \\ 3 & 2 & 1 & 2 \\ 4 & 1 & 2 & 3 \\ 0 & 1 & 2 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{5})$
	$\left\langle \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 9 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 15 & 8 \\ 4 & 7 \end{array}\right), \left(\begin{array}{rr} 9 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 3 & 12 \\ 4 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Direct product:	$C_4$ $\, \times\, $ $(C_2^3.S_4)$
Semidirect product:	$(C_4\wr C_3)$ $\,\rtimes\,$ $C_4$	$C_4^3$ $\,\rtimes\,$ $(C_3:C_4)$	$C_4^2$ $\,\rtimes\,$ $(C_{12}:C_4)$	$(C_4^2:C_3)$ $\,\rtimes\,$ $C_4^2$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(C_4\wr S_3)$	$(C_2^4.S_4)$ . $C_2$	$(C_2\times C_4^3)$ . $S_3$	$(C_2^3\times C_4)$ . $S_4$	all 18

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

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

Subgroups

There are 2088 subgroups in 448 conjugacy classes, 39 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_4$	$G/Z \simeq$ $C_4^2:S_3$
Commutator:	$G' \simeq$ $C_4^2:C_3$	$G/G' \simeq$ $C_4^2$
Frattini:	$\Phi \simeq$ $C_2^4$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4^3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_4\times C_2^3.S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^3:C_4$
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

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 768: $C_4\times C_2^3.S_4$

Order 384: $C_2^4.S_4$ x 2, $C_4^3:C_6$

Order 256: $C_4^3:C_4$

Order 192: $C_2^3.S_4$ x 4, $C_4\wr C_3$ x 2, $C_2^4.A_4$, $A_4:C_4^2$

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

Order 96: $C_4^2:C_6$ x 3, $C_2^2.S_4$ x 2, $C_2^3:C_{12}$

Order 64: $C_4^2:C_4$ x 16, $C_4\times \OD_{16}$ x 6, $C_4^3$ x 5, $C_4:C_4^2$ x 4, $C_2^2\times C_4^2$ x 3, $C_2^2:C_4^2$ x 2, $C_2^2\times \OD_{16}$ x 2, $C_4^2:C_2^2$ x 2, $C_4.C_4^2$, $C_2\times C_4\times C_8$, $C_4^2:C_4$

Order 48: $A_4:C_4$ x 4, $C_4\times A_4$ x 2, $C_2^2\times A_4$, $C_4^2:C_3$, $C_{12}:C_4$

Order 32: $C_2\times C_4^2$ x 33, $C_2\times \OD_{16}$ x 18, $C_4^2:C_2$ x 12, $C_8:C_4$ x 4, $C_4\times C_8$ x 4, $C_2^2.D_4$ x 4, $C_2.C_4^2$ x 4, $C_2^3\times C_4$ x 3, $C_2^2\times C_8$ x 2, $C_2^3:C_4$ x 2

Order 24: $C_2\times A_4$ x 3, $C_6:C_4$ x 2, $C_2\times C_{12}$

Order 16: $C_4^2$ x 47, $C_2^2\times C_4$ x 34, $\OD_{16}$ x 20, $C_2\times C_8$ x 12, $C_4:C_4$ x 12, $C_2^2:C_4$ x 8, $C_2^4$

Order 12: $C_3:C_4$ x 4, $C_{12}$ x 2, $C_2\times C_6$, $A_4$

Order 8: $C_2\times C_4$ x 71, $C_8$ x 8, $C_2^3$ x 7

Order 6: $C_6$ x 3

Order 4: $C_4$ x 26, $C_2^2$ x 12

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_4\times C_2^3.S_4$

Order 384: $C_4^3:C_6$, $C_2^4.S_4$

Order 256: $C_4^3:C_4$

Order 192: $C_2^4.A_4$, $A_4:C_4^2$, $C_4\wr C_3$, $C_2^3.S_4$

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

Order 96: $C_4^2:C_6$ x 2, $C_2^3:C_{12}$, $C_2^2.S_4$

Order 64: $C_4:C_4^2$ x 3, $C_4^3$ x 3, $C_4^2:C_4$ x 3, $C_2^2\times C_4^2$ x 3, $C_4\times \OD_{16}$ x 2, $C_2^2:C_4^2$ x 2, $C_4.C_4^2$, $C_2\times C_4\times C_8$, $C_4^2:C_4$, $C_2^2\times \OD_{16}$, $C_4^2:C_2^2$

Order 48: $C_2^2\times A_4$, $C_4\times A_4$, $A_4:C_4$, $C_4^2:C_3$, $C_{12}:C_4$

Order 32: $C_2\times C_4^2$ x 20, $C_2\times \OD_{16}$ x 6, $C_4^2:C_2$ x 4, $C_2^3\times C_4$ x 3, $C_2.C_4^2$ x 3, $C_2^2.D_4$ x 2, $C_8:C_4$, $C_2^2\times C_8$, $C_4\times C_8$, $C_2^3:C_4$

Order 24: $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 16: $C_4^2$ x 21, $C_2^2\times C_4$ x 21, $\OD_{16}$ x 4, $C_4:C_4$ x 4, $C_2\times C_8$ x 3, $C_2^2:C_4$ x 2, $C_2^4$

Order 12: $C_2\times C_6$, $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 36, $C_2^3$ x 5, $C_8$

Order 6: $C_6$ x 2

Order 4: $C_4$ x 12, $C_2^2$ x 9

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_4\times C_2^3.S_4$ ($C_1$)

Order 384: $C_2^4.S_4$ ($C_2$) x 2, $C_4^3:C_6$ ($C_2$)

Order 192: $C_2^3.S_4$ ($C_4$) x 4, $C_4\wr C_3$ ($C_4$) x 2, $C_2^4.A_4$ ($C_2^2$)

Order 128: $C_2\times C_4^3$ ($S_3$)

Order 96: $C_4^2:C_6$ ($C_2\times C_4$) x 3

Order 64: $C_4^3$ ($C_3:C_4$) x 2, $C_2^2\times C_4^2$ ($D_6$)

Order 48: $C_4^2:C_3$ ($C_4^2$)

Order 32: $C_2\times C_4^2$ ($C_4\times S_3$) x 2, $C_2^3\times C_4$ ($S_4$), $C_2\times C_4^2$ ($C_6:C_4$)

Order 16: $C_2^2\times C_4$ ($A_4:C_4$) x 2, $C_4^2$ ($C_{12}:C_4$), $C_2^4$ ($C_2\times S_4$)

Order 8: $C_2^3$ ($C_4\times S_4$) x 2, $C_2^3$ ($C_2^2.S_4$), $C_2\times C_4$ ($C_4^2:S_3$)

Order 4: $C_4$ ($C_2^3.S_4$) x 2, $C_2^2$ ($A_4:C_4^2$), $C_2^2$ ($C_4^2:D_6$)

Order 2: $C_2$ ($C_4\wr S_3$) x 2, $C_2$ ($C_2^4.S_4$)

Order 1: $C_1$ ($C_4\times C_2^3.S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_4\times C_2^3.S_4$ ($C_1$)

Order 384: $C_4^3:C_6$ ($C_2$), $C_2^4.S_4$ ($C_2$)

Order 192: $C_2^4.A_4$ ($C_2^2$), $C_4\wr C_3$ ($C_4$), $C_2^3.S_4$ ($C_4$)

Order 128: $C_2\times C_4^3$ ($S_3$)

Order 96: $C_4^2:C_6$ ($C_2\times C_4$) x 2

Order 64: $C_4^3$ ($C_3:C_4$), $C_2^2\times C_4^2$ ($D_6$)

Order 48: $C_4^2:C_3$ ($C_4^2$)

Order 32: $C_2^3\times C_4$ ($S_4$), $C_2\times C_4^2$ ($C_6:C_4$), $C_2\times C_4^2$ ($C_4\times S_3$)

Order 16: $C_4^2$ ($C_{12}:C_4$), $C_2^4$ ($C_2\times S_4$), $C_2^2\times C_4$ ($A_4:C_4$)

Order 8: $C_2^3$ ($C_2^2.S_4$), $C_2^3$ ($C_4\times S_4$), $C_2\times C_4$ ($C_4^2:S_3$)

Order 4: $C_2^2$ ($A_4:C_4^2$), $C_2^2$ ($C_4^2:D_6$), $C_4$ ($C_2^3.S_4$)

Order 2: $C_2$ ($C_2^4.S_4$), $C_2$ ($C_4\wr S_3$)

Order 1: $C_1$ ($C_4\times C_2^3.S_4$)

Series

Derived series

$C_4\times C_2^3.S_4$

$\rhd$

$C_4^2:C_3$

$\rhd$

$C_4^2$

$\rhd$

$C_1$

Chief series

$C_4\times C_2^3.S_4$

$\rhd$

$C_2^4.S_4$

$\rhd$

$C_2^4.A_4$

$\rhd$

$C_4^2:C_6$

$\rhd$

$C_4^2:C_3$

$\rhd$

$C_4^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_4\times C_2^3.S_4$

$\rhd$

$C_4^2:C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 16 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 $80 \times 80$ 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.