Abstract group 18.5: $C_3\times C_6$

• Label: 18.5
• Order: \( 2 \cdot 3^{2} \)
• Exponent: \( 2 \cdot 3 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $8$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3\times C_6$
Order:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$\GL(2,3)$, of order \(48\)\(\medspace = 2^{4} \cdot 3 \)
Composition factors:	$C_2$, $C_3$ x 2
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
Elements	1	1	8	8	18
Conjugacy classes	1	1	8	8	18
Divisions	1	1	4	4	10
Autjugacy classes	1	1	1	1	4

Dimension	1	2
Irr. complex chars.	18	0	18
Irr. rational chars.	2	8	10

Minimal presentations

Permutation degree:	$8$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$3$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{3}=b^{6}=1 \rangle$
Permutation group:	$\langle(1,2), (3,5,4), (6,8,7)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & -1 & 0 & 0 \\ 1 & -1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right) \right\rangle \subseteq \GL_{4}(\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 2 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{7})$
Transitive group:	18T2				more information
Direct product:	$C_2$ $\, \times\, $ $C_3$ ${}^2$
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 12 subgroups, all normal (4 characteristic).

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

Special subgroups

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

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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Classes of subgroups up to conjugation

Order 18: $C_3\times C_6$

Order 9: $C_3^2$

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 18: $C_3\times C_6$

Order 9: $C_3^2$

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 18: $C_3\times C_6$ ($C_1$)

Order 9: $C_3^2$ ($C_2$)

Order 6: $C_6$ ($C_3$) x 4

Order 3: $C_3$ ($C_6$) x 4

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

Order 1: $C_1$ ($C_3\times C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 18: $C_3\times C_6$ ($C_1$)

Order 9: $C_3^2$ ($C_2$)

Order 6: $C_6$ ($C_3$)

Order 3: $C_3$ ($C_6$)

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

Order 1: $C_1$ ($C_3\times C_6$)

Series

Derived series	$C_3\times C_6$	$\rhd$	$C_1$
Chief series	$C_3\times C_6$	$\rhd$	$C_6$	$\rhd$	$C_3$	$\rhd$	$C_1$
Lower central series	$C_3\times C_6$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_3\times C_6$

Supergroups

This group is a maximal subgroup of 163 larger groups in the database.

This group is a maximal quotient of 171 larger groups in the database.

Character theory

Complex character table

See the $18 \times 18$ 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
	Size	1	1	2	2	2	2	2	2	2	2
	2 P	1A	1A	3A	3B	3C	3D	3A	3B	3C	3D
	3 P	1A	2A	1A	1A	1A	1A	2A	2A	2A	2A
18.5.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
18.5.1b		$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
18.5.1c		$2$	$2$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
18.5.1d		$2$	$2$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$2$	$-1$
18.5.1e		$2$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$2$	$-1$	$-1$
18.5.1f		$2$	$2$	$2$	$-1$	$-1$	$2$	$-1$	$-1$	$-1$	$2$
18.5.1g		$2$	$-2$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
18.5.1h		$2$	$-2$	$-1$	$2$	$-1$	$-1$	$-2$	$1$	$-2$	$1$
18.5.1i		$2$	$-2$	$-1$	$-1$	$2$	$-1$	$1$	$-2$	$1$	$1$
18.5.1j		$2$	$-2$	$2$	$-1$	$-1$	$2$	$1$	$1$	$1$	$-2$