Abstract group 972.750: $(C_6\times \He_3).S_3$

• Label: 972.750
• Order: \( 2^{2} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{8} \cdot 3^{4} \cdot 5 \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $5$

Group information

Description:	$(C_6\times \He_3).S_3$
Order:	\(972\)\(\medspace = 2^{2} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_3:(C_3^3:C_2).\SU(4,2).C_2$, of order \(16796160\)\(\medspace = 2^{9} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1	242	162	242	324	972
Conjugacy classes	1	1	42	2	42	4	92
Divisions	1	1	41	1	41	1	86
Autjugacy classes	1	1	2	1	2	1	8

Dimension	1	2	9	18	36
Irr. complex chars.	4	80	8	0	0	92
Irr. rational chars.	2	81	0	2	1	86

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$5$
Inequivalent generating 5-tuples:	$28203552$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{4}=b^{3}=c^{3}=d^{3}=e^{3}=f^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $31$ $\langle(2,6)(3,8)(4,11)(7,15)(9,19)(10,22)(12,24)(14,16)(17,20)(18,23)(21,25)(26,27) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^3:C_3^2)$ $\,\rtimes\,$ $C_4$	$C_3^3$ $\,\rtimes\,$ $(C_3^2:C_4)$	$\He_3$ $\,\rtimes\,$ $(C_3^2:C_4)$	$(C_3\times \He_3)$ $\,\rtimes\,$ $(C_3:C_4)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_3^3:S_3)$	$(C_6\times \He_3)$ . $S_3$	$(C_6.C_3^4)$ . $C_2$	$C_3$ . $(C_3^4:C_4)$	all 9

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_{3}^{5}$
Commutator length:	$1$

Subgroups

There are 5359 subgroups in 879 conjugacy classes, 427 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_3^3:S_3$
Commutator:	$G' \simeq$ $C_3^3:C_3^2$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3^3:S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_6\times \He_3).S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^3:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3:C_3^2$

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.

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

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

Classes of subgroups up to conjugation

Order 972: $(C_6\times \He_3).S_3$

Order 486: $C_6.C_3^4$

Order 324: $(C_3\times \He_3):C_4$ x 40

Order 243: $C_3^3:C_3^2$

Order 162: $C_6\times \He_3$ x 40

Order 108: $\He_3:C_4$ x 90, $C_3^2:C_{12}$ x 40

Order 81: $C_3\times \He_3$ x 40

Order 54: $C_2\times \He_3$ x 90, $C_3^2\times C_6$ x 40

Order 36: $C_3^2:C_4$ x 40, $C_3:C_{12}$ x 40

Order 27: $\He_3$ x 90, $C_3^3$ x 40

Order 18: $C_3\times C_6$ x 80

Order 12: $C_3:C_4$ x 40, $C_{12}$

Order 9: $C_3^2$ x 80

Order 6: $C_6$ x 41

Order 4: $C_4$

Order 3: $C_3$ x 41

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 972: $(C_6\times \He_3).S_3$

Order 486: $C_6.C_3^4$

Order 324: $(C_3\times \He_3):C_4$

Order 243: $C_3^3:C_3^2$

Order 162: $C_6\times \He_3$

Order 108: $C_3^2:C_{12}$, $\He_3:C_4$

Order 81: $C_3\times \He_3$

Order 54: $C_3^2\times C_6$, $C_2\times \He_3$

Order 36: $C_3^2:C_4$, $C_3:C_{12}$

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

Order 18: $C_3\times C_6$ x 2

Order 12: $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 2

Order 6: $C_6$ x 2

Order 4: $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 972: $(C_6\times \He_3).S_3$ ($C_1$)

Order 486: $C_6.C_3^4$ ($C_2$)

Order 243: $C_3^3:C_3^2$ ($C_4$)

Order 162: $C_6\times \He_3$ ($S_3$) x 40

Order 81: $C_3\times \He_3$ ($C_3:C_4$) x 40

Order 54: $C_2\times \He_3$ ($C_3:S_3$) x 90, $C_3^2\times C_6$ ($C_3:S_3$) x 40

Order 27: $\He_3$ ($C_3^2:C_4$) x 90, $C_3^3$ ($C_3^2:C_4$) x 40

Order 18: $C_3\times C_6$ ($C_3^2:S_3$) x 40

Order 9: $C_3^2$ ($C_3^3:C_4$) x 40

Order 6: $C_6$ ($C_3^3:S_3$)

Order 3: $C_3$ ($C_3^4:C_4$)

Order 2: $C_2$ ($(C_3\times \He_3):S_3$)

Order 1: $C_1$ ($(C_6\times \He_3).S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 972: $(C_6\times \He_3).S_3$ ($C_1$)

Order 486: $C_6.C_3^4$ ($C_2$)

Order 243: $C_3^3:C_3^2$ ($C_4$)

Order 162: $C_6\times \He_3$ ($S_3$)

Order 81: $C_3\times \He_3$ ($C_3:C_4$)

Order 54: $C_3^2\times C_6$ ($C_3:S_3$), $C_2\times \He_3$ ($C_3:S_3$)

Order 27: $C_3^3$ ($C_3^2:C_4$), $\He_3$ ($C_3^2:C_4$)

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

Order 9: $C_3^2$ ($C_3^3:C_4$)

Order 6: $C_6$ ($C_3^3:S_3$)

Order 3: $C_3$ ($C_3^4:C_4$)

Order 2: $C_2$ ($(C_3\times \He_3):S_3$)

Order 1: $C_1$ ($(C_6\times \He_3).S_3$)

Series

Derived series

$(C_6\times \He_3).S_3$

$\rhd$

$C_3^3:C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$(C_6\times \He_3).S_3$

$\rhd$

$C_6.C_3^4$

$\rhd$

$C_3^3:C_3^2$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_6\times \He_3).S_3$

$\rhd$

$C_3^3:C_3^2$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $92 \times 92$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $86 \times 86$ rational character table.