Abstract group $C_2^2:C_{18}$

• Label: 72.16
• Order: \( 2^{3} \cdot 3^{2} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{3} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_2^2:C_{18}$
Order:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3\times S_4$, of order \(72\)\(\medspace = 2^{3} \cdot 3^{2} \) (generators)
Outer automorphisms:	$C_6$, of order \(6\)\(\medspace = 2 \cdot 3 \)
Composition factors:	$C_2$ x 3, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	7	2	14	24	24	72
Conjugacy classes	1	3	2	6	6	6	24
Divisions	1	3	1	3	1	1	10
Autjugacy classes	1	3	1	3	1	1	10

Dimension	1	2	3	6
Irr. complex chars.	18	0	6	0	24
Irr. rational chars.	2	2	2	4	10

Minimal Presentations

Permutation degree:	$15$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	6
Arbitrary	3	5	6

Constructions

Presentation:	$\langle a, b, c \mid a^{18}=b^{2}=c^{2}=[b,c]=1, b^{a}=c, c^{a}=bc \rangle$
Permutation group:	Degree $15$ $\langle(10,11), (1,2,4,3,5,7,6,8,9)(13,14,15), (1,3,6)(2,5,8)(4,7,9), (12,13)(14,15), (12,14)(13,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right), \left(\begin{array}{rrrrrr} 0 & 0 & 0 & 0 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ -1 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrr} 5 & 5 & 3 \\ 4 & 4 & 0 \\ 1 & 3 & 5 \end{array}\right), \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 5 & 5 & 2 \end{array}\right), \left(\begin{array}{rrr} 5 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 5 \end{array}\right), \left(\begin{array}{rrr} 4 & 1 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{array}\right), \left(\begin{array}{rrr} 4 & 0 & 0 \\ 0 & 4 & 0 \\ 0 & 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{7})$
Transitive group:	18T26	36T16	36T30		more information
Direct product:	$C_2$ $\, \times\, $ $(C_2^2:C_9)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $C_9$	$C_2^2$ $\,\rtimes\,$ $C_{18}$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $A_4$	$C_3$ . $(C_2\times A_4)$	$(C_2\times C_6)$ . $C_6$	$(C_2^2\times C_6)$ . $C_3$	more information

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

Homology

Abelianization:	$C_{18} \simeq C_{2} \times C_{9}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 42 subgroups in 20 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $A_4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_{18}$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_2\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_6$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^2:C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$

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 72: $C_2^2:C_{18}$

Order 36: $C_2^2:C_9$

Order 24: $C_2^2\times C_6$

Order 18: $C_{18}$

Order 12: $C_2\times C_6$ x 3

Order 9: $C_9$

Order 8: $C_2^3$

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 72: $C_2^2:C_{18}$

Order 36: $C_2^2:C_9$

Order 24: $C_2^2\times C_6$

Order 18: $C_{18}$

Order 12: $C_2\times C_6$ x 3

Order 9: $C_9$

Order 8: $C_2^3$

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 72: $C_2^2:C_{18}$ ($C_1$)

Order 36: $C_2^2:C_9$ ($C_2$)

Order 24: $C_2^2\times C_6$ ($C_3$)

Order 12: $C_2\times C_6$ ($C_6$)

Order 8: $C_2^3$ ($C_9$)

Order 6: $C_6$ ($A_4$)

Order 4: $C_2^2$ ($C_{18}$)

Order 3: $C_3$ ($C_2\times A_4$)

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

Order 1: $C_1$ ($C_2^2:C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 72: $C_2^2:C_{18}$ ($C_1$)

Order 36: $C_2^2:C_9$ ($C_2$)

Order 24: $C_2^2\times C_6$ ($C_3$)

Order 12: $C_2\times C_6$ ($C_6$)

Order 8: $C_2^3$ ($C_9$)

Order 6: $C_6$ ($A_4$)

Order 4: $C_2^2$ ($C_{18}$)

Order 3: $C_3$ ($C_2\times A_4$)

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

Order 1: $C_1$ ($C_2^2:C_{18}$)

Series

Derived series	$C_2^2:C_{18}$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Chief series	$C_2^2:C_{18}$	$\rhd$	$C_2^2:C_9$	$\rhd$	$C_2\times C_6$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Lower central series	$C_2^2:C_{18}$	$\rhd$	$C_2^2$
Upper central series	$C_1$	$\lhd$	$C_6$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 48 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	6A	6B	6C	9A	18A
	Size	1	1	3	3	2	2	6	6	24	24
	2 P	1A	1A	1A	1A	3A	3A	3A	3A	9A	9A
	3 P	1A	2A	2B	2C	1A	2A	2C	2B	3A	6A
72.16.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.16.1b		$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
72.16.1c		$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$-1$	$-1$
72.16.1d		$2$	$-2$	$-2$	$2$	$2$	$-2$	$2$	$-2$	$-1$	$1$
72.16.1e		$6$	$6$	$6$	$6$	$-3$	$-3$	$-3$	$-3$	$0$	$0$
72.16.1f		$6$	$-6$	$-6$	$6$	$-3$	$3$	$-3$	$3$	$0$	$0$
72.16.3a		$3$	$3$	$-1$	$-1$	$3$	$3$	$-1$	$-1$	$0$	$0$
72.16.3b		$3$	$-3$	$1$	$-1$	$3$	$-3$	$-1$	$1$	$0$	$0$
72.16.3c		$6$	$6$	$-2$	$-2$	$-3$	$-3$	$1$	$1$	$0$	$0$
72.16.3d		$6$	$-6$	$2$	$-2$	$-3$	$3$	$1$	$-1$	$0$	$0$