Abstract group 1296.2136: $(C_3\times C_{18}).S_4$

• Label: 1296.2136
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3^{5} \)
• Perm deg.: $20$
• Trans deg.: $324$
• Rank: $4$

Group information

Description:	$(C_3\times C_{18}).S_4$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_3^4.A_4.C_3.S_3^3$, of order \(1259712\)\(\medspace = 2^{6} \cdot 3^{9} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	18
Elements	1	7	80	648	128	162	270	1296
Conjugacy classes	1	3	13	4	21	27	45	114
Divisions	1	3	13	2	21	9	15	64
Autjugacy classes	1	3	3	2	7	2	4	22

Dimension	1	2	3	6	18
Irr. complex chars.	4	80	4	26	0	114
Irr. rational chars.	2	27	2	27	6	64

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$324$
Rank:	$4$
Inequivalent generating quadruples:	$1179360$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{4}=b^{3}=c^{18}=d^{6}=[c,d]=1, b^{a}=b^{2}, c^{a}=c^{17}d^{3}, d^{a}=d^{5}, c^{b}=cd^{3}, d^{b}=c^{9}d^{4} \rangle$
Permutation group:	Degree $20$ $\langle(2,4)(3,6)(5,9)(7,8)(11,12)(14,16)(17,18,19,20), (17,19)(18,20), (1,2,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 54 \\ 54 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 36 \\ 72 & 73 \end{array}\right), \left(\begin{array}{rr} 55 & 0 \\ 54 & 55 \end{array}\right), \left(\begin{array}{rr} 1 & 81 \\ 81 & 82 \end{array}\right), \left(\begin{array}{rr} 53 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 36 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 54 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/108\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_9$ $\,\rtimes\,$ $(C_6.S_4)$	$A_4$ $\,\rtimes\,$ $(C_6.D_9)$	$C_3$ $\,\rtimes\,$ $(C_{18}.S_4)$	$(C_6^2.C_3^2)$ $\,\rtimes\,$ $C_4$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6$ . $(C_9:S_4)$	$(C_6\times A_4)$ . $D_9$	$(A_4\times C_{18})$ . $S_3$	$(C_3\times C_{18})$ . $S_4$	all 26

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_{3}^{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 4366 subgroups in 370 conjugacy classes, 121 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_3\times C_9):S_4$
Commutator:	$G' \simeq$ $C_6^2.C_3^2$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_3^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6\times C_{18}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(C_3\times C_{18}).S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times C_9$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1296: $(C_3\times C_{18}).S_4$

Order 648: $C_3\times A_4\times C_{18}$

Order 432: $C_{18}.S_4$ x 9, $(C_3\times C_6).S_4$ x 2, $(C_3^2\times A_4):C_4$, $C_6^2.D_6$

Order 324: $C_6^2.C_3^2$, $(C_3^2\times C_9):C_4$

Order 216: $A_4\times C_{18}$ x 9, $C_6.D_{18}$ x 2, $C_6^2.C_6$ x 2, $C_6^2:C_6$, $C_2\times C_6\times C_{18}$

Order 162: $C_3^2\times C_{18}$

Order 144: $C_6.S_4$ x 12, $C_6.S_4$ x 6, $C_{18}.D_4$ x 3, $C_6^2:C_4$

Order 108: $C_6.D_9$ x 13, $C_9\times A_4$ x 9, $C_6\times C_{18}$ x 3, $C_3^2.A_4$ x 2, $C_3^2\times A_4$, $C_3^3:C_4$

Order 81: $C_3^2\times C_9$

Order 72: $C_6\times A_4$ x 12, $C_{18}:C_4$ x 6, $C_2^2:C_{18}$ x 6, $C_2^2\times C_{18}$ x 3, $C_6.D_6$ x 2, $C_2\times C_6^2$

Order 54: $C_3\times C_{18}$ x 14, $C_3^2\times C_6$

Order 48: $A_4:C_4$ x 9, $C_6.D_4$ x 4

Order 36: $C_3^2:C_4$ x 14, $C_3\times A_4$ x 12, $C_9:C_4$ x 12, $C_2\times C_{18}$ x 9, $C_2^2:C_9$ x 6, $C_6^2$ x 3

Order 27: $C_3\times C_9$ x 12, $C_3^3$

Order 24: $C_2\times A_4$ x 9, $C_6:C_4$ x 8, $C_2^2\times C_6$ x 4

Order 18: $C_3\times C_6$ x 15, $C_{18}$ x 15

Order 16: $C_2^2:C_4$

Order 12: $C_3:C_4$ x 17, $C_2\times C_6$ x 12, $A_4$ x 9

Order 9: $C_3^2$ x 13, $C_9$ x 9

Order 8: $C_2\times C_4$ x 2, $C_2^3$

Order 6: $C_6$ x 21

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 13

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $(C_3\times C_{18}).S_4$

Order 648: $C_3\times A_4\times C_{18}$

Order 432: $(C_3^2\times A_4):C_4$, $(C_3\times C_6).S_4$, $C_{18}.S_4$, $C_6^2.D_6$

Order 324: $C_6^2.C_3^2$, $(C_3^2\times C_9):C_4$

Order 216: $C_6.D_{18}$ x 2, $C_6^2:C_6$, $C_2\times C_6\times C_{18}$, $C_6^2.C_6$, $A_4\times C_{18}$

Order 162: $C_3^2\times C_{18}$

Order 144: $C_6.S_4$ x 2, $C_6.S_4$, $C_{18}.D_4$, $C_6^2:C_4$

Order 108: $C_6.D_9$ x 4, $C_6\times C_{18}$ x 3, $C_3^2\times A_4$, $C_3^3:C_4$, $C_3^2.A_4$, $C_9\times A_4$

Order 81: $C_3^2\times C_9$

Order 72: $C_{18}:C_4$ x 2, $C_6\times A_4$ x 2, $C_6.D_6$ x 2, $C_2\times C_6^2$, $C_2^2\times C_{18}$, $C_2^2:C_{18}$

Order 54: $C_3\times C_{18}$ x 5, $C_3^2\times C_6$

Order 48: $C_6.D_4$ x 2, $A_4:C_4$

Order 36: $C_3^2:C_4$ x 4, $C_2\times C_{18}$ x 3, $C_6^2$ x 3, $C_9:C_4$ x 3, $C_3\times A_4$ x 2, $C_2^2:C_9$

Order 27: $C_3\times C_9$ x 3, $C_3^3$

Order 24: $C_6:C_4$ x 4, $C_2^2\times C_6$ x 2, $C_2\times A_4$

Order 18: $C_3\times C_6$ x 5, $C_{18}$ x 4

Order 16: $C_2^2:C_4$

Order 12: $C_2\times C_6$ x 6, $C_3:C_4$ x 5, $A_4$

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

Order 8: $C_2\times C_4$ x 2, $C_2^3$

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $(C_3\times C_{18}).S_4$ ($C_1$)

Order 648: $C_3\times A_4\times C_{18}$ ($C_2$)

Order 324: $C_6^2.C_3^2$ ($C_4$)

Order 216: $A_4\times C_{18}$ ($S_3$) x 9, $C_6^2.C_6$ ($S_3$) x 2, $C_6^2:C_6$ ($S_3$), $C_2\times C_6\times C_{18}$ ($S_3$)

Order 108: $C_9\times A_4$ ($C_3:C_4$) x 9, $C_3^2.A_4$ ($C_3:C_4$) x 2, $C_3^2\times A_4$ ($C_3:C_4$), $C_6\times C_{18}$ ($C_3:C_4$)

Order 72: $C_6\times A_4$ ($D_9$) x 9, $C_2^2:C_{18}$ ($C_3:S_3$) x 6, $C_6\times A_4$ ($C_3:S_3$) x 3, $C_2^2\times C_{18}$ ($C_3:S_3$) x 3, $C_2\times C_6^2$ ($C_3:S_3$)

Order 54: $C_3\times C_{18}$ ($S_4$)

Order 36: $C_3\times A_4$ ($C_9:C_4$) x 9, $C_2^2:C_9$ ($C_3^2:C_4$) x 6, $C_2\times C_{18}$ ($C_3^2:C_4$) x 3, $C_3\times A_4$ ($C_3^2:C_4$) x 3, $C_6^2$ ($C_3^2:C_4$)

Order 27: $C_3\times C_9$ ($A_4:C_4$)

Order 24: $C_2\times A_4$ ($C_3:D_9$) x 9, $C_2^2\times C_6$ ($C_3:D_9$) x 3, $C_2^2\times C_6$ ($C_3^2:S_3$)

Order 18: $C_{18}$ ($C_3:S_4$) x 3, $C_3\times C_6$ ($C_3:S_4$)

Order 12: $A_4$ ($C_6.D_9$) x 9, $C_2\times C_6$ ($C_6.D_9$) x 3, $C_2\times C_6$ ($C_3^3:C_4$)

Order 9: $C_9$ ($C_6.S_4$) x 3, $C_3^2$ ($C_6.S_4$)

Order 8: $C_2^3$ ($C_3^2:D_9$)

Order 6: $C_6$ ($C_9:S_4$) x 3, $C_6$ ($C_3^2:S_4$)

Order 4: $C_2^2$ ($(C_3^2\times C_9):C_4$)

Order 3: $C_3$ ($C_{18}.S_4$) x 3, $C_3$ ($(C_3^2\times A_4):C_4$)

Order 2: $C_2$ ($(C_3\times C_9):S_4$)

Order 1: $C_1$ ($(C_3\times C_{18}).S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $(C_3\times C_{18}).S_4$ ($C_1$)

Order 648: $C_3\times A_4\times C_{18}$ ($C_2$)

Order 324: $C_6^2.C_3^2$ ($C_4$)

Order 216: $C_6^2:C_6$ ($S_3$), $C_2\times C_6\times C_{18}$ ($S_3$), $C_6^2.C_6$ ($S_3$), $A_4\times C_{18}$ ($S_3$)

Order 108: $C_3^2\times A_4$ ($C_3:C_4$), $C_6\times C_{18}$ ($C_3:C_4$), $C_3^2.A_4$ ($C_3:C_4$), $C_9\times A_4$ ($C_3:C_4$)

Order 72: $C_2\times C_6^2$ ($C_3:S_3$), $C_6\times A_4$ ($C_3:S_3$), $C_6\times A_4$ ($D_9$), $C_2^2\times C_{18}$ ($C_3:S_3$), $C_2^2:C_{18}$ ($C_3:S_3$)

Order 54: $C_3\times C_{18}$ ($S_4$)

Order 36: $C_2\times C_{18}$ ($C_3^2:C_4$), $C_2^2:C_9$ ($C_3^2:C_4$), $C_6^2$ ($C_3^2:C_4$), $C_3\times A_4$ ($C_3^2:C_4$), $C_3\times A_4$ ($C_9:C_4$)

Order 27: $C_3\times C_9$ ($A_4:C_4$)

Order 24: $C_2^2\times C_6$ ($C_3:D_9$), $C_2^2\times C_6$ ($C_3^2:S_3$), $C_2\times A_4$ ($C_3:D_9$)

Order 18: $C_3\times C_6$ ($C_3:S_4$), $C_{18}$ ($C_3:S_4$)

Order 12: $C_2\times C_6$ ($C_3^3:C_4$), $C_2\times C_6$ ($C_6.D_9$), $A_4$ ($C_6.D_9$)

Order 9: $C_3^2$ ($C_6.S_4$), $C_9$ ($C_6.S_4$)

Order 8: $C_2^3$ ($C_3^2:D_9$)

Order 6: $C_6$ ($C_9:S_4$), $C_6$ ($C_3^2:S_4$)

Order 4: $C_2^2$ ($(C_3^2\times C_9):C_4$)

Order 3: $C_3$ ($(C_3^2\times A_4):C_4$), $C_3$ ($C_{18}.S_4$)

Order 2: $C_2$ ($(C_3\times C_9):S_4$)

Order 1: $C_1$ ($(C_3\times C_{18}).S_4$)

Series

Derived series

$(C_3\times C_{18}).S_4$

$\rhd$

$C_6^2.C_3^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$(C_3\times C_{18}).S_4$

$\rhd$

$C_3\times A_4\times C_{18}$

$\rhd$

$C_6^2.C_3^2$

$\rhd$

$C_9\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_3\times C_{18}).S_4$

$\rhd$

$C_6^2.C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2$

Character theory

Complex character table

See the $114 \times 114$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $64 \times 64$ rational character table.