Abstract group 648.572: $C_3^3:S_4$

• Label: 648.572
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $16$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_3^3:S_4$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_3\times C_6^2:D_6$, of order \(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Composition factors:	$C_2$ x 3, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	21	242	18	222	144	648
Conjugacy classes	1	2	20	1	25	8	57
Divisions	1	2	12	1	13	4	33
Autjugacy classes	1	2	6	1	7	2	19

Dimension	1	2	3	4	6	12
Irr. complex chars.	6	12	30	0	9	0	57
Irr. rational chars.	2	6	2	4	15	4	33

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$49140$

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:	$\langle a, b, c, d \mid a^{6}=b^{3}=c^{6}=d^{6}=[a,c]=[c,d]=1, b^{a}=b^{2}, d^{a}=cd^{5}, c^{b}=cd^{3}, d^{b}=cd^{4} \rangle$
Permutation group:	Degree $16$ $\langle(2,3)(10,13)(12,15)(14,16), (5,6,7)(8,9,11)(10,12,14)(13,15,16), (2,3,4) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_3^2:S_4)$
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $S_4$	$(C_3^2\times A_4)$ $\,\rtimes\,$ $S_3$	$(C_3\times C_6^2)$ $\,\rtimes\,$ $S_3$	$C_6^2$ $\,\rtimes\,$ $(C_3\times S_3)$	all 10
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(C_3:S_3)$	$C_3^2$ . $(C_3:S_4)$	$C_3$ . $(C_3^2:S_4)$	$(C_2\times C_6)$ . $(C_3^2:C_6)$	more information

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

Homology

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

Subgroups

There are 1160 subgroups in 196 conjugacy classes, 26 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3^2$	$G/Z \simeq$ $C_3:S_4$
Commutator:	$G' \simeq$ $C_3^2:A_4$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_3^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $C_3:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$

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

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: $C_3^3:S_4$

Order 324: $C_3^3:A_4$

Order 216: $C_3^2\times S_4$ x 3, $C_3^2:S_4$, $C_6^2:C_6$

Order 162: $C_3^3:S_3$

Order 108: $C_3^2:A_4$ x 5, $C_3^2\times A_4$ x 3, $C_3\times C_6^2$, $C_3^2\times D_6$, $C_3^2:C_{12}$

Order 81: $C_3\times \He_3$

Order 72: $C_3\times S_4$ x 12, $C_6\wr C_2$ x 4, $D_4\times C_3^2$

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

Order 36: $C_3\times A_4$ x 15, $C_6^2$ x 7, $C_3:C_{12}$ x 4, $C_6\times S_3$ x 4, $C_3\times C_{12}$

Order 27: $\He_3$ x 5, $C_3^3$ x 4

Order 24: $C_3\times D_4$ x 4, $S_4$ x 3, $C_3:D_4$

Order 18: $C_3\times S_3$ x 16, $C_3\times C_6$ x 10

Order 12: $C_2\times C_6$ x 10, $A_4$ x 6, $C_{12}$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$ x 21

Order 8: $D_4$

Order 6: $C_6$ x 13, $S_3$ x 4

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

Order 3: $C_3$ x 12

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $C_3^3:S_4$

Order 324: $C_3^3:A_4$

Order 216: $C_3^2:S_4$, $C_3^2\times S_4$, $C_6^2:C_6$

Order 162: $C_3^3:S_3$

Order 108: $C_3^2:A_4$ x 3, $C_3\times C_6^2$, $C_3^2\times D_6$, $C_3^2\times A_4$, $C_3^2:C_{12}$

Order 81: $C_3\times \He_3$

Order 72: $C_3\times S_4$ x 2, $C_6\wr C_2$ x 2, $D_4\times C_3^2$

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

Order 36: $C_6^2$ x 5, $C_3\times A_4$ x 3, $C_3:C_{12}$ x 2, $C_6\times S_3$ x 2, $C_3\times C_{12}$

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

Order 24: $C_3\times D_4$ x 2, $C_3:D_4$, $S_4$

Order 18: $C_3\times C_6$ x 6, $C_3\times S_3$ x 4

Order 12: $C_2\times C_6$ x 6, $A_4$ x 2, $C_{12}$ x 2, $D_6$, $C_3:C_4$

Order 9: $C_3^2$ x 7

Order 8: $D_4$

Order 6: $C_6$ x 7, $S_3$ x 2

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

Order 3: $C_3$ x 6

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $C_3^3:S_4$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_2$)

Order 216: $C_3^2:S_4$ ($C_3$)

Order 108: $C_3^2\times A_4$ ($S_3$) x 3, $C_3\times C_6^2$ ($S_3$), $C_3^2:A_4$ ($C_6$)

Order 36: $C_3\times A_4$ ($C_3\times S_3$) x 3, $C_6^2$ ($C_3:S_3$), $C_6^2$ ($C_3\times S_3$)

Order 27: $C_3^3$ ($S_4$)

Order 12: $C_2\times C_6$ ($C_3^2:S_3$) x 3, $C_2\times C_6$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3:S_4$), $C_3^2$ ($C_3\times S_4$)

Order 4: $C_2^2$ ($C_3^3:S_3$)

Order 3: $C_3$ ($C_3^2:S_4$) x 3, $C_3$ ($C_3^2:S_4$)

Order 1: $C_1$ ($C_3^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $C_3^3:S_4$ ($C_1$)

Order 324: $C_3^3:A_4$ ($C_2$)

Order 216: $C_3^2:S_4$ ($C_3$)

Order 108: $C_3\times C_6^2$ ($S_3$), $C_3^2\times A_4$ ($S_3$), $C_3^2:A_4$ ($C_6$)

Order 36: $C_6^2$ ($C_3:S_3$), $C_6^2$ ($C_3\times S_3$), $C_3\times A_4$ ($C_3\times S_3$)

Order 27: $C_3^3$ ($S_4$)

Order 12: $C_2\times C_6$ ($C_3^2:S_3$), $C_2\times C_6$ ($C_3^2:C_6$)

Order 9: $C_3^2$ ($C_3:S_4$), $C_3^2$ ($C_3\times S_4$)

Order 4: $C_2^2$ ($C_3^3:S_3$)

Order 3: $C_3$ ($C_3^2:S_4$), $C_3$ ($C_3^2:S_4$)

Order 1: $C_1$ ($C_3^3:S_4$)

Series

Derived series

$C_3^3:S_4$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

Chief series

$C_3^3:S_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^2:A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_3^3:S_4$

$\rhd$

$C_3^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_3^2$

Supergroups

This group is a maximal subgroup of 20 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

See the $57 \times 57$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.