Abstract group 648.750: $C_3^4:C_2^3$

• Label: 648.750
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3^{2} \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^4:C_2^3$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$(C_3^2\times C_6^2).\GL(2,3)\wr C_2$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	6
Elements	1	199	80	368	648
Conjugacy classes	1	7	24	40	72
Divisions	1	7	24	40	72
Autjugacy classes	1	3	2	3	9

Dimension	1	2	4
Irr. complex chars.	8	32	32	72
Irr. rational chars.	8	32	32	72

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$21$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{3}=c^{6}=d^{3}=e^{6}=[a,d]=[a,e]=[b,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,3)(4,5)(7,8), (10,11)(13,14), (4,5), (9,10,11)(12,13,14), (9,11,10), (1,2,3)(6,7,8), (6,7,8)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 1 & 2 & 2 & 1 \\ 2 & 2 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 2 & 0 \\ 0 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 0 \\ 2 & 2 & 1 & 0 \\ 0 & 0 & 2 & 0 \\ 2 & 0 & 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	36T1063				more information
Direct product:	$C_2$ $\, \times\, $ $(C_3:S_3)$ ${}^2$
Semidirect product:	$C_3^4$ $\,\rtimes\,$ $C_2^3$	$(C_3^2:D_6)$ $\,\rtimes\,$ $S_3$	$C_3^2$ $\,\rtimes\,$ $(S_3\times D_6)$	$(C_3\times C_6)$ $\,\rtimes\,$ $S_3^2$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 10992 subgroups in 1016 conjugacy classes, 116 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_3^2:S_3^2$
Commutator:	$G' \simeq$ $C_3^4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^4:C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4:C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 648: $C_3^4:C_2^3$

Order 324: $C_3^2:S_3^2$ x 4, $C_3^2:C_6^2$ x 2, $C_3^3:D_6$

Order 216: $C_6:S_3^2$ x 8

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 108: $C_3:S_3^2$ x 32, $C_3^2:D_6$ x 24, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 8

Order 81: $C_3^4$

Order 72: $S_3\times D_6$ x 16, $C_6:D_6$ x 2

Order 54: $C_3^2:S_3$ x 48, $C_3^2\times C_6$ x 24, $C_3^2:C_6$ x 16, $S_3\times C_3^2$ x 16

Order 36: $C_6:S_3$ x 84, $S_3^2$ x 64, $C_6\times S_3$ x 32, $C_6^2$ x 2

Order 27: $C_3^3$ x 24

Order 24: $C_2\times D_6$ x 8

Order 18: $C_3:S_3$ x 152, $C_3\times C_6$ x 78, $C_3\times S_3$ x 64

Order 12: $D_6$ x 64, $C_2\times C_6$ x 8

Order 9: $C_3^2$ x 74

Order 8: $C_2^3$

Order 6: $S_3$ x 64, $C_6$ x 40

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 24

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_3^4:C_2^3$

Order 324: $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^2:S_3^2$

Order 216: $C_6:S_3^2$

Order 162: $C_3^3\times C_6$, $C_3^3:S_3$, $C_3^2\wr C_2$

Order 108: $C_3^2:D_6$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3:S_3^2$

Order 81: $C_3^4$

Order 72: $C_6:D_6$, $S_3\times D_6$

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

Order 36: $C_6:S_3$ x 6, $C_6^2$, $C_6\times S_3$, $S_3^2$

Order 27: $C_3^3$ x 2

Order 24: $C_2\times D_6$

Order 18: $C_3\times C_6$ x 5, $C_3:S_3$ x 5, $C_3\times S_3$

Order 12: $D_6$ x 4, $C_2\times C_6$

Order 9: $C_3^2$ x 4

Order 8: $C_2^3$

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

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_3^4:C_2^3$ ($C_1$)

Order 324: $C_3^2:S_3^2$ ($C_2$) x 4, $C_3^2:C_6^2$ ($C_2$) x 2, $C_3^3:D_6$ ($C_2$)

Order 162: $C_3^2\wr C_2$ ($C_2^2$) x 4, $C_3^3:S_3$ ($C_2^2$) x 2, $C_3^3\times C_6$ ($C_2^2$)

Order 108: $C_3^2:D_6$ ($S_3$) x 8

Order 81: $C_3^4$ ($C_2^3$)

Order 54: $C_3^2:C_6$ ($D_6$) x 16, $C_3^2\times C_6$ ($D_6$) x 8

Order 36: $C_6:S_3$ ($C_3:S_3$) x 2

Order 27: $C_3^3$ ($C_2\times D_6$) x 8

Order 18: $C_3\times C_6$ ($S_3^2$) x 16, $C_3:S_3$ ($C_6:S_3$) x 4, $C_3\times C_6$ ($C_6:S_3$) x 2

Order 9: $C_3^2$ ($S_3\times D_6$) x 16, $C_3^2$ ($C_6:D_6$) x 2

Order 6: $C_6$ ($C_3:S_3^2$) x 8

Order 3: $C_3$ ($C_6:S_3^2$) x 8

Order 2: $C_2$ ($C_3^2:S_3^2$)

Order 1: $C_1$ ($C_3^4:C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_3^4:C_2^3$ ($C_1$)

Order 324: $C_3^3:D_6$ ($C_2$), $C_3^2:C_6^2$ ($C_2$), $C_3^2:S_3^2$ ($C_2$)

Order 162: $C_3^3\times C_6$ ($C_2^2$), $C_3^3:S_3$ ($C_2^2$), $C_3^2\wr C_2$ ($C_2^2$)

Order 108: $C_3^2:D_6$ ($S_3$)

Order 81: $C_3^4$ ($C_2^3$)

Order 54: $C_3^2\times C_6$ ($D_6$), $C_3^2:C_6$ ($D_6$)

Order 36: $C_6:S_3$ ($C_3:S_3$)

Order 27: $C_3^3$ ($C_2\times D_6$)

Order 18: $C_3\times C_6$ ($C_6:S_3$), $C_3\times C_6$ ($S_3^2$), $C_3:S_3$ ($C_6:S_3$)

Order 9: $C_3^2$ ($C_6:D_6$), $C_3^2$ ($S_3\times D_6$)

Order 6: $C_6$ ($C_3:S_3^2$)

Order 3: $C_3$ ($C_6:S_3^2$)

Order 2: $C_2$ ($C_3^2:S_3^2$)

Order 1: $C_1$ ($C_3^4:C_2^3$)

Series

Derived series

$C_3^4:C_2^3$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^4:C_2^3$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^2\wr C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^4:C_2^3$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 38 larger groups in the database.

This group is a maximal quotient of 30 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $72 \times 72$ rational character table. Alternatively, you may search for characters of this group with desired properties.