Abstract group 1296.879: $C_6^2.D_{18}$

• Label: 1296.879
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $144$
• Rank: $3$

Group information

Description:	$C_6^2.D_{18}$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^2.C_6^3.C_2^4$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	111	26	144	294	54	288	162	216	1296
Conjugacy classes	1	7	5	8	19	6	8	18	12	84
Divisions	1	7	5	4	19	2	4	6	2	50
Autjugacy classes	1	3	5	2	11	2	2	4	1	31

Dimension	1	2	4	6	12
Irr. complex chars.	16	40	16	8	4	84
Irr. rational chars.	8	12	8	12	10	50

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$144$
Rank:	$3$
Inequivalent generating triples:	$18144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	8	10	14

Constructions

Presentation:	$\langle a, b, c, d \mid b^{12}=c^{3}=d^{18}=[a,d]=[c,d]=1, a^{2}=b^{6}, b^{a}=b^{5}, c^{a}=c^{2}, c^{b}=c^{2}, d^{b}=c^{2}d^{17} \rangle$
Permutation group:	Degree $24$ $\langle(2,4)(3,7)(5,8)(6,9)(10,11,13,12)(17,19)(18,22)(20,23)(21,24), (10,12,13,11) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_3^2:(C_4\times D_9))$
Semidirect product:	$(C_6.D_9)$ $\,\rtimes\,$ $D_6$ (2)	$(C_6.D_6)$ $\,\rtimes\,$ $D_9$	$(C_6.D_{18})$ $\,\rtimes\,$ $S_3$	$(C_6^2.D_9)$ $\,\rtimes\,$ $C_2$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $D_{18}$	$C_6^2$ . $S_3^2$	$C_6$ . $(S_3\times D_{18})$	$(C_6\times C_{18})$ . $D_6$	all 28

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

Homology

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

Subgroups

There are 3491 subgroups in 377 conjugacy classes, 71 normal (31 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^2:D_{18}$
Commutator:	$G' \simeq$ $C_3^2:C_9$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $S_3\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^2:C_9$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^2.D_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_6^2$	$G/\operatorname{soc} \simeq$ $S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2:C_9$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1296: $C_6^2.D_{18}$

Order 648: $C_3^2:(C_4\times D_9)$ x 4, $C_6^2:D_9$, $C_6^2.D_9$, $C_6^2.C_{18}$

Order 432: $C_6^2.D_6$, $D_{18}.D_6$

Order 324: $C_3^2:D_{18}$ x 6, $(C_3\times C_9):C_{12}$ x 2, $C_3^2:C_{36}$ x 2, $C_6^2:C_9$

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

Order 162: $C_3^2:D_9$ x 4, $C_3^2:C_{18}$ x 3

Order 144: $D_6.D_6$ x 2, $C_4\times D_{18}$, $C_2^2.S_3^2$

Order 108: $C_3\times D_{18}$ x 12, $C_3^2:D_6$ x 6, $C_3^2:C_{12}$ x 4, $C_3:C_{36}$ x 2, $C_6\times C_{18}$ x 2, $C_6.D_9$ x 2, $C_3\times C_6^2$

Order 81: $C_3^2:C_9$

Order 72: $C_6.D_6$ x 8, $C_6:C_{12}$ x 5, $C_4\times D_9$ x 4, $C_6.D_6$ x 4, $C_6\times D_6$ x 3, $C_6.D_6$ x 2, $C_2\times D_{18}$ x 2, $C_2\times C_{36}$, $C_{18}:C_4$, $C_6:D_6$

Order 54: $C_3\times D_9$ x 8, $C_3\times C_{18}$ x 6, $C_3^2:C_6$ x 4, $C_3^2\times C_6$ x 3

Order 48: $C_4\times D_6$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 18, $D_{18}$ x 12, $C_3:C_{12}$ x 10, $C_6:S_3$ x 6, $C_6^2$ x 5, $C_3^2:C_4$ x 4, $C_2\times C_{18}$ x 2, $C_{36}$ x 2, $C_9:C_4$ x 2

Order 27: $C_3\times C_9$ x 2, $C_3^3$

Order 24: $C_6:C_4$ x 9, $C_4\times S_3$ x 8, $C_2\times D_6$ x 3, $C_2\times C_{12}$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 15, $C_3\times S_3$ x 12, $D_9$ x 8, $C_{18}$ x 6, $C_3:S_3$ x 4

Order 16: $C_2^2\times C_4$

Order 12: $D_6$ x 18, $C_2\times C_6$ x 11, $C_3:C_4$ x 10, $C_{12}$ x 4

Order 9: $C_3^2$ x 5, $C_9$ x 2

Order 8: $C_2\times C_4$ x 6, $C_2^3$

Order 6: $C_6$ x 19, $S_3$ x 12

Order 4: $C_2^2$ x 7, $C_4$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_6^2.D_{18}$

Order 648: $C_3^2:(C_4\times D_9)$, $C_6^2:D_9$, $C_6^2.D_9$, $C_6^2.C_{18}$

Order 432: $C_6^2.D_6$, $D_{18}.D_6$

Order 324: $C_3^2:D_{18}$ x 2, $C_6^2:C_9$, $(C_3\times C_9):C_{12}$, $C_3^2:C_{36}$

Order 216: $C_6^2.C_6$ x 2, $C_6\times D_{18}$ x 2, $C_6.D_{18}$, $C_6:C_{36}$, $C_6.D_{18}$, $C_6^2:C_6$, $C_6.S_3^2$

Order 162: $C_3^2:C_{18}$ x 2, $C_3^2:D_9$

Order 144: $D_6.D_6$ x 2, $C_4\times D_{18}$, $C_2^2.S_3^2$

Order 108: $C_3\times D_{18}$ x 4, $C_3^2:D_6$ x 2, $C_3^2:C_{12}$ x 2, $C_6\times C_{18}$ x 2, $C_3:C_{36}$, $C_3\times C_6^2$, $C_6.D_9$

Order 81: $C_3^2:C_9$

Order 72: $C_6:C_{12}$ x 5, $C_6\times D_6$ x 3, $C_6.D_6$ x 2, $C_6.D_6$ x 2, $C_2\times D_{18}$ x 2, $C_2\times C_{36}$, $C_{18}:C_4$, $C_4\times D_9$, $C_6:D_6$, $C_6.D_6$

Order 54: $C_3\times C_{18}$ x 4, $C_3\times D_9$ x 2, $C_3^2\times C_6$ x 2, $C_3^2:C_6$

Order 48: $C_4\times D_6$ x 2, $C_6.C_2^3$

Order 36: $C_6\times S_3$ x 6, $C_3:C_{12}$ x 5, $C_6^2$ x 5, $D_{18}$ x 4, $C_3^2:C_4$ x 2, $C_2\times C_{18}$ x 2, $C_6:S_3$ x 2, $C_{36}$, $C_9:C_4$

Order 27: $C_3\times C_9$ x 2, $C_3^3$

Order 24: $C_6:C_4$ x 6, $C_2\times D_6$ x 3, $C_2\times C_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times C_6$ x 10, $C_{18}$ x 4, $C_3\times S_3$ x 3, $D_9$ x 2, $C_3:S_3$

Order 16: $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 7, $D_6$ x 6, $C_3:C_4$ x 5, $C_{12}$ x 2

Order 9: $C_3^2$ x 5, $C_9$ x 2

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

Order 6: $C_6$ x 11, $S_3$ x 3

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_6^2.D_{18}$ ($C_1$)

Order 648: $C_3^2:(C_4\times D_9)$ ($C_2$) x 4, $C_6^2:D_9$ ($C_2$), $C_6^2.D_9$ ($C_2$), $C_6^2.C_{18}$ ($C_2$)

Order 324: $C_3^2:D_{18}$ ($C_4$) x 4, $(C_3\times C_9):C_{12}$ ($C_2^2$) x 2, $C_3^2:D_{18}$ ($C_2^2$) x 2, $C_3^2:C_{36}$ ($C_2^2$) x 2, $C_6^2:C_9$ ($C_2^2$)

Order 216: $C_6.D_{18}$ ($S_3$), $C_6^2.C_6$ ($S_3$)

Order 162: $C_3^2:D_9$ ($C_2\times C_4$) x 4, $C_3^2:C_{18}$ ($C_2\times C_4$) x 2, $C_3^2:C_{18}$ ($C_2^3$)

Order 108: $C_3^2:C_{12}$ ($D_6$) x 2, $C_6.D_9$ ($D_6$) x 2, $C_3\times C_6^2$ ($D_6$), $C_6\times C_{18}$ ($D_6$)

Order 81: $C_3^2:C_9$ ($C_2^2\times C_4$)

Order 72: $C_6.D_6$ ($D_9$)

Order 54: $C_3\times C_{18}$ ($C_4\times S_3$) x 2, $C_3^2\times C_6$ ($C_4\times S_3$) x 2, $C_3\times C_{18}$ ($C_2\times D_6$), $C_3^2\times C_6$ ($C_2\times D_6$)

Order 36: $C_3^2:C_4$ ($D_{18}$) x 2, $C_6^2$ ($D_{18}$), $C_6^2$ ($S_3^2$)

Order 27: $C_3^3$ ($C_4\times D_6$), $C_3\times C_9$ ($C_4\times D_6$)

Order 18: $C_3\times C_6$ ($C_4\times D_9$) x 2, $C_3\times C_6$ ($C_6.D_6$) x 2, $C_3\times C_6$ ($S_3\times D_6$), $C_3\times C_6$ ($C_2\times D_{18}$)

Order 12: $C_2\times C_6$ ($C_3^2:D_6$), $C_2\times C_6$ ($S_3\times D_9$)

Order 9: $C_3^2$ ($C_4\times D_{18}$), $C_3^2$ ($C_2^2.S_3^2$)

Order 6: $C_6$ ($C_6.S_3^2$) x 2, $C_6$ ($C_6.D_{18}$) x 2, $C_6$ ($C_6.S_3^2$), $C_6$ ($S_3\times D_{18}$)

Order 4: $C_2^2$ ($C_3^2:D_{18}$)

Order 3: $C_3$ ($C_6^2.D_6$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3^2:(C_4\times D_9)$) x 2, $C_2$ ($C_2\times C_3^2:D_{18}$)

Order 1: $C_1$ ($C_6^2.D_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_6^2.D_{18}$ ($C_1$)

Order 648: $C_3^2:(C_4\times D_9)$ ($C_2$), $C_6^2:D_9$ ($C_2$), $C_6^2.D_9$ ($C_2$), $C_6^2.C_{18}$ ($C_2$)

Order 324: $C_6^2:C_9$ ($C_2^2$), $(C_3\times C_9):C_{12}$ ($C_2^2$), $C_3^2:D_{18}$ ($C_2^2$), $C_3^2:D_{18}$ ($C_4$), $C_3^2:C_{36}$ ($C_2^2$)

Order 216: $C_6.D_{18}$ ($S_3$), $C_6^2.C_6$ ($S_3$)

Order 162: $C_3^2:C_{18}$ ($C_2^3$), $C_3^2:C_{18}$ ($C_2\times C_4$), $C_3^2:D_9$ ($C_2\times C_4$)

Order 108: $C_3\times C_6^2$ ($D_6$), $C_3^2:C_{12}$ ($D_6$), $C_6\times C_{18}$ ($D_6$), $C_6.D_9$ ($D_6$)

Order 81: $C_3^2:C_9$ ($C_2^2\times C_4$)

Order 72: $C_6.D_6$ ($D_9$)

Order 54: $C_3\times C_{18}$ ($C_4\times S_3$), $C_3\times C_{18}$ ($C_2\times D_6$), $C_3^2\times C_6$ ($C_4\times S_3$), $C_3^2\times C_6$ ($C_2\times D_6$)

Order 36: $C_3^2:C_4$ ($D_{18}$), $C_6^2$ ($D_{18}$), $C_6^2$ ($S_3^2$)

Order 27: $C_3^3$ ($C_4\times D_6$), $C_3\times C_9$ ($C_4\times D_6$)

Order 18: $C_3\times C_6$ ($C_4\times D_9$), $C_3\times C_6$ ($S_3\times D_6$), $C_3\times C_6$ ($C_6.D_6$), $C_3\times C_6$ ($C_2\times D_{18}$)

Order 12: $C_2\times C_6$ ($C_3^2:D_6$), $C_2\times C_6$ ($S_3\times D_9$)

Order 9: $C_3^2$ ($C_4\times D_{18}$), $C_3^2$ ($C_2^2.S_3^2$)

Order 6: $C_6$ ($C_6.S_3^2$), $C_6$ ($C_6.D_{18}$), $C_6$ ($C_6.S_3^2$), $C_6$ ($S_3\times D_{18}$)

Order 4: $C_2^2$ ($C_3^2:D_{18}$)

Order 3: $C_3$ ($C_6^2.D_6$), $C_3$ ($C_6^2.D_6$)

Order 2: $C_2$ ($C_3^2:(C_4\times D_9)$), $C_2$ ($C_2\times C_3^2:D_{18}$)

Order 1: $C_1$ ($C_6^2.D_{18}$)

Series

Derived series

$C_6^2.D_{18}$

$\rhd$

$C_3^2:C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Chief series

$C_6^2.D_{18}$

$\rhd$

$C_3^2:(C_4\times D_9)$

$\rhd$

$C_3^2:C_{36}$

$\rhd$

$C_3^2:C_{18}$

$\rhd$

$C_3^2:C_9$

$\rhd$

$C_3\times C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^2.D_{18}$

$\rhd$

$C_3^2:C_9$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Character theory

Complex character table

See the $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.