Abstract group 648.717: $C_3^4:(C_2\times C_4)$

• Label: 648.717
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $12$
• Trans deg.: $12$
• Rank: $2$

Group information

Description:	$C_3^4:(C_2\times C_4)$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$\SOPlus(4,2)^2.D_4$, of order \(41472\)\(\medspace = 2^{9} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	4	6
Elements	1	99	80	324	144	648
Conjugacy classes	1	3	12	4	4	24
Divisions	1	3	12	2	4	22
Autjugacy classes	1	2	2	1	1	7

Dimension	1	2	4	8
Irr. complex chars.	8	0	8	8	24
Irr. rational chars.	4	2	8	8	22

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$12$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{4}=b^{3}=c^{6}=d^{3}=e^{3}=[b,c]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,7)(2,9,3,8)(4,10)(5,12,6,11), (1,4)(2,5)(3,6)(7,10)(8,11)(9,12), (2,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 1 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 2 & 2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 0 & 2 & 0 \\ 2 & 2 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 2 & 0 & 1 & 0 \\ 2 & 2 & 2 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 1 & 1 & 2 \\ 0 & 2 & 0 & 2 \\ 0 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 2 & 2 & 0 \\ 2 & 2 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	12T171	18T190	18T192	24T1523	all 9
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:C_4)$ $\,\rtimes\,$ $C_2$	$C_3^4$ $\,\rtimes\,$ $(C_2\times C_4)$	$(C_3^2\wr C_2)$ $\,\rtimes\,$ $C_4$	$(C_3:S_3)$ $\,\rtimes\,$ $(C_3^2:C_4)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^2:S_3^2)$ . $C_2$	$(C_3^3:S_3)$ . $C_2^2$			more information

Elements of the group are displayed as matrices in $\GL_{4}(\F_{3})$.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 3856 subgroups in 216 conjugacy classes, 13 normal (5 characteristic).

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

Special subgroups

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

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 648: $C_3^4:(C_2\times C_4)$

Order 324: $C_3^4:C_4$ x 2, $C_3^2:S_3^2$

Order 162: $C_3^2\wr C_2$ x 2, $C_3^3:S_3$

Order 108: $C_3:S_3^2$ x 4

Order 81: $C_3^4$

Order 72: $C_2\times C_3^2:C_4$ x 2

Order 54: $C_3^2:S_3$ x 12, $C_3^2:C_6$ x 4, $S_3\times C_3^2$ x 4

Order 36: $C_3^2:C_4$ x 12, $S_3^2$ x 8, $C_6:S_3$ x 2

Order 27: $C_3^3$ x 12

Order 18: $C_3:S_3$ x 44, $C_3\times S_3$ x 16, $C_3\times C_6$ x 2

Order 12: $D_6$ x 4

Order 9: $C_3^2$ x 42

Order 8: $C_2\times C_4$

Order 6: $S_3$ x 16, $C_6$ x 4

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

Order 3: $C_3$ x 12

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_3^4:(C_2\times C_4)$

Order 324: $C_3^2:S_3^2$, $C_3^4:C_4$

Order 162: $C_3^3:S_3$, $C_3^2\wr C_2$

Order 108: $C_3:S_3^2$

Order 81: $C_3^4$

Order 72: $C_2\times C_3^2:C_4$

Order 54: $C_3^2:S_3$ x 2, $C_3^2:C_6$, $S_3\times C_3^2$

Order 36: $C_3^2:C_4$ x 2, $C_6:S_3$, $S_3^2$

Order 27: $C_3^3$ x 2

Order 18: $C_3:S_3$ x 6, $C_3\times C_6$, $C_3\times S_3$

Order 12: $D_6$

Order 9: $C_3^2$ x 5

Order 8: $C_2\times C_4$

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

Order 4: $C_2^2$, $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_3^4:(C_2\times C_4)$ ($C_1$)

Order 324: $C_3^4:C_4$ ($C_2$) x 2, $C_3^2:S_3^2$ ($C_2$)

Order 162: $C_3^2\wr C_2$ ($C_4$) x 2, $C_3^3:S_3$ ($C_2^2$)

Order 81: $C_3^4$ ($C_2\times C_4$)

Order 18: $C_3:S_3$ ($C_3^2:C_4$) x 2

Order 9: $C_3^2$ ($C_2\times C_3^2:C_4$) x 2

Order 1: $C_1$ ($C_3^4:(C_2\times C_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_3^4:(C_2\times C_4)$ ($C_1$)

Order 324: $C_3^2:S_3^2$ ($C_2$), $C_3^4:C_4$ ($C_2$)

Order 162: $C_3^3:S_3$ ($C_2^2$), $C_3^2\wr C_2$ ($C_4$)

Order 81: $C_3^4$ ($C_2\times C_4$)

Order 18: $C_3:S_3$ ($C_3^2:C_4$)

Order 9: $C_3^2$ ($C_2\times C_3^2:C_4$)

Order 1: $C_1$ ($C_3^4:(C_2\times C_4)$)

Series

Derived series

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 19 larger groups in the database.

This group is a maximal quotient of 16 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.