Abstract group 648.557: $S_4\times \He_3$

• Label: 648.557
• Order: \( 2^{3} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$S_4\times \He_3$
Order:	\(648\)\(\medspace = 2^{3} \cdot 3^{4} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$S_4\times C_3^2:\GL(2,3)$, of order \(10368\)\(\medspace = 2^{7} \cdot 3^{4} \)
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	9	242	6	234	156	648
Conjugacy classes	1	2	21	1	20	10	55
Divisions	1	2	11	1	10	5	30
Autjugacy classes	1	2	5	1	4	2	15

Dimension	1	2	3	4	6	9	12	18
Irr. complex chars.	18	9	22	0	2	4	0	0	55
Irr. rational chars.	2	9	2	4	10	0	1	2	30

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{3}=c^{6}=d^{6}=[c,d]=1, b^{a}=b^{2}, c^{a}=cd^{3}, d^{a}=c^{4}d, c^{b}=c^{4}d^{3}, d^{b}=c^{3}d \rangle$
Permutation group:	Degree $13$ $\langle(11,12), (1,2,5)(3,4,6)(7,8,9), (1,3,2)(4,7,6)(5,9,8), (1,4,8)(2,6,9)(3,7,5), (11,12,13), (10,11)(12,13), (10,12)(11,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 8 & 9 \\ 9 & 8 \end{array}\right), \left(\begin{array}{rr} 7 & 29 \\ 3 & 4 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 27 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Transitive group:	36T1210				more information
Direct product:	$S_4$ $\, \times\, $ $\He_3$
Semidirect product:	$(C_3^2\times S_4)$ $\,\rtimes\,$ $C_3$	$C_6^2$ $\,\rtimes\,$ $(C_3\times S_3)$	$C_3^2$ $\,\rtimes\,$ $(C_3\times S_4)$	$(C_3^2\times A_4)$ $\,\rtimes\,$ $C_6$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times S_4)$ . $C_3^2$	$C_3$ . $(C_3^2\times S_4)$	$(C_3\times A_4)$ . $(C_3\times C_6)$	$(C_2\times C_6)$ . $(S_3\times C_3^2)$	more information

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

Homology

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

Subgroups

There are 900 subgroups in 155 conjugacy classes, 28 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^2\times S_4$
Commutator:	$G' \simeq$ $C_3\times A_4$	$G/G' \simeq$ $C_3\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times \He_3$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $S_4\times \He_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times C_3^2$
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

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 648: $S_4\times \He_3$

Order 324: $A_4\times \He_3$

Order 216: $C_3^2\times S_4$ x 4, $D_4\times \He_3$

Order 162: $S_3\times \He_3$

Order 108: $C_3^2\times A_4$ x 4, $C_3^2:A_4$ x 4, $C_2^2\times \He_3$ x 2, $C_4\times \He_3$

Order 81: $C_3\times \He_3$

Order 72: $C_3\times S_4$ x 5, $D_4\times C_3^2$ x 4

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

Order 36: $C_3\times A_4$ x 13, $C_6^2$ x 8, $C_3\times C_{12}$ x 4

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

Order 24: $C_3\times D_4$ x 5, $S_4$

Order 18: $C_3\times C_6$ x 8, $C_3\times S_3$ x 5

Order 12: $C_2\times C_6$ x 10, $A_4$ x 6, $C_{12}$ x 5

Order 9: $C_3^2$ x 17

Order 8: $D_4$

Order 6: $C_6$ x 10, $S_3$

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

Order 3: $C_3$ x 11

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 648: $S_4\times \He_3$

Order 324: $A_4\times \He_3$

Order 216: $D_4\times \He_3$, $C_3^2\times S_4$

Order 162: $S_3\times \He_3$

Order 108: $C_2^2\times \He_3$ x 2, $C_3^2\times A_4$, $C_3^2:A_4$, $C_4\times \He_3$

Order 81: $C_3\times \He_3$

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

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

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

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

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

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

Order 12: $C_2\times C_6$ x 4, $A_4$ x 3, $C_{12}$ x 2

Order 9: $C_3^2$ x 5

Order 8: $D_4$

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

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 648: $S_4\times \He_3$ ($C_1$)

Order 324: $A_4\times \He_3$ ($C_2$)

Order 216: $C_3^2\times S_4$ ($C_3$) x 4

Order 108: $C_3^2\times A_4$ ($C_6$) x 4, $C_2^2\times \He_3$ ($S_3$)

Order 72: $C_3\times S_4$ ($C_3^2$)

Order 36: $C_6^2$ ($C_3\times S_3$) x 4, $C_3\times A_4$ ($C_3\times C_6$)

Order 27: $\He_3$ ($S_4$)

Order 24: $S_4$ ($\He_3$)

Order 12: $C_2\times C_6$ ($S_3\times C_3^2$), $A_4$ ($C_2\times \He_3$)

Order 9: $C_3^2$ ($C_3\times S_4$) x 4

Order 4: $C_2^2$ ($S_3\times \He_3$)

Order 3: $C_3$ ($C_3^2\times S_4$)

Order 1: $C_1$ ($S_4\times \He_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 648: $S_4\times \He_3$ ($C_1$)

Order 324: $A_4\times \He_3$ ($C_2$)

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

Order 108: $C_3^2\times A_4$ ($C_6$), $C_2^2\times \He_3$ ($S_3$)

Order 72: $C_3\times S_4$ ($C_3^2$)

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

Order 27: $\He_3$ ($S_4$)

Order 24: $S_4$ ($\He_3$)

Order 12: $C_2\times C_6$ ($S_3\times C_3^2$), $A_4$ ($C_2\times \He_3$)

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

Order 4: $C_2^2$ ($S_3\times \He_3$)

Order 3: $C_3$ ($C_3^2\times S_4$)

Order 1: $C_1$ ($S_4\times \He_3$)

Series

Derived series

$S_4\times \He_3$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$S_4\times \He_3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$S_4\times \He_3$

$\rhd$

$C_3\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_3$

$\lhd$

$\He_3$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 17 larger groups in the database.

Character theory

Complex character table

See the $55 \times 55$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.