Abstract group 486.247: $C_3^3:D_9$

• Label: 486.247
• Order: \( 2 \cdot 3^{5} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{15} \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3^{10} \cdot 13 \)
• Perm deg.: $18$
• Trans deg.: $243$
• Rank: $5$

Group information

Description:	$C_3^3:D_9$
Order:	\(486\)\(\medspace = 2 \cdot 3^{5} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3^4.C_3^3.C_2^2.\SL(3,3)$, of order \(11938290624\)\(\medspace = 2^{6} \cdot 3^{15} \cdot 13 \)
Composition factors:	$C_2$, $C_3$ x 5
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	9
Elements	1	243	80	162	486
Conjugacy classes	1	1	40	81	123
Divisions	1	1	40	27	69
Autjugacy classes	1	1	2	1	5

Dimension	1	2	6
Irr. complex chars.	2	121	0	123
Irr. rational chars.	2	40	27	69

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$243$
Rank:	$5$
Inequivalent generating 5-tuples:	$1240$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{3}=c^{3}=d^{3}=e^{9}=[b,c]=[b,d]=[b,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,3)(5,6)(8,9)(11,14)(12,13)(15,17)(16,18), (7,8,9)(10,11,15,12,16,18,13,17,14) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^3$ $\,\rtimes\,$ $D_9$	$(C_3^2\times C_9)$ $\,\rtimes\,$ $S_3$	$C_9$ $\,\rtimes\,$ $(C_3^2:S_3)$	$C_3^2$ $\,\rtimes\,$ $(C_3:D_9)$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $S_3$	$C_3^3$ . $(C_3:S_3)$	$C_3$ . $(C_3^3:S_3)$	$C_3^2$ . $(C_3^2:S_3)$	more information

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

Homology

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

Subgroups

There are 9652 subgroups in 792 conjugacy classes, 397 normal (5 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:D_9$
Commutator:	$G' \simeq$ $C_3^3\times C_9$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^3:S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_9$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_3^3:D_9$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3\times C_9$

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: $C_3^3:D_9$

Order 243: $C_3^3\times C_9$

Order 162: $C_3^2:D_9$ x 39, $C_3^3:S_3$

Order 81: $C_3^2\times C_9$ x 39, $C_3^4$

Order 54: $C_3:D_9$ x 117, $C_3^2:S_3$ x 40

Order 27: $C_3\times C_9$ x 117, $C_3^3$ x 40

Order 18: $C_3:S_3$ x 130, $D_9$ x 27

Order 9: $C_3^2$ x 130, $C_9$ x 27

Order 6: $S_3$ x 40

Order 3: $C_3$ x 40

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 486: $C_3^3:D_9$

Order 243: $C_3^3\times C_9$

Order 162: $C_3^3:S_3$, $C_3^2:D_9$

Order 81: $C_3^4$, $C_3^2\times C_9$

Order 54: $C_3^2:S_3$ x 2, $C_3:D_9$

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

Order 18: $C_3:S_3$ x 2, $D_9$

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

Order 6: $S_3$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 486: $C_3^3:D_9$ ($C_1$)

Order 243: $C_3^3\times C_9$ ($C_2$)

Order 81: $C_3^2\times C_9$ ($S_3$) x 39, $C_3^4$ ($S_3$)

Order 27: $C_3\times C_9$ ($C_3:S_3$) x 117, $C_3^3$ ($D_9$) x 27, $C_3^3$ ($C_3:S_3$) x 13

Order 9: $C_3^2$ ($C_3:D_9$) x 117, $C_9$ ($C_3^2:S_3$) x 27, $C_3^2$ ($C_3^2:S_3$) x 13

Order 3: $C_3$ ($C_3^2:D_9$) x 39, $C_3$ ($C_3^3:S_3$)

Order 1: $C_1$ ($C_3^3:D_9$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 486: $C_3^3:D_9$ ($C_1$)

Order 243: $C_3^3\times C_9$ ($C_2$)

Order 81: $C_3^4$ ($S_3$), $C_3^2\times C_9$ ($S_3$)

Order 27: $C_3^3$ ($C_3:S_3$), $C_3^3$ ($D_9$), $C_3\times C_9$ ($C_3:S_3$)

Order 9: $C_3^2$ ($C_3:D_9$), $C_3^2$ ($C_3^2:S_3$), $C_9$ ($C_3^2:S_3$)

Order 3: $C_3$ ($C_3^3:S_3$), $C_3$ ($C_3^2:D_9$)

Order 1: $C_1$ ($C_3^3:D_9$)

Series

Derived series

$C_3^3:D_9$

$\rhd$

$C_3^3\times C_9$

$\rhd$

$C_1$

Chief series

$C_3^3:D_9$

$\rhd$

$C_3^3\times C_9$

$\rhd$

$C_3^2\times C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^3:D_9$

$\rhd$

$C_3^3\times C_9$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 28 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $123 \times 123$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $69 \times 69$ rational character table.