Abstract group 486.235: $(C_3^2\times C_9):S_3$

• Label: 486.235
• Order: \( 2 \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3^{3} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{4} \)
• Perm deg.: $30$
• Trans deg.: $81$
• Rank: $3$

Group information

Description:	$(C_3^2\times C_9):S_3$
Order:	\(486\)\(\medspace = 2 \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3:S_3.C_6^2.S_3^2$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Composition factors:	$C_2$, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	9	80	72	162	162	486
Conjugacy classes	1	1	20	8	42	18	90
Divisions	1	1	12	4	15	3	36
Autjugacy classes	1	1	4	2	2	1	11

Dimension	1	2	3	4	18
Irr. complex chars.	18	36	36	0	0	90
Irr. rational chars.	2	12	0	16	6	36

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$81$
Rank:	$3$
Inequivalent generating triples:	$2184$

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:	$\langle a, b, c, d \mid a^{6}=b^{3}=c^{3}=d^{9}=[a,d]=[b,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{2}, c^{b}=cd^{6} \rangle$
Permutation group:	Degree $30$ $\langle(4,9)(5,13)(8,15)(11,18)(12,21)(17,23)(19,25)(20,26)(24,27), (1,2,6)(3,7,14) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(\He_3.C_6)$
Semidirect product:	$(C_3^2\times C_9)$ $\,\rtimes\,$ $S_3$	$(C_9.C_3^3)$ $\,\rtimes\,$ $C_2$	$(C_9.C_3^2)$ $\,\rtimes\,$ $C_6$	$(C_3\times C_9)$ $\,\rtimes\,$ $(C_3\times S_3)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $C_3$	$C_3^3$ . $(C_3\times S_3)$	$C_9$ . $(C_3^2:C_6)$	$(C_3\times \He_3)$ . $C_6$	all 10

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:	$C_{3}$
Commutator length:	$1$

Subgroups

There are 504 subgroups in 204 conjugacy classes, 46 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_9$	$G/Z \simeq$ $C_3:S_3$
Commutator:	$G' \simeq$ $\He_3$	$G/G' \simeq$ $C_3\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^2\wr C_2$
Fitting:	$\operatorname{Fit} \simeq$ $C_9.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $(C_3^2\times C_9):S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3^2:C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9.C_3^3$

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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Classes of subgroups up to conjugation

Order 486: $(C_3^2\times C_9):S_3$

Order 243: $C_9.C_3^3$

Order 162: $C_3^2:C_{18}$ x 4, $\He_3.C_6$ x 3, $C_3^3:S_3$

Order 81: $C_9.C_3^2$ x 15, $C_9:C_3^2$ x 4, $C_3^2\times C_9$ x 4, $C_3\times \He_3$

Order 54: $S_3\times C_9$ x 12, $S_3\times C_3^2$ x 4, $C_3\times C_{18}$, $C_3^2:S_3$

Order 27: $C_9:C_3$ x 36, $C_3\times C_9$ x 29, $\He_3$ x 5, $C_3^3$ x 4

Order 18: $C_3\times S_3$ x 16, $C_{18}$ x 3, $C_3\times C_6$

Order 9: $C_3^2$ x 21, $C_9$ x 15

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

Order 3: $C_3$ x 12

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 486: $(C_3^2\times C_9):S_3$

Order 243: $C_9.C_3^3$

Order 162: $\He_3.C_6$, $C_3^3:S_3$, $C_3^2:C_{18}$

Order 81: $C_9.C_3^2$ x 2, $C_9:C_3^2$, $C_3\times \He_3$, $C_3^2\times C_9$

Order 54: $C_3\times C_{18}$, $C_3^2:S_3$, $S_3\times C_9$, $S_3\times C_3^2$

Order 27: $C_3\times C_9$ x 4, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3^3$

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

Order 9: $C_3^2$ x 4, $C_9$ x 2

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

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 486: $(C_3^2\times C_9):S_3$ ($C_1$)

Order 243: $C_9.C_3^3$ ($C_2$)

Order 162: $\He_3.C_6$ ($C_3$) x 3, $C_3^3:S_3$ ($C_3$)

Order 81: $C_3^2\times C_9$ ($S_3$) x 4, $C_9.C_3^2$ ($C_6$) x 3, $C_3\times \He_3$ ($C_6$)

Order 54: $C_3^2:S_3$ ($C_3^2$)

Order 27: $C_3\times C_9$ ($C_3\times S_3$) x 12, $C_3^3$ ($C_3\times S_3$) x 4, $\He_3$ ($C_3\times C_6$), $C_3\times C_9$ ($C_3:S_3$)

Order 9: $C_3^2$ ($S_3\times C_3^2$) x 4, $C_9$ ($C_3^2:C_6$) x 3, $C_3^2$ ($C_3^2:C_6$)

Order 3: $C_3$ ($\He_3.C_6$) x 3, $C_3$ ($C_3^2\wr C_2$)

Order 1: $C_1$ ($(C_3^2\times C_9):S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 486: $(C_3^2\times C_9):S_3$ ($C_1$)

Order 243: $C_9.C_3^3$ ($C_2$)

Order 162: $\He_3.C_6$ ($C_3$), $C_3^3:S_3$ ($C_3$)

Order 81: $C_9.C_3^2$ ($C_6$), $C_3\times \He_3$ ($C_6$), $C_3^2\times C_9$ ($S_3$)

Order 54: $C_3^2:S_3$ ($C_3^2$)

Order 27: $C_3^3$ ($C_3\times S_3$), $\He_3$ ($C_3\times C_6$), $C_3\times C_9$ ($C_3:S_3$), $C_3\times C_9$ ($C_3\times S_3$)

Order 9: $C_3^2$ ($C_3^2:C_6$), $C_3^2$ ($S_3\times C_3^2$), $C_9$ ($C_3^2:C_6$)

Order 3: $C_3$ ($C_3^2\wr C_2$), $C_3$ ($\He_3.C_6$)

Order 1: $C_1$ ($(C_3^2\times C_9):S_3$)

Series

Derived series

$(C_3^2\times C_9):S_3$

$\rhd$

$\He_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$(C_3^2\times C_9):S_3$

$\rhd$

$C_9.C_3^3$

$\rhd$

$C_9.C_3^2$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$(C_3^2\times C_9):S_3$

$\rhd$

$\He_3$

Upper central series

$C_1$

$\lhd$

$C_3\times C_9$

Supergroups

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

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

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $36 \times 36$ rational character table.