Abstract group 1296.2305: $C_3:D_{12}\times C_{18}$

• Label: 1296.2305
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $21$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_3:D_{12}\times C_{18}$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_6^3.C_2^5$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	51	26	12	246	54	96	594	216	1296
Conjugacy classes	1	7	11	2	53	24	16	120	36	270
Divisions	1	7	7	2	31	4	6	20	4	82
Autjugacy classes	1	4	7	1	18	4	3	11	2	51

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	72	162	36	0	0	0	0	270
Irr. rational chars.	8	18	22	8	6	14	6	82

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$78624$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	6	10

Constructions

Presentation:	$\langle a, b, c \mid a^{18}=b^{12}=c^{6}=[a,c]=1, b^{a}=b^{11}, c^{b}=c^{5} \rangle$
Permutation group:	Degree $21$ $\langle(11,12)(16,17)(19,21), (14,15)(16,17)(18,19,20,21), (16,17)(18,20)(19,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 53 & 13 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 22 \\ 16 & 49 \end{array}\right), \left(\begin{array}{rr} 37 & 36 \\ 18 & 19 \end{array}\right), \left(\begin{array}{rr} 25 & 0 \\ 0 & 25 \end{array}\right), \left(\begin{array}{rr} 1 & 18 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 27 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 53 & 0 \\ 0 & 53 \end{array}\right), \left(\begin{array}{rr} 19 & 0 \\ 0 & 19 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/54\Z)$
Direct product:	$C_2$ $\, \times\, $ $C_9$ $\, \times\, $ $(C_3:D_{12})$
Semidirect product:	$(D_6\times C_{18})$ $\,\rtimes\,$ $S_3$	$(S_3\times C_{18})$ $\,\rtimes\,$ $D_6$ (2)	$D_6$ $\,\rtimes\,$ $(S_3\times C_{18})$ (2)	$(C_6:C_{36})$ $\,\rtimes\,$ $S_3$	all 32
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{18}$ . $(S_3\times D_6)$	$(C_6\times C_{18})$ . $D_6$ (2)	$C_6$ . $(D_6\times C_{18})$ (2)	$(S_3\times C_6^2)$ . $C_6$	all 38

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

Homology

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

Subgroups

There are 1776 subgroups in 488 conjugacy classes, 120 normal (54 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_{18}$	$G/Z \simeq$ $S_3^2$
Commutator:	$G' \simeq$ $C_3\times C_6$	$G/G' \simeq$ $C_2^2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_6$	$G/\Phi \simeq$ $C_6\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\times C_6\times C_{18}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3:D_{12}\times C_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1296: $C_3:D_{12}\times C_{18}$

Order 648: $C_{18}.S_3^2$ x 4, $C_6^2:C_{18}$, $C_6^2:C_{18}$, $C_6^2.C_{18}$

Order 432: $C_6^2.D_6$, $C_2\times D_6:C_{18}$, $C_{18}\times D_{12}$

Order 324: $C_3:S_3\times C_{18}$ x 4, $C_3^2.C_6^2$ x 4, $C_3^2:C_{36}$ x 2, $C_3\times C_6\times C_{18}$

Order 216: $D_6\times C_{18}$ x 5, $D_6:C_{18}$ x 4, $C_9\times D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_6:C_{36}$ x 2, $C_6\times C_{36}$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6.C_6^2$, $C_2\times C_6\times C_{18}$

Order 162: $C_3^2\times C_{18}$ x 3, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$ x 2

Order 144: $D_4\times C_{18}$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_6:D_{12}$

Order 108: $S_3\times C_{18}$ x 24, $C_6\times C_{18}$ x 10, $C_3:C_{36}$ x 4, $C_3^2:D_6$ x 4, $C_3^2\times D_6$ x 4, $C_3^2:C_{12}$ x 2, $C_3\times C_{36}$ x 2, $C_3\times C_6^2$

Order 81: $C_3^2\times C_9$

Order 72: $C_6\times D_6$ x 6, $C_6\wr C_2$ x 4, $C_3\times D_{12}$ x 4, $C_3:D_{12}$ x 4, $D_4\times C_9$ x 4, $C_6:C_{12}$ x 3, $C_2^2\times C_{18}$ x 3, $C_2\times C_{36}$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $C_6\times C_{12}$

Order 54: $C_3\times C_{18}$ x 20, $S_3\times C_9$ x 14, $C_3^2\times C_6$ x 3, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2

Order 48: $C_6\times D_4$, $C_6:D_4$, $C_2\times D_{12}$

Order 36: $C_6\times S_3$ x 28, $C_2\times C_{18}$ x 18, $C_6^2$ x 11, $C_3:C_{12}$ x 6, $C_{36}$ x 4, $C_6:S_3$ x 4, $C_3\times C_{12}$ x 2

Order 27: $C_3\times C_9$ x 6, $C_3^3$

Order 24: $C_3:D_4$ x 4, $D_{12}$ x 4, $C_2^2\times C_6$ x 4, $C_2\times D_6$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$

Order 18: $C_3\times C_6$ x 23, $C_{18}$ x 20, $C_3\times S_3$ x 16, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 25, $D_6$ x 18, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 7, $C_9$ x 4

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 31, $S_3$ x 10

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_3:D_{12}\times C_{18}$

Order 648: $C_6^2:C_{18}$, $C_6^2:C_{18}$, $C_6^2.C_{18}$, $C_{18}.S_3^2$

Order 432: $C_6^2.D_6$, $C_2\times D_6:C_{18}$, $C_{18}\times D_{12}$

Order 324: $C_3:S_3\times C_{18}$ x 2, $C_3^2.C_6^2$ x 2, $C_3^2:C_{36}$, $C_3\times C_6\times C_{18}$

Order 216: $D_6\times C_{18}$ x 5, $C_6:C_{36}$ x 2, $C_6\times C_{36}$, $D_6:C_{18}$, $C_9\times D_{12}$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6.C_6^2$, $C_3^2:D_{12}$, $C_2\times C_6\times C_{18}$

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

Order 144: $D_4\times C_{18}$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_6:D_{12}$

Order 108: $S_3\times C_{18}$ x 10, $C_6\times C_{18}$ x 8, $C_3:C_{36}$ x 2, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3\times C_6^2$, $C_3^2:C_{12}$, $C_3\times C_{36}$

Order 81: $C_3^2\times C_9$

Order 72: $C_6\times D_6$ x 6, $C_6:C_{12}$ x 3, $C_2^2\times C_{18}$ x 3, $C_2\times C_{36}$ x 2, $C_2\times C_6^2$, $C_6:D_6$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $C_3:D_{12}$, $D_4\times C_9$

Order 54: $C_3\times C_{18}$ x 13, $S_3\times C_9$ x 5, $C_3^2\times C_6$ x 2, $C_3^2:C_6$, $S_3\times C_3^2$

Order 48: $C_6\times D_4$, $C_6:D_4$, $C_2\times D_{12}$

Order 36: $C_6\times S_3$ x 12, $C_2\times C_{18}$ x 10, $C_6^2$ x 9, $C_3:C_{12}$ x 3, $C_{36}$ x 2, $C_6:S_3$ x 2, $C_3\times C_{12}$

Order 27: $C_3\times C_9$ x 6, $C_3^3$

Order 24: $C_2^2\times C_6$ x 4, $C_2\times D_6$ x 4, $C_2\times C_{12}$ x 3, $C_3:D_4$, $C_6:C_4$, $D_{12}$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 15, $C_{18}$ x 11, $C_3\times S_3$ x 6, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 15, $D_6$ x 8, $C_{12}$ x 3, $C_3:C_4$

Order 9: $C_3^2$ x 7, $C_9$ x 4

Order 8: $C_2^3$ x 2, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 18, $S_3$ x 4

Order 4: $C_2^2$ x 5, $C_4$

Order 3: $C_3$ x 7

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_3:D_{12}\times C_{18}$ ($C_1$)

Order 648: $C_{18}.S_3^2$ ($C_2$) x 4, $C_6^2:C_{18}$ ($C_2$), $C_6^2:C_{18}$ ($C_2$), $C_6^2.C_{18}$ ($C_2$)

Order 432: $C_6^2.D_6$ ($C_3$)

Order 324: $C_3^2:C_{36}$ ($C_2^2$) x 2, $C_3:S_3\times C_{18}$ ($C_2^2$) x 2, $C_3^2.C_6^2$ ($C_2^2$) x 2, $C_3\times C_6\times C_{18}$ ($C_2^2$)

Order 216: $C_3^2:D_{12}$ ($C_6$) x 4, $C_6:C_{36}$ ($S_3$), $C_6^2:C_6$ ($C_6$), $S_3\times C_6^2$ ($C_6$), $C_6.C_6^2$ ($C_6$), $D_6\times C_{18}$ ($S_3$)

Order 162: $C_3^2\times C_{18}$ ($D_4$) x 2, $C_3^2\times C_{18}$ ($C_2^3$)

Order 144: $C_6:D_{12}$ ($C_9$)

Order 108: $C_3:C_{36}$ ($D_6$) x 2, $C_3^2:D_6$ ($C_2\times C_6$) x 2, $C_3^2\times D_6$ ($C_2\times C_6$) x 2, $C_3^2:C_{12}$ ($C_2\times C_6$) x 2, $C_6\times C_{18}$ ($D_6$) x 2, $S_3\times C_{18}$ ($D_6$) x 2, $C_3\times C_6^2$ ($C_2\times C_6$)

Order 81: $C_3^2\times C_9$ ($C_2\times D_4$)

Order 72: $C_3:D_{12}$ ($C_{18}$) x 4, $C_6:D_6$ ($C_{18}$), $C_6\times D_6$ ($C_3\times S_3$), $C_6\times D_6$ ($C_{18}$), $C_6:C_{12}$ ($C_3\times S_3$), $C_6:C_{12}$ ($C_{18}$)

Order 54: $C_3\times C_{18}$ ($C_3:D_4$) x 2, $C_3\times C_{18}$ ($D_{12}$) x 2, $C_3\times C_{18}$ ($C_2\times D_6$) x 2, $C_3^2\times C_6$ ($C_3\times D_4$) x 2, $C_3^2\times C_6$ ($C_2^2\times C_6$)

Order 36: $C_3:C_{12}$ ($C_2\times C_{18}$) x 2, $C_3:C_{12}$ ($C_6\times S_3$) x 2, $C_6^2$ ($C_6\times S_3$) x 2, $C_6:S_3$ ($C_2\times C_{18}$) x 2, $C_6\times S_3$ ($C_2\times C_{18}$) x 2, $C_6\times S_3$ ($C_6\times S_3$) x 2, $C_2\times C_{18}$ ($S_3^2$), $C_6^2$ ($C_2\times C_{18}$)

Order 27: $C_3^3$ ($C_6\times D_4$), $C_3\times C_9$ ($C_6:D_4$), $C_3\times C_9$ ($C_2\times D_{12}$)

Order 24: $C_6:C_4$ ($S_3\times C_9$), $C_2\times D_6$ ($S_3\times C_9$)

Order 18: $C_3\times C_6$ ($C_6\times D_6$) x 2, $C_3\times C_6$ ($C_6\wr C_2$) x 2, $C_3\times C_6$ ($C_3\times D_{12}$) x 2, $C_3\times C_6$ ($D_4\times C_9$) x 2, $C_{18}$ ($C_3:D_{12}$) x 2, $C_3\times C_6$ ($C_2^2\times C_{18}$), $C_{18}$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($S_3\times C_{18}$) x 2, $D_6$ ($S_3\times C_{18}$) x 2, $C_3:C_4$ ($S_3\times C_{18}$) x 2, $C_2\times C_6$ ($C_3\times S_3^2$)

Order 9: $C_3^2$ ($D_4\times C_{18}$), $C_3^2$ ($C_6^2:C_2^2$), $C_3^2$ ($C_6\times D_{12}$), $C_9$ ($C_6:D_{12}$)

Order 6: $C_6$ ($D_6:C_{18}$) x 2, $C_6$ ($C_9\times D_{12}$) x 2, $C_6$ ($C_3^2:D_{12}$) x 2, $C_6$ ($D_6\times C_{18}$) x 2, $C_6$ ($C_6\times S_3^2$)

Order 4: $C_2^2$ ($C_9\times S_3^2$)

Order 3: $C_3$ ($C_6^2.D_6$), $C_3$ ($C_2\times D_6:C_{18}$), $C_3$ ($C_{18}\times D_{12}$)

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

Order 1: $C_1$ ($C_3:D_{12}\times C_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_3:D_{12}\times C_{18}$ ($C_1$)

Order 648: $C_6^2:C_{18}$ ($C_2$), $C_6^2:C_{18}$ ($C_2$), $C_6^2.C_{18}$ ($C_2$), $C_{18}.S_3^2$ ($C_2$)

Order 432: $C_6^2.D_6$ ($C_3$)

Order 324: $C_3^2:C_{36}$ ($C_2^2$), $C_3\times C_6\times C_{18}$ ($C_2^2$), $C_3:S_3\times C_{18}$ ($C_2^2$), $C_3^2.C_6^2$ ($C_2^2$)

Order 216: $C_6:C_{36}$ ($S_3$), $C_6^2:C_6$ ($C_6$), $S_3\times C_6^2$ ($C_6$), $C_6.C_6^2$ ($C_6$), $C_3^2:D_{12}$ ($C_6$), $D_6\times C_{18}$ ($S_3$)

Order 162: $C_3^2\times C_{18}$ ($C_2^3$), $C_3^2\times C_{18}$ ($D_4$)

Order 144: $C_6:D_{12}$ ($C_9$)

Order 108: $C_6\times C_{18}$ ($D_6$) x 2, $C_3:C_{36}$ ($D_6$), $C_3\times C_6^2$ ($C_2\times C_6$), $C_3^2:D_6$ ($C_2\times C_6$), $C_3^2\times D_6$ ($C_2\times C_6$), $C_3^2:C_{12}$ ($C_2\times C_6$), $S_3\times C_{18}$ ($D_6$)

Order 81: $C_3^2\times C_9$ ($C_2\times D_4$)

Order 72: $C_6:D_6$ ($C_{18}$), $C_6\times D_6$ ($C_3\times S_3$), $C_6\times D_6$ ($C_{18}$), $C_6:C_{12}$ ($C_3\times S_3$), $C_6:C_{12}$ ($C_{18}$), $C_3:D_{12}$ ($C_{18}$)

Order 54: $C_3\times C_{18}$ ($C_2\times D_6$) x 2, $C_3\times C_{18}$ ($C_3:D_4$), $C_3\times C_{18}$ ($D_{12}$), $C_3^2\times C_6$ ($C_2^2\times C_6$), $C_3^2\times C_6$ ($C_3\times D_4$)

Order 36: $C_6^2$ ($C_6\times S_3$) x 2, $C_3:C_{12}$ ($C_2\times C_{18}$), $C_3:C_{12}$ ($C_6\times S_3$), $C_2\times C_{18}$ ($S_3^2$), $C_6^2$ ($C_2\times C_{18}$), $C_6:S_3$ ($C_2\times C_{18}$), $C_6\times S_3$ ($C_2\times C_{18}$), $C_6\times S_3$ ($C_6\times S_3$)

Order 27: $C_3^3$ ($C_6\times D_4$), $C_3\times C_9$ ($C_6:D_4$), $C_3\times C_9$ ($C_2\times D_{12}$)

Order 24: $C_6:C_4$ ($S_3\times C_9$), $C_2\times D_6$ ($S_3\times C_9$)

Order 18: $C_3\times C_6$ ($C_6\times D_6$) x 2, $C_3\times C_6$ ($C_6\wr C_2$), $C_3\times C_6$ ($C_3\times D_{12}$), $C_3\times C_6$ ($C_2^2\times C_{18}$), $C_3\times C_6$ ($D_4\times C_9$), $C_{18}$ ($S_3\times D_6$), $C_{18}$ ($C_3:D_{12}$)

Order 12: $C_2\times C_6$ ($S_3\times C_{18}$) x 2, $C_2\times C_6$ ($C_3\times S_3^2$), $D_6$ ($S_3\times C_{18}$), $C_3:C_4$ ($S_3\times C_{18}$)

Order 9: $C_3^2$ ($D_4\times C_{18}$), $C_3^2$ ($C_6^2:C_2^2$), $C_3^2$ ($C_6\times D_{12}$), $C_9$ ($C_6:D_{12}$)

Order 6: $C_6$ ($D_6\times C_{18}$) x 2, $C_6$ ($D_6:C_{18}$), $C_6$ ($C_9\times D_{12}$), $C_6$ ($C_6\times S_3^2$), $C_6$ ($C_3^2:D_{12}$)

Order 4: $C_2^2$ ($C_9\times S_3^2$)

Order 3: $C_3$ ($C_6^2.D_6$), $C_3$ ($C_2\times D_6:C_{18}$), $C_3$ ($C_{18}\times D_{12}$)

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

Order 1: $C_1$ ($C_3:D_{12}\times C_{18}$)

Series

Derived series

$C_3:D_{12}\times C_{18}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_1$

Chief series

$C_3:D_{12}\times C_{18}$

$\rhd$

$C_6^2.C_{18}$

$\rhd$

$C_6.C_6^2$

$\rhd$

$C_6:C_{12}$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3:D_{12}\times C_{18}$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

Upper central series

$C_1$

$\lhd$

$C_2\times C_{18}$

Character theory

Complex character table

See the $270 \times 270$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $82 \times 82$ rational character table.