Abstract group 243.58: $\He_3:C_3^2$

• Label: 243.58
• Order: \( 3^{5} \)
• Exponent: \( 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $3$

Group information

Description:	$\He_3:C_3^2$
Order:	\(243\)\(\medspace = 3^{5} \)
Exponent:	\(9\)\(\medspace = 3^{2} \)
Automorphism group:	$C_3^5:C_3.S_3^2$, of order \(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Composition factors:	$C_3$ x 5
Nilpotency class:	$3$
Derived length:	$2$

This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.

Group statistics

Order	1	3	9
Elements	1	188	54	243
Conjugacy classes	1	28	6	35
Divisions	1	14	3	18
Autjugacy classes	1	5	1	7

Dimension	1	2	3	6	9	18
Irr. complex chars.	27	0	6	0	2	0	35
Irr. rational chars.	1	13	0	3	0	1	18

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$3$
Inequivalent generating triples:	$312$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	9	18	18

Constructions

Presentation:	$\langle a, b, c, d \mid a^{3}=b^{3}=c^{3}=d^{9}=[a,b]=[a,c]=[c,d]=1, d^{a}=d^{4}, c^{b}=cd^{3}, d^{b}=c^{2}d^{7} \rangle$
Permutation group:	Degree $27$ $\langle(1,10,22)(2,11,23)(3,12,24)(4,14,27)(5,15,25)(6,13,26)(7,18,20)(8,16,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 2 & 0 & 2 & 0 \\ 1 & 0 & 2 & 1 \end{array}\right), \left(\begin{array}{rrrr} 2 & 2 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 2 & 1 & 1 \\ 2 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrr} 2 & 1 & 0 & 1 \\ 0 & 2 & 2 & 2 \\ 1 & 0 & 2 & 2 \\ 2 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 2 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 2 & 0 & 0 & 1 \\ 1 & 1 & 2 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 0 & 2 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{3})$
Transitive group:	27T96	27T97			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$\He_3$ $\,\rtimes\,$ $C_3^2$ (9)	$(C_9:C_3^2)$ $\,\rtimes\,$ $C_3$	$(C_3\times C_9)$ $\,\rtimes\,$ $C_3^2$ (3)	$(C_3\times \He_3)$ $\,\rtimes\,$ $C_3$ (3)	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^3$ . $C_3^2$	$C_3^2$ . $C_3^3$	$C_3^2$ . $\He_3$ (3)	$C_3$ . $(C_3\times \He_3)$	more information

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

Homology

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

Subgroups

There are 288 subgroups in 80 conjugacy classes, 33 normal (6 characteristic).

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

Special subgroups

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

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Subgroup information

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

Classes of subgroups up to conjugation

Order 243: $\He_3:C_3^2$

Order 81: $\He_3:C_3$ x 9, $C_3\times \He_3$ x 3, $C_9:C_3^2$

Order 27: $\He_3$ x 15, $C_3^3$ x 4, $C_9:C_3$ x 3, $C_3\times C_9$ x 3

Order 9: $C_3^2$ x 23, $C_9$ x 3

Order 3: $C_3$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 243: $\He_3:C_3^2$

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

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

Order 9: $C_3^2$ x 5, $C_9$

Order 3: $C_3$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 243: $\He_3:C_3^2$ ($C_1$)

Order 81: $\He_3:C_3$ ($C_3$) x 9, $C_3\times \He_3$ ($C_3$) x 3, $C_9:C_3^2$ ($C_3$)

Order 27: $\He_3$ ($C_3^2$) x 9, $C_3\times C_9$ ($C_3^2$) x 3, $C_3^3$ ($C_3^2$)

Order 9: $C_3^2$ ($\He_3$) x 3, $C_3^2$ ($C_3^3$)

Order 3: $C_3$ ($C_3\times \He_3$)

Order 1: $C_1$ ($\He_3:C_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 243: $\He_3:C_3^2$ ($C_1$)

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

Order 27: $C_3^3$ ($C_3^2$), $\He_3$ ($C_3^2$), $C_3\times C_9$ ($C_3^2$)

Order 9: $C_3^2$ ($C_3^3$), $C_3^2$ ($\He_3$)

Order 3: $C_3$ ($C_3\times \He_3$)

Order 1: $C_1$ ($\He_3:C_3^2$)

Series

Derived series

$\He_3:C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$\He_3:C_3^2$

$\rhd$

$\He_3:C_3$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$\He_3:C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3^3$

$\lhd$

$\He_3:C_3^2$

Supergroups

This group is a maximal subgroup of 31 larger groups in the database.

This group is a maximal quotient of 52 larger groups in the database.

Character theory

Complex character table

See the $35 \times 35$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $18 \times 18$ rational character table.