Abstract group 162.50: $C_6.C_3^3$

• Label: 162.50
• Order: \( 2 \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $29$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_6.C_3^3$
Order:	\(162\)\(\medspace = 2 \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^3:\GL(2,3)$, of order \(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Composition factors:	$C_2$, $C_3$ x 4
Nilpotency class:	$2$
Derived length:	$2$

This group is nonabelian, elementary for $p = 3$ (hence nilpotent, solvable, supersolvable, monomial, and hyperelementary), and metabelian.

Group statistics

Order	1	2	3	6	9	18
Elements	1	1	26	26	54	54	162
Conjugacy classes	1	1	10	10	22	22	66
Divisions	1	1	5	5	9	9	30
Autjugacy classes	1	1	2	2	2	2	10

Dimension	1	2	3	18
Irr. complex chars.	54	0	12	0	66
Irr. rational chars.	2	26	0	2	30

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$1638$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	18
Arbitrary	3	6	18

Constructions

Presentation:	$\langle a, b, c \mid a^{3}=b^{3}=c^{18}=[a,c]=[b,c]=1, b^{a}=bc^{6} \rangle$
Permutation group:	Degree $29$ $\langle(1,2), (3,12,22)(4,13,23)(5,14,21)(6,15,25)(7,16,26)(8,17,24)(9,18,28)(10,19,29) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 19 & 13 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 26 & 0 \\ 0 & 26 \end{array}\right), \left(\begin{array}{rr} 25 & 11 \\ 3 & 10 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/27\Z)$
	$\left\langle \left(\begin{array}{rr} 19 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 10 \\ 42 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 31 & 38 \\ 30 & 31 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_9.C_3^2)$
Semidirect product:	$(C_9:C_6)$ $\,\rtimes\,$ $C_3$ (8)	$(C_3\times C_9)$ $\,\rtimes\,$ $C_6$ (4)	$(C_9:C_3)$ $\,\rtimes\,$ $C_6$ (8)	$(C_3\times C_{18})$ $\,\rtimes\,$ $C_3$ (4)	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $C_3^3$	$\He_3$ . $C_6$	$C_{18}$ . $C_3^2$ (9)	$C_9$ . $(C_3\times C_6)$ (9)	all 8

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

Homology

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

Subgroups

There are 82 subgroups in 66 conjugacy classes, 58 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{18}$	$G/Z \simeq$ $C_3^2$
Commutator:	$G' \simeq$ $C_3$	$G/G' \simeq$ $C_3^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^2\times C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_6.C_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6$	$G/\operatorname{soc} \simeq$ $C_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9.C_3^2$

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 162: $C_6.C_3^3$

Order 81: $C_9.C_3^2$

Order 54: $C_9:C_6$ x 8, $C_3\times C_{18}$ x 4, $C_2\times \He_3$

Order 27: $C_9:C_3$ x 8, $C_3\times C_9$ x 4, $\He_3$

Order 18: $C_{18}$ x 9, $C_3\times C_6$ x 4

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

Order 6: $C_6$ x 5

Order 3: $C_3$ x 5

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 162: $C_6.C_3^3$

Order 81: $C_9.C_3^2$

Order 54: $C_3\times C_{18}$, $C_9:C_6$, $C_2\times \He_3$

Order 27: $C_9:C_3$, $\He_3$, $C_3\times C_9$

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

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

Order 6: $C_6$ x 2

Order 3: $C_3$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 162: $C_6.C_3^3$ ($C_1$)

Order 81: $C_9.C_3^2$ ($C_2$)

Order 54: $C_9:C_6$ ($C_3$) x 8, $C_3\times C_{18}$ ($C_3$) x 4, $C_2\times \He_3$ ($C_3$)

Order 27: $C_9:C_3$ ($C_6$) x 8, $C_3\times C_9$ ($C_6$) x 4, $\He_3$ ($C_6$)

Order 18: $C_{18}$ ($C_3^2$) x 9, $C_3\times C_6$ ($C_3^2$) x 4

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

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

Order 3: $C_3$ ($C_3^2\times C_6$)

Order 2: $C_2$ ($C_9.C_3^2$)

Order 1: $C_1$ ($C_6.C_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 162: $C_6.C_3^3$ ($C_1$)

Order 81: $C_9.C_3^2$ ($C_2$)

Order 54: $C_3\times C_{18}$ ($C_3$), $C_9:C_6$ ($C_3$), $C_2\times \He_3$ ($C_3$)

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

Order 18: $C_{18}$ ($C_3^2$) x 2, $C_3\times C_6$ ($C_3^2$)

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

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

Order 3: $C_3$ ($C_3^2\times C_6$)

Order 2: $C_2$ ($C_9.C_3^2$)

Order 1: $C_1$ ($C_6.C_3^3$)

Series

Derived series

$C_6.C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_6.C_3^3$

$\rhd$

$C_3\times C_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6.C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_{18}$

$\lhd$

$C_6.C_3^3$

Supergroups

This group is a maximal subgroup of 32 larger groups in the database.

This group is a maximal quotient of 30 larger groups in the database.

Character theory

Complex character table

See the $66 \times 66$ 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.