Abstract group 54.9: $C_3\times C_{18}$

• Label: 54.9
• Order: \( 2 \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{2} \cdot 3^{3} \)
• Perm deg.: $14$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_3\times C_{18}$
Order:	\(54\)\(\medspace = 2 \cdot 3^{3} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^2:D_6$, of order \(108\)\(\medspace = 2^{2} \cdot 3^{3} \)
Composition factors:	$C_2$, $C_3$ x 3
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 3$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	2	3	6	9	18
Elements	1	1	8	8	18	18	54
Conjugacy classes	1	1	8	8	18	18	54
Divisions	1	1	4	4	3	3	16
Autjugacy classes	1	1	2	2	1	1	8

Dimension	1	2	6
Irr. complex chars.	54	0	0	54
Irr. rational chars.	2	8	6	16

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	2	4	8

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{3}=b^{18}=1 \rangle$
Permutation group:	Degree $14$ $\langle(1,2), (6,14,11,8,13,10,7,12,9), (3,5,4), (6,8,7)(9,11,10)(12,14,13)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 11 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{19})$
Direct product:	$C_2$ $\, \times\, $ $C_3$ $\, \times\, $ $C_9$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $C_6$	$C_6$ . $C_3^2$	$(C_3\times C_6)$ . $C_3$	$C_3$ . $(C_3\times C_6)$	more information

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

Homology

Primary decomposition:	$C_{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{3}$
Commutator length:	$0$

Subgroups

There are 20 subgroups, all normal (8 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 54: $C_3\times C_{18}$

Order 27: $C_3\times C_9$

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

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

Order 6: $C_6$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 54: $C_3\times C_{18}$

Order 27: $C_3\times C_9$

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

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

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 54: $C_3\times C_{18}$ ($C_1$)

Order 27: $C_3\times C_9$ ($C_2$)

Order 18: $C_{18}$ ($C_3$) x 3, $C_3\times C_6$ ($C_3$)

Order 9: $C_9$ ($C_6$) x 3, $C_3^2$ ($C_6$)

Order 6: $C_6$ ($C_9$) x 3, $C_6$ ($C_3^2$)

Order 3: $C_3$ ($C_{18}$) x 3, $C_3$ ($C_3\times C_6$)

Order 2: $C_2$ ($C_3\times C_9$)

Order 1: $C_1$ ($C_3\times C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 54: $C_3\times C_{18}$ ($C_1$)

Order 27: $C_3\times C_9$ ($C_2$)

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

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

Order 6: $C_6$ ($C_3^2$), $C_6$ ($C_9$)

Order 3: $C_3$ ($C_3\times C_6$), $C_3$ ($C_{18}$)

Order 2: $C_2$ ($C_3\times C_9$)

Order 1: $C_1$ ($C_3\times C_{18}$)

Series

Derived series	$C_3\times C_{18}$	$\rhd$	$C_1$
Chief series	$C_3\times C_{18}$	$\rhd$	$C_{18}$	$\rhd$	$C_9$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3\times C_{18}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_3\times C_{18}$

Supergroups

This group is a maximal subgroup of 126 larger groups in the database.

This group is a maximal quotient of 113 larger groups in the database.

Character theory

Complex character table

See the $54 \times 54$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	3A	3B	3C	3D	6A	6B	6C	6D	9A	9B	9C	18A	18B	18C
	Size	1	1	2	2	2	2	2	2	2	2	6	6	6	6	6	6
	2 P	1A	1A	3A	3B	3C	3D	3A	3B	3C	3D	9A	9B	9C	9A	9B	9C
	3 P	1A	2A	1A	1A	1A	1A	2A	2A	2A	2A	3D	3D	3D	6D	6D	6D
54.9.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
54.9.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$-1$
54.9.1c		$2$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$2$
54.9.1d		$2$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$
54.9.1e		$2$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$2$	$-1$	$-1$	$2$	$2$	$-1$
54.9.1f		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
54.9.1g		$2$	$-2$	$-1$	$-1$	$2$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$2$	$1$	$1$	$-2$
54.9.1h		$2$	$-2$	$-1$	$-1$	$2$	$-1$	$1$	$1$	$1$	$1$	$-1$	$2$	$-1$	$1$	$1$	$1$
54.9.1i		$2$	$-2$	$-1$	$-1$	$2$	$-1$	$1$	$1$	$1$	$1$	$2$	$-1$	$-1$	$-2$	$-2$	$1$
54.9.1j		$2$	$-2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$-2$	$-2$	$-1$	$-1$	$-1$	$1$	$1$	$1$
54.9.1k		$6$	$6$	$-3$	$6$	$-3$	$-3$	$-3$	$6$	$-3$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$
54.9.1l		$6$	$6$	$-3$	$-3$	$-3$	$-3$	$-3$	$-3$	$6$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$
54.9.1m		$6$	$6$	$6$	$-3$	$-3$	$6$	$6$	$-3$	$-3$	$6$	$0$	$0$	$0$	$0$	$0$	$0$
54.9.1n		$6$	$-6$	$-3$	$6$	$-3$	$-3$	$3$	$-6$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$
54.9.1o		$6$	$-6$	$-3$	$-3$	$-3$	$-3$	$3$	$3$	$-6$	$3$	$0$	$0$	$0$	$0$	$0$	$0$
54.9.1p		$6$	$-6$	$6$	$-3$	$-3$	$6$	$-6$	$3$	$3$	$-6$	$0$	$0$	$0$	$0$	$0$	$0$