Abstract group 648.555: $C_3.S_3^3$

• Label: 648.555
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $12$
• Trans deg.: $18$
• Rank: $3$

Group information

Description:	$C_3.S_3^3$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_3\times \He_3:D_4$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$3$

This group is nonabelian, supersolvable (hence solvable and monomial), and rational.

Group statistics

Order	1	2	3	6
Elements	1	111	80	456	648
Conjugacy classes	1	7	9	16	33
Divisions	1	7	9	16	33
Autjugacy classes	1	5	7	11	24

Dimension	1	2	4	6	8	12
Irr. complex chars.	8	12	6	4	1	2	33
Irr. rational chars.	8	12	6	4	1	2	33

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$18$
Rank:	$3$
Inequivalent generating triples:	$48384$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	8	8	8

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{3}=d^{6}=e^{3}=[a,c]=[a,e]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(11,12), (3,7)(5,6)(8,9), (2,4)(3,7)(5,8)(6,9), (1,2,4)(3,6,8)(5,9,7), (2,5,6)(4,8,9), (10,11,12), (1,3,7)(2,6,5)(4,8,9)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 2 & 1 & 1 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ 2 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 0 & 1 & 0 & 2 \\ 0 & 2 & 2 & 2 \\ 2 & 0 & 2 & 0 \\ 1 & 1 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 2 & 2 \\ 1 & 0 & 0 & 1 \\ 2 & 1 & 1 & 2 \\ 2 & 0 & 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	18T191	27T218	36T1084	36T1085	all 6
Direct product:	$S_3$ $\, \times\, $ $(C_3^2:D_6)$
Semidirect product:	$(C_3:S_3^2)$ $\,\rtimes\,$ $S_3$ (2)	$(C_3:S_3)$ $\,\rtimes\,$ $S_3^2$ (2)	$(S_3\times C_3^2)$ $\,\rtimes\,$ $D_6$ (2)	$(C_3^2:S_3)$ $\,\rtimes\,$ $D_6$	all 22
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $S_3^3$	$(C_3\times S_3)$ . $S_3^2$	$C_3^2$ . $(S_3\times D_6)$		more information
Aut. group:	$\Aut(C_3^3:C_6)$	$\Aut(C_3^2:S_3^2)$

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

Homology

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

Subgroups

There are 3156 subgroups in 297 conjugacy classes, 41 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3.S_3^3$
Commutator:	$G' \simeq$ $C_3\times \He_3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times \He_3$	$G/\operatorname{Fit} \simeq$ $C_2^3$
Radical:	$R \simeq$ $C_3.S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $S_3\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 648: $C_3.S_3^3$

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

Order 216: $S_3^3$ x 2, $C_6.S_3^2$

Order 162: $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$

Order 108: $C_3\times S_3^2$ x 7, $C_3:S_3^2$ x 6, $C_3^2:D_6$ x 6, $C_3:S_3^2$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$

Order 81: $C_3\times \He_3$

Order 72: $S_3\times D_6$ x 5

Order 54: $C_3^2:C_6$ x 8, $S_3\times C_3^2$ x 8, $C_3^2:C_6$ x 7, $C_3^2:S_3$ x 5, $C_3^2:S_3$ x 2, $C_2\times \He_3$

Order 36: $S_3^2$ x 25, $C_6\times S_3$ x 11, $C_6:S_3$ x 5

Order 27: $\He_3$ x 4, $C_3^3$ x 3

Order 24: $C_2\times D_6$ x 4

Order 18: $C_3\times S_3$ x 34, $C_3:S_3$ x 15, $C_3\times C_6$ x 6

Order 12: $D_6$ x 28, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 13

Order 8: $C_2^3$

Order 6: $S_3$ x 25, $C_6$ x 16

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_3.S_3^3$

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

Order 216: $S_3^3$, $C_6.S_3^2$

Order 162: $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3:C_6$, $S_3\times \He_3$, $C_3^3:C_6$

Order 108: $C_3\times S_3^2$ x 4, $C_3^2:D_6$ x 4, $C_3:S_3^2$ x 3, $C_3:S_3^2$, $C_3^2:D_6$, $C_3^2:D_6$

Order 81: $C_3\times \He_3$

Order 72: $S_3\times D_6$ x 3

Order 54: $S_3\times C_3^2$ x 5, $C_3^2:S_3$ x 4, $C_3^2:C_6$ x 4, $C_3^2:C_6$ x 4, $C_3^2:S_3$, $C_2\times \He_3$

Order 36: $S_3^2$ x 13, $C_6\times S_3$ x 7, $C_6:S_3$ x 3

Order 27: $\He_3$ x 3, $C_3^3$ x 2

Order 24: $C_2\times D_6$ x 3

Order 18: $C_3\times S_3$ x 21, $C_3:S_3$ x 8, $C_3\times C_6$ x 4

Order 12: $D_6$ x 17, $C_2\times C_6$ x 3

Order 9: $C_3^2$ x 9

Order 8: $C_2^3$

Order 6: $S_3$ x 15, $C_6$ x 11

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_3.S_3^3$ ($C_1$)

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

Order 162: $C_3^3:C_6$ ($C_2^2$) x 2, $C_3^3:C_6$ ($C_2^2$) x 2, $C_3^3:S_3$ ($C_2^2$), $C_3^3:S_3$ ($C_2^2$), $S_3\times \He_3$ ($C_2^2$)

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

Order 81: $C_3\times \He_3$ ($C_2^3$)

Order 54: $C_3^2:C_6$ ($D_6$) x 2, $C_3^2:S_3$ ($D_6$) x 2, $C_3^2:C_6$ ($D_6$) x 2, $S_3\times C_3^2$ ($D_6$) x 2, $C_3^2:S_3$ ($D_6$)

Order 27: $C_3^3$ ($C_2\times D_6$) x 2, $\He_3$ ($C_2\times D_6$)

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

Order 9: $C_3^2$ ($S_3\times D_6$) x 3

Order 6: $S_3$ ($C_3^2:D_6$)

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

Order 1: $C_1$ ($C_3.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_3.S_3^3$ ($C_1$)

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

Order 162: $C_3^3:S_3$ ($C_2^2$), $C_3^3:S_3$ ($C_2^2$), $C_3^3:C_6$ ($C_2^2$), $S_3\times \He_3$ ($C_2^2$), $C_3^3:C_6$ ($C_2^2$)

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

Order 81: $C_3\times \He_3$ ($C_2^3$)

Order 54: $C_3^2:S_3$ ($D_6$), $C_3^2:C_6$ ($D_6$), $C_3^2:S_3$ ($D_6$), $C_3^2:C_6$ ($D_6$), $S_3\times C_3^2$ ($D_6$)

Order 27: $C_3^3$ ($C_2\times D_6$), $\He_3$ ($C_2\times D_6$)

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

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

Order 6: $S_3$ ($C_3^2:D_6$)

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

Order 1: $C_1$ ($C_3.S_3^3$)

Series

Derived series

$C_3.S_3^3$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_3.S_3^3$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^3:C_6$

$\rhd$

$C_3\times \He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3.S_3^3$

$\rhd$

$C_3\times \He_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 57 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 $33 \times 33$ rational character table. Alternatively, you may search for characters of this group with desired properties.