Abstract group 2592.nl: $C_6^3:D_6$

• Label: 2592.nl
• Order: \( 2^{5} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $4$

Group information

Description:	$C_6^3:D_6$
Order:	\(2592\)\(\medspace = 2^{5} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^2.C_6^3.C_2^6$, of order \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ 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
Elements	1	295	44	216	1316	36	432	252	2592
Conjugacy classes	1	15	6	4	74	1	4	7	112
Divisions	1	15	6	4	58	1	4	5	94
Autjugacy classes	1	7	4	2	18	1	2	3	38

Dimension	1	2	4	6	8	12
Irr. complex chars.	16	28	10	56	0	2	112
Irr. rational chars.	16	20	10	32	2	14	94

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$4$
Inequivalent generating quadruples:	$103488840$

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:	${\langle a, b, c, d, e \mid a^{2}=b^{6}=c^{6}=d^{6}=e^{6}=[a,d]=[a,e]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(7,11)(8,10)(9,12), (10,11,12)(13,14,15), (7,14,11)(8,13,10)(9,15,12), (2,3) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $(C_6^2.S_3^2)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $D_6$	$(C_6^2:D_6)$ $\,\rtimes\,$ $S_3$	$(C_6^2:S_3)$ $\,\rtimes\,$ $D_6$ (2)	$(C_6^2:S_3)$ $\,\rtimes\,$ $D_6$	all 50
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:S_3)$ . $D_6$	$C_6^2$ . $(S_3\times D_6)$ (2)	$(C_3\times C_6)$ . $D_6^2$	$C_6$ . $(C_6^2:D_6)$	all 22

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

Homology

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

Subgroups

There are 29242 subgroups in 1942 conjugacy classes, 148 normal (44 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2\times C_3^3:D_6$
Commutator:	$G' \simeq$ $C_3^3:C_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^3:C_3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_6^3:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_6.S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 2592: $C_6^3:D_6$

Order 1296: $C_6^2.S_3^2$ x 8, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3:S_3$, $C_6^2.S_3^2$, $C_6^3:S_3$, $C_6^3:C_6$

Order 864: $C_6^3:C_2^2$, $C_6.D_6^2$

Order 648: $C_2^2\times C_3\wr S_3$ x 10, $C_2\times C_3^3:D_6$ x 8, $C_3\wr C_3:D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $(C_3\times C_6^2):S_3$ x 4, $(C_3\times C_6^2):C_6$ x 4, $C_6^3:C_3$, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $(C_3\times C_6^2).S_3$, $C_2\times C_3^3:C_{12}$

Order 432: $C_6^2:D_6$ x 8, $C_6^2:D_6$ x 8, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $D_6\times C_6^2$, $C_3:D_6^2$, $C_6^3:C_2$, $C_6^3:C_2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 324: $C_3^3:D_6$ x 18, $C_3^3:D_6$ x 8, $C_3^2.C_6^2$ x 5, $C_3^3:D_6$ x 4, $\He_3:D_6$ x 4, $C_3\wr C_3:C_4$ x 2, $C_3^3:C_{12}$ x 2

Order 288: $D_6^2:C_2$ x 5, $C_6^2:D_4$

Order 216: $S_3\times C_6^2$ x 10, $C_6^2:S_3$ x 10, $C_6^2:C_6$ x 8, $C_3^2:D_{12}$ x 8, $C_6:S_3^2$ x 8, $C_6.S_3^2$ x 8, $D_{18}:C_6$ x 4, $C_6.S_3^2$ x 4, $C_6^2:S_3$ x 4, $C_6^2:C_6$ x 4, $C_3^2:D_{12}$ x 4, $C_3^3:D_4$ x 4, $C_6.S_3^2$ x 4, $C_6^2.C_6$ x 2, $C_6^2:C_6$ x 2, $C_{18}:C_{12}$, $C_6^3$, $C_6^2:S_3$, $C_6^2:C_6$, $C_6^2.S_3$, $C_6^2.C_6$, $C_6.C_6^2$, $C_2^3\times \He_3$, $D_{18}:C_6$

Order 162: $C_3\wr S_3$ x 6, $C_3^3:C_6$ x 5, $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2

Order 144: $D_6:D_6$ x 40, $C_6^2:C_2^2$ x 17, $C_6^2:C_2^2$ x 5, $D_6^2$ x 5, $C_6^2:C_2^2$ x 5, $C_6:D_{12}$ x 5, $D_6:D_6$ x 5, $D_6.D_6$ x 5, $C_{18}:D_4$, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 108: $C_3^2\times D_6$ x 18, $C_3^2:D_6$ x 18, $C_3:S_3^2$ x 8, $C_3^2:D_6$ x 8, $C_3^2:D_6$ x 8, $C_3\times C_6^2$ x 5, $C_{18}:C_6$ x 5, $C_2^2\times \He_3$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 4, $C_3^2:D_6$ x 4, $C_{18}:C_6$ x 4, $C_9:C_{12}$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2

Order 96: $C_2^3:D_6$ x 4, $D_4\times D_6$ x 2

Order 81: $C_3\wr C_3$

Order 72: $C_6\times D_6$ x 55, $S_3\times D_6$ x 40, $C_6^2:C_2$ x 36, $C_6\wr C_2$ x 20, $C_3:D_{12}$ x 20, $D_6:S_3$ x 20, $C_6.D_6$ x 20, $C_2\times C_6^2$ x 16, $C_6:D_6$ x 15, $C_6.D_6$ x 11, $C_6:C_{12}$ x 5, $C_9:D_4$ x 4, $C_{18}:C_4$, $C_2^2\times C_{18}$, $C_2\times D_{18}$

Order 54: $C_3^2:S_3$ x 6, $S_3\times C_3^2$ x 6, $C_3^2\times C_6$ x 5, $C_9:C_6$ x 5, $C_2\times \He_3$ x 5, $C_3^2:C_6$ x 4, $C_9:C_6$ x 2, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2

Order 48: $C_6:D_4$ x 52, $S_3\times D_4$ x 16, $C_2^2\times D_6$ x 8, $C_2^3\times C_6$ x 4, $C_6.C_2^3$ x 4, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $C_6\times S_3$ x 110, $C_6^2$ x 48, $S_3^2$ x 40, $C_6:S_3$ x 36, $C_3^2:C_4$ x 14, $C_3:C_{12}$ x 10, $C_2\times C_{18}$ x 5, $D_{18}$ x 4, $C_9:C_4$ x 2

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_3:D_4$ x 80, $C_2\times D_6$ x 78, $C_2^2\times C_6$ x 48, $C_6:C_4$ x 26, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 2

Order 18: $C_3\times S_3$ x 40, $C_3\times C_6$ x 36, $C_3:S_3$ x 14, $C_{18}$ x 5, $D_9$ x 2

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 124, $C_2\times C_6$ x 110, $C_3:C_4$ x 20, $C_{12}$ x 4

Order 9: $C_3^2$ x 6, $C_9$

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 58, $S_3$ x 32

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 6

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2592: $C_6^3:D_6$

Order 1296: $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3:S_3$, $C_6^2.S_3^2$, $C_6^3:S_3$, $C_6^3:C_6$

Order 864: $C_6^3:C_2^2$, $C_6.D_6^2$

Order 648: $C_2^2\times C_3\wr S_3$ x 4, $C_2\times C_3^3:D_6$ x 2, $C_3\wr C_3:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $C_6^3:C_3$, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $(C_3\times C_6^2):S_3$, $(C_3\times C_6^2).S_3$, $(C_3\times C_6^2):C_6$, $C_2\times C_3^3:C_{12}$

Order 432: $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $D_6\times C_6^2$, $C_3:D_6^2$, $C_6^3:C_2$, $C_6^3:C_2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2:D_6$, $C_6^2.D_6$

Order 324: $C_3^3:D_6$ x 5, $C_3^2.C_6^2$ x 3, $C_3^3:D_6$ x 2, $\He_3:D_6$ x 2, $C_3^3:D_6$, $C_3\wr C_3:C_4$, $C_3^3:C_{12}$

Order 288: $D_6^2:C_2$ x 3, $C_6^2:D_4$

Order 216: $S_3\times C_6^2$ x 4, $C_6^2:S_3$ x 4, $C_6^2:C_6$ x 2, $C_6^2.C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_6.S_3^2$ x 2, $D_{18}:C_6$, $C_{18}:C_{12}$, $C_6.S_3^2$, $C_6^3$, $C_6^2:S_3$, $C_6^2:C_6$, $C_6^2:S_3$, $C_6^2.S_3$, $C_6^2:C_6$, $C_6^2.C_6$, $C_3^2:D_{12}$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6.C_6^2$, $C_2^3\times \He_3$, $D_{18}:C_6$

Order 162: $C_3^3:C_6$ x 3, $C_3\wr S_3$ x 2, $C_3^3:S_3$, $C_3^3:C_6$

Order 144: $C_6^2:C_2^2$ x 8, $C_6^2:C_2^2$ x 3, $D_6^2$ x 3, $C_6^2:C_2^2$ x 3, $D_6:D_6$ x 3, $C_6:D_{12}$ x 3, $D_6:D_6$ x 3, $D_6.D_6$ x 3, $C_{18}:D_4$, $C_2^2\times C_6^2$, $C_6^2:C_2^2$, $C_6^2:C_4$

Order 108: $C_3^2\times D_6$ x 5, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 4, $C_3\times C_6^2$ x 3, $C_{18}:C_6$ x 3, $C_2^2\times \He_3$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$ x 2, $C_{18}:C_6$ x 2, $C_9:C_{12}$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 96: $C_2^3:D_6$ x 2, $D_4\times D_6$ x 2

Order 81: $C_3\wr C_3$

Order 72: $C_6\times D_6$ x 15, $C_2\times C_6^2$ x 8, $C_6:D_6$ x 7, $C_6^2:C_2$ x 7, $S_3\times D_6$ x 6, $C_6.D_6$ x 6, $C_6\wr C_2$ x 3, $C_6:C_{12}$ x 3, $C_3:D_{12}$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_9:D_4$, $C_{18}:C_4$, $C_2^2\times C_{18}$, $C_2\times D_{18}$

Order 54: $C_3^2\times C_6$ x 3, $C_9:C_6$ x 3, $C_2\times \He_3$ x 3, $C_3^2:S_3$ x 2, $C_3^2:C_6$ x 2, $S_3\times C_3^2$ x 2, $C_9:C_6$, $C_3^2:S_3$, $C_3^2:C_6$

Order 48: $C_6:D_4$ x 14, $C_2^2\times D_6$ x 6, $C_2^3\times C_6$ x 2, $C_6\times D_4$ x 2, $C_6.C_2^3$ x 2, $S_3\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $C_6\times S_3$ x 21, $C_6^2$ x 17, $C_6:S_3$ x 11, $C_3^2:C_4$ x 5, $C_3:C_{12}$ x 3, $C_2\times C_{18}$ x 3, $S_3^2$ x 3, $D_{18}$ x 2, $C_9:C_4$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$, $C_9:C_3$, $\He_3$

Order 24: $C_2\times D_6$ x 22, $C_2^2\times C_6$ x 14, $C_3:D_4$ x 12, $C_6:C_4$ x 8, $C_2\times C_{12}$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$ x 14, $C_3\times S_3$ x 9, $C_3:S_3$ x 5, $C_{18}$ x 3, $D_9$

Order 16: $C_2\times D_4$ x 5, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 26, $D_6$ x 26, $C_3:C_4$ x 6, $C_{12}$ x 2

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

Order 8: $C_2^3$ x 9, $D_4$ x 4, $C_2\times C_4$ x 3

Order 6: $C_6$ x 18, $S_3$ x 10

Order 4: $C_2^2$ x 13, $C_4$ x 2

Order 3: $C_3$ x 4

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2592: $C_6^3:D_6$ ($C_1$)

Order 1296: $C_6^2.S_3^2$ ($C_2$) x 8, $C_6^2.S_3^2$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^3:S_3$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^3:S_3$ ($C_2$), $C_6^3:C_6$ ($C_2$)

Order 648: $C_2^2\times C_3\wr S_3$ ($C_2^2$) x 6, $C_3\wr C_3:D_4$ ($C_2^2$) x 4, $C_3^3:D_{12}$ ($C_2^2$) x 4, $C_3^3:(C_4\times S_3)$ ($C_2^2$) x 4, $C_2\times C_3^3:D_6$ ($C_2^2$) x 4, $(C_3\times C_6^2):S_3$ ($C_2^2$) x 4, $(C_3\times C_6^2):C_6$ ($C_2^2$) x 4, $C_6^3:C_3$ ($C_2^2$), $C_2\times C_3^3:D_6$ ($C_2^2$), $C_2^2\times C_3^3:C_6$ ($C_2^2$), $(C_3\times C_6^2).S_3$ ($C_2^2$), $C_2\times C_3^3:C_{12}$ ($C_2^2$)

Order 432: $C_6^3:C_2$ ($S_3$), $C_6^2:D_6$ ($S_3$)

Order 324: $C_3^3:D_6$ ($C_2^3$) x 4, $C_3^3:D_6$ ($D_4$) x 4, $C_3^2.C_6^2$ ($C_2^3$) x 3, $C_3^3:D_6$ ($C_2^3$) x 2, $\He_3:D_6$ ($C_2^3$) x 2, $C_3\wr C_3:C_4$ ($C_2^3$) x 2, $C_3^3:C_{12}$ ($C_2^3$) x 2

Order 216: $C_6^2:S_3$ ($D_6$) x 6, $C_6^2:S_3$ ($D_6$) x 4, $C_6^3$ ($D_6$), $C_6^2:S_3$ ($D_6$), $C_6^2.S_3$ ($D_6$), $C_2^3\times \He_3$ ($D_6$)

Order 162: $C_3\wr S_3$ ($C_2\times D_4$) x 4, $C_3^3:C_6$ ($C_2\times D_4$) x 2, $C_3^3:C_6$ ($C_2^4$)

Order 108: $C_3^2:D_6$ ($C_3:D_4$) x 4, $C_3^2:D_6$ ($C_2\times D_6$) x 4, $C_3\times C_6^2$ ($C_2\times D_6$) x 3, $C_2^2\times \He_3$ ($C_2\times D_6$) x 3, $C_3^2:D_6$ ($C_2\times D_6$) x 2, $C_3^3:C_4$ ($C_2\times D_6$) x 2

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

Order 72: $C_2\times C_6^2$ ($S_3^2$)

Order 54: $C_3^2:S_3$ ($C_6:D_4$) x 4, $C_3^2\times C_6$ ($S_3\times D_4$) x 2, $C_2\times \He_3$ ($C_6:D_4$) x 2, $C_3^2\times C_6$ ($C_2^2\times D_6$), $C_2\times \He_3$ ($C_2^2\times D_6$)

Order 36: $C_6^2$ ($S_3\times D_6$) x 3

Order 27: $C_3^3$ ($D_4\times D_6$), $\He_3$ ($C_2^3:D_6$)

Order 24: $C_2^2\times C_6$ ($C_3^2:D_6$)

Order 18: $C_3\times C_6$ ($D_6:D_6$) x 2, $C_3\times C_6$ ($D_6^2$)

Order 12: $C_2\times C_6$ ($C_6.S_3^2$) x 3

Order 9: $C_3^2$ ($D_6^2:C_2$)

Order 8: $C_2^3$ ($C_3^3:D_6$)

Order 6: $C_6$ ($C_6^2:D_6$) x 2, $C_6$ ($C_6^2:D_6$)

Order 4: $C_2^2$ ($C_2\times C_3^3:D_6$) x 3

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

Order 2: $C_2$ ($C_6^2.S_3^2$) x 2, $C_2$ ($C_6^2.S_3^2$)

Order 1: $C_1$ ($C_6^3:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2592: $C_6^3:D_6$ ($C_1$)

Order 1296: $C_6^2.S_3^2$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^3:S_3$ ($C_2$), $C_6^2.S_3^2$ ($C_2$), $C_6^3:S_3$ ($C_2$), $C_6^3:C_6$ ($C_2$)

Order 648: $C_2^2\times C_3\wr S_3$ ($C_2^2$) x 3, $C_3\wr C_3:D_4$ ($C_2^2$), $C_3^3:D_{12}$ ($C_2^2$), $C_3^3:(C_4\times S_3)$ ($C_2^2$), $C_6^3:C_3$ ($C_2^2$), $C_2\times C_3^3:D_6$ ($C_2^2$), $C_2^2\times C_3^3:C_6$ ($C_2^2$), $C_2\times C_3^3:D_6$ ($C_2^2$), $(C_3\times C_6^2):S_3$ ($C_2^2$), $(C_3\times C_6^2).S_3$ ($C_2^2$), $(C_3\times C_6^2):C_6$ ($C_2^2$), $C_2\times C_3^3:C_{12}$ ($C_2^2$)

Order 432: $C_6^3:C_2$ ($S_3$), $C_6^2:D_6$ ($S_3$)

Order 324: $C_3^2.C_6^2$ ($C_2^3$) x 2, $C_3^3:D_6$ ($C_2^3$) x 2, $C_3^3:D_6$ ($C_2^3$), $\He_3:D_6$ ($C_2^3$), $C_3^3:D_6$ ($D_4$), $C_3\wr C_3:C_4$ ($C_2^3$), $C_3^3:C_{12}$ ($C_2^3$)

Order 216: $C_6^2:S_3$ ($D_6$) x 3, $C_6^3$ ($D_6$), $C_6^2:S_3$ ($D_6$), $C_6^2:S_3$ ($D_6$), $C_6^2.S_3$ ($D_6$), $C_2^3\times \He_3$ ($D_6$)

Order 162: $C_3^3:C_6$ ($C_2^4$), $C_3^3:C_6$ ($C_2\times D_4$), $C_3\wr S_3$ ($C_2\times D_4$)

Order 108: $C_3\times C_6^2$ ($C_2\times D_6$) x 2, $C_2^2\times \He_3$ ($C_2\times D_6$) x 2, $C_3^2:D_6$ ($C_2\times D_6$) x 2, $C_3^2:D_6$ ($C_2\times D_6$), $C_3^3:C_4$ ($C_2\times D_6$), $C_3^2:D_6$ ($C_3:D_4$)

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

Order 72: $C_2\times C_6^2$ ($S_3^2$)

Order 54: $C_3^2:S_3$ ($C_6:D_4$), $C_3^2\times C_6$ ($C_2^2\times D_6$), $C_3^2\times C_6$ ($S_3\times D_4$), $C_2\times \He_3$ ($C_2^2\times D_6$), $C_2\times \He_3$ ($C_6:D_4$)

Order 36: $C_6^2$ ($S_3\times D_6$) x 2

Order 27: $C_3^3$ ($D_4\times D_6$), $\He_3$ ($C_2^3:D_6$)

Order 24: $C_2^2\times C_6$ ($C_3^2:D_6$)

Order 18: $C_3\times C_6$ ($D_6^2$), $C_3\times C_6$ ($D_6:D_6$)

Order 12: $C_2\times C_6$ ($C_6.S_3^2$) x 2

Order 9: $C_3^2$ ($D_6^2:C_2$)

Order 8: $C_2^3$ ($C_3^3:D_6$)

Order 6: $C_6$ ($C_6^2:D_6$), $C_6$ ($C_6^2:D_6$)

Order 4: $C_2^2$ ($C_2\times C_3^3:D_6$) x 2

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

Order 2: $C_2$ ($C_6^2.S_3^2$), $C_2$ ($C_6^2.S_3^2$)

Order 1: $C_1$ ($C_6^3:D_6$)

Series

Derived series

$C_6^3:D_6$

$\rhd$

$C_3^3:C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^3:D_6$

$\rhd$

$C_6^2.S_3^2$

$\rhd$

$C_2\times C_3^3:D_6$

$\rhd$

$C_3^3:D_6$

$\rhd$

$C_3^3:C_6$

$\rhd$

$C_3\wr C_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3:D_6$

$\rhd$

$C_3^3:C_6$

$\rhd$

$C_3\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $94 \times 94$ rational character table.