Abstract group 216.103: $A_4\times C_{18}$

• Label: 216.103
• Order: \( 2^{3} \cdot 3^{3} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$A_4\times C_{18}$
Order:	\(216\)\(\medspace = 2^{3} \cdot 3^{3} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^2:D_6$, of order \(432\)\(\medspace = 2^{4} \cdot 3^{3} \)
Composition factors:	$C_2$ x 3, $C_3$ x 3
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	26	38	54	90	216
Conjugacy classes	1	3	8	12	18	30	72
Divisions	1	3	4	6	3	5	22
Autjugacy classes	1	3	2	4	2	4	16

Dimension	1	2	3	6	18
Irr. complex chars.	54	0	18	0	0	72
Irr. rational chars.	2	8	2	8	2	22

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	18
Arbitrary	3	5	9

Constructions

Presentation:	$\langle a, b, c \mid a^{6}=b^{2}=c^{18}=[b,c]=1, b^{a}=c^{9}, c^{a}=bc \rangle$
Permutation group:	Degree $15$ $\langle(10,11), (13,14,15), (1,2,4,3,5,7,6,8,9), (1,3,6)(2,5,8)(4,7,9), (12,13)(14,15), (12,14)(13,15)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 25 & 27 \\ 27 & 34 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 19 & 18 \\ 18 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 18 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/36\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_9$ $\, \times\, $ $A_4$
Semidirect product:	$(C_2\times C_{18})$ $\,\rtimes\,$ $C_6$	$(C_2^2:C_9)$ $\,\rtimes\,$ $C_6$ (2)	$C_2^3$ $\,\rtimes\,$ $(C_3\times C_9)$	$(C_2^2\times C_{18})$ $\,\rtimes\,$ $C_3$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6\times A_4)$ . $C_3$	$(C_3\times A_4)$ . $C_6$	$C_6$ . $(C_3\times A_4)$	$C_3$ . $(C_6\times A_4)$	all 6

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

Homology

Abelianization:	$C_{3} \times C_{18} \simeq C_{2} \times C_{3} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 118 subgroups in 52 conjugacy classes, 26 normal (16 characteristic).

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

Special subgroups

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

Order 108: $C_9\times A_4$

Order 72: $C_2^2:C_{18}$ x 2, $C_6\times A_4$, $C_2^2\times C_{18}$

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

Order 36: $C_2\times C_{18}$ x 3, $C_2^2:C_9$ x 2, $C_3\times A_4$

Order 27: $C_3\times C_9$

Order 24: $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 18: $C_{18}$ x 5, $C_3\times C_6$

Order 12: $C_2\times C_6$ x 3, $A_4$ x 3

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

Order 8: $C_2^3$

Order 6: $C_6$ x 6

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 4

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 216: $A_4\times C_{18}$

Order 108: $C_9\times A_4$

Order 72: $C_6\times A_4$, $C_2^2\times C_{18}$, $C_2^2:C_{18}$

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

Order 36: $C_2\times C_{18}$ x 3, $C_2^2:C_9$, $C_3\times A_4$

Order 27: $C_3\times C_9$

Order 24: $C_2^2\times C_6$, $C_2\times A_4$

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

Order 12: $C_2\times C_6$ x 3, $A_4$

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

Order 8: $C_2^3$

Order 6: $C_6$ x 4

Order 4: $C_2^2$ x 3

Order 3: $C_3$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 216: $A_4\times C_{18}$ ($C_1$)

Order 108: $C_9\times A_4$ ($C_2$)

Order 72: $C_2^2:C_{18}$ ($C_3$) x 2, $C_6\times A_4$ ($C_3$), $C_2^2\times C_{18}$ ($C_3$)

Order 36: $C_2^2:C_9$ ($C_6$) x 2, $C_2\times C_{18}$ ($C_6$), $C_3\times A_4$ ($C_6$)

Order 24: $C_2\times A_4$ ($C_9$) x 3, $C_2^2\times C_6$ ($C_3^2$)

Order 18: $C_{18}$ ($A_4$)

Order 12: $A_4$ ($C_{18}$) x 3, $C_2\times C_6$ ($C_3\times C_6$)

Order 9: $C_9$ ($C_2\times A_4$)

Order 8: $C_2^3$ ($C_3\times C_9$)

Order 6: $C_6$ ($C_3\times A_4$)

Order 4: $C_2^2$ ($C_3\times C_{18}$)

Order 3: $C_3$ ($C_6\times A_4$)

Order 2: $C_2$ ($C_9\times A_4$)

Order 1: $C_1$ ($A_4\times C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 216: $A_4\times C_{18}$ ($C_1$)

Order 108: $C_9\times A_4$ ($C_2$)

Order 72: $C_6\times A_4$ ($C_3$), $C_2^2\times C_{18}$ ($C_3$), $C_2^2:C_{18}$ ($C_3$)

Order 36: $C_2\times C_{18}$ ($C_6$), $C_2^2:C_9$ ($C_6$), $C_3\times A_4$ ($C_6$)

Order 24: $C_2^2\times C_6$ ($C_3^2$), $C_2\times A_4$ ($C_9$)

Order 18: $C_{18}$ ($A_4$)

Order 12: $C_2\times C_6$ ($C_3\times C_6$), $A_4$ ($C_{18}$)

Order 9: $C_9$ ($C_2\times A_4$)

Order 8: $C_2^3$ ($C_3\times C_9$)

Order 6: $C_6$ ($C_3\times A_4$)

Order 4: $C_2^2$ ($C_3\times C_{18}$)

Order 3: $C_3$ ($C_6\times A_4$)

Order 2: $C_2$ ($C_9\times A_4$)

Order 1: $C_1$ ($A_4\times C_{18}$)

Series

Derived series

$A_4\times C_{18}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$A_4\times C_{18}$

$\rhd$

$C_9\times A_4$

$\rhd$

$C_2\times C_{18}$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$A_4\times C_{18}$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_{18}$

Supergroups

This group is a maximal subgroup of 40 larger groups in the database.

This group is a maximal quotient of 29 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.