Abstract group 26244.nm: $C_3^6.S_3^2$

• Label: 26244.nm
• Order: \( 2^{2} \cdot 3^{8} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3^{2} \)
• Perm deg.: $36$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_3^6.S_3^2$
Order:	\(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^4.C_3^5.C_2^3$, of order \(157464\)\(\medspace = 2^{3} \cdot 3^{9} \)
Composition factors:	$C_2$ x 2, $C_3$ x 8
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	351	3644	16416	2916	2916	26244
Conjugacy classes	1	3	83	60	12	6	165
Divisions	1	3	57	35	6	3	105
Autjugacy classes	1	3	34	21	2	1	62

Dimension	1	2	3	4	6	8	12	18	24	36	72
Irr. complex chars.	12	30	24	12	32	0	22	22	0	11	0	165
Irr. rational chars.	4	14	0	14	20	4	28	2	3	11	5	105

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$38211264$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{6}=b^{6}=c^{3}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(5,8,14)(10,17,23)(11,16,22)(12,18,19)(20,26,25)(21,24,27)(28,29,30), (1,2,6) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_3^6$ . $S_3^2$	$(C_3^6.C_3)$ . $D_6$	$(C_3^5:C_3)$ . $S_3^2$	$C_3^5$ . $(C_3\times S_3^2)$	all 43
Aut. group:	$\Aut(C_3^2.C_3^3)$	$\Aut((C_3^2\times C_{18}):C_3)$

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

Homology

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

Subgroups

There are 444827 subgroups in 8214 conjugacy classes, 78 normal (57 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3.C_3^5.C_6.C_2$
Commutator:	$G' \simeq$ $C_3^3.C_3^4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^4$	$G/\Phi \simeq$ $C_3^2:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^6.S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3^5.C_6.C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5.C_3^3$

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 26244: $C_3^6.S_3^2$

Order 13122: $C_3^3.C_3^4.C_6$, $C_3^5.C_3^3.C_2$, $C_3^4.C_3^3.C_6$

Order 8748: $C_3^4.\He_3.C_2^2$ x 3, $C_3.C_3^5.D_6$, $C_3\times C_3^3.C_3^3.C_2^2$, $C_3^5.(C_6\times S_3)$

Order 6561: $C_3^5.C_3^3$

Order 4374: $C_3^4.\He_3.C_2$ x 6, $C_3^4.\He_3.C_2$ x 3, $C_3^4.C_3^3.C_2$ x 3, $C_3^3.C_3^3.C_6$ x 3, $C_3^3.\He_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3.C_3^5.C_6$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3\times (C_3\times C_3^4:C_3):C_2$, $C_3^5.C_3.C_6$, $C_3^3.C_3^4.C_2$, $C_3\times (C_3\times C_3^4:C_3):C_2$, $C_3^6.C_6$, $C_3.C_3^5.S_3$, $C_3\times (C_3\times C_3^4:C_3):C_2$

Order 2916: $C_3\times C_3^4:C_3:C_2^2$ x 3, $C_3^4:C_3.D_6$ x 3, $C_3^4.(C_6\times S_3)$ x 3, $C_3^3.C_3^3.C_2^2$ x 3, $C_3^3.C_3^3.C_2^2$, $C_3\times C_3^3:C_3^2:C_2^2$, $C_3\times (C_3^2\times \He_3):C_2^2$, $C_3^2\times C_3^3:C_2:S_3$, $C_3^5.D_6$, $C_3\times (C_3^2\times \He_3):C_2^2$, $C_3^4:C_3.D_6$, $(C_3^2\times \He_3).D_6$, $C_3^5.D_6$

Order 2187: $C_3^4.C_3^3$ x 12, $C_3^6.C_3$ x 2, $C_3^5.C_3^2$ x 2, $C_3^3.C_3^4$, $C_3^2\times C_3^4:C_3$, $C_3^4.C_3^3$

Order 1458: $C_3^5:S_3$ x 15, $\He_3^2:C_2$ x 14, $C_3^5:S_3$ x 10, $(C_3^2\times \He_3):S_3$ x 9, $C_3^5:S_3$ x 7, $C_3^5:C_6$ x 7, $C_3^5:C_6$ x 6, $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 5, $C_3^4:(C_3\times S_3)$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $C_3^5:S_3$ x 4, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $(C_3^2\times \He_3):S_3$ x 3, $C_3^4.(C_3\times C_6)$ x 3, $C_3^5:S_3$ x 3, $C_3^3\wr C_2$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^3:(C_3\times C_{18})$, $(C_3^2\times \He_3):C_6$, $C_3^5:C_6$, $(C_3^2\times \He_3):S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $\He_3^2:C_2$, $\He_3^2:C_2$, $C_3^5.C_6$, $C_3^4.(C_3\times C_6)$, $C_3^5.C_6$

Order 972: $C_3^3:S_3^2$ x 13, $C_3^3:S_3^2$ x 11, $C_3^4:D_6$ x 9, $C_3^3:S_3^2$ x 5, $C_3^3:C_6^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 729: $C_3^5:C_3$ x 24, $C_3^5:C_3$ x 21, $C_3^3.C_3^3$ x 15, $\He_3^2$ x 14, $C_3^4:C_3^2$ x 9, $C_3^3.C_3^3$ x 9, $C_3^3.C_3^3$ x 6, $C_3^5:C_3$ x 5, $C_3^3.C_3^3$ x 5, $C_3^3\times \He_3$ x 4, $C_3^5.C_3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^5.C_3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^6$, $C_3^3:C_3^3$

Order 486: $C_3^4:S_3$ x 173, $C_3^4:S_3$ x 57, $C_3^4:C_6$ x 50, $C_3^4:S_3$ x 31, $C_3^4:C_6$ x 31, $C_3^4:C_6$ x 29, $C_3^4:C_6$ x 20, $C_3^4:S_3$ x 19, $C_3^4:C_6$ x 18, $C_3^4:S_3$ x 18, $C_3^4:S_3$ x 15, $C_3^3:(C_3\times C_6)$ x 7, $C_3^4:C_6$ x 7, $C_3^4:C_6$ x 6, $C_3^4.C_6$ x 5, $C_3^4:C_6$ x 4, $(C_3\times \He_3):C_6$ x 3, $C_3^3:C_{18}$ x 3, $S_3\times C_3^4$ x 2, $C_3^4:S_3$ x 2, $C_3^3:C_{18}$, $(C_3\times \He_3):S_3$, $C_3^4.C_6$, $C_3^4:C_6$

Order 324: $C_3\wr C_2^2$ x 20, $C_3^3:D_6$ x 12, $C_3^2:S_3^2$ x 11, $C_3^2:S_3^2$ x 11, $C_3^2\times S_3^2$ x 9, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^3:D_6$ x 3, $C_3^2:S_3^2$ x 2, $C_3^2:C_6^2$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^4:C_3$ x 254, $C_3^2\times \He_3$ x 146, $C_3^4:C_3$ x 58, $C_3^3.C_3^2$ x 42, $C_3^5$ x 32, $C_3^4.C_3$ x 19, $C_3^3:C_9$ x 15, $C_3^3:C_3^2$ x 10, $C_3^3:C_9$ x 5, $C_9:C_3^3$ x 2

Order 162: $C_3^2\wr C_2$ x 223, $C_3^3:S_3$ x 195, $C_3^3:C_6$ x 140, $C_3^3:S_3$ x 63, $S_3\times C_3^3$ x 59, $C_3\wr S_3$ x 45, $C_3^3:C_6$ x 41, $S_3\times \He_3$ x 27, $C_3^3:C_6$ x 25, $C_3^3:C_6$ x 6, $C_3^3.C_6$ x 5, $C_3^2:C_{18}$ x 3, $C_6\times \He_3$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_3^2:C_{18}$

Order 108: $C_3\times S_3^2$ x 44, $C_3^2:D_6$ x 12, $C_3:S_3^2$ x 8, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 6, $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 3

Order 81: $C_3\times \He_3$ x 828, $C_3^4$ x 497, $C_3\wr C_3$ x 84, $C_3^2:C_9$ x 24, $C_9:C_3^2$ x 24, $C_3^2\times C_9$ x 2

Order 54: $S_3\times C_3^2$ x 319, $C_3^2:C_6$ x 292, $C_3^2:S_3$ x 199, $C_3^2:C_6$ x 94, $C_3^2\times C_6$ x 17, $C_2\times \He_3$ x 11, $C_3^2:S_3$ x 6, $C_9:C_6$ x 5, $S_3\times C_9$ x 3, $C_3\times C_{18}$

Order 36: $C_6\times S_3$ x 28, $S_3^2$ x 17, $C_6:S_3$ x 3, $C_6^2$

Order 27: $C_3^3$ x 1494, $\He_3$ x 524, $C_9:C_3$ x 25, $C_3\times C_9$ x 11

Order 18: $C_3\times S_3$ x 291, $C_3\times C_6$ x 62, $C_3:S_3$ x 52, $C_{18}$ x 3

Order 12: $D_6$ x 10, $C_2\times C_6$ x 4

Order 9: $C_3^2$ x 661, $C_9$ x 6

Order 6: $S_3$ x 41, $C_6$ x 35

Order 4: $C_2^2$

Order 3: $C_3$ x 57

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 26244: $C_3^6.S_3^2$

Order 13122: $C_3^3.C_3^4.C_6$, $C_3^5.C_3^3.C_2$, $C_3^4.C_3^3.C_6$

Order 8748: $C_3.C_3^5.D_6$, $C_3^4.\He_3.C_2^2$, $C_3\times C_3^3.C_3^3.C_2^2$, $C_3^5.(C_6\times S_3)$

Order 6561: $C_3^5.C_3^3$

Order 4374: $C_3^4.\He_3.C_2$ x 2, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.\He_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3.C_3^5.C_6$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3\times (C_3\times C_3^4:C_3):C_2$, $C_3^5.C_3.C_6$, $C_3^3.C_3^4.C_2$, $C_3^3.C_3^3.C_6$, $C_3\times (C_3\times C_3^4:C_3):C_2$, $C_3^6.C_6$, $C_3.C_3^5.S_3$, $C_3\times (C_3\times C_3^4:C_3):C_2$

Order 2916: $C_3\times C_3^4:C_3:C_2^2$ x 3, $C_3^3.C_3^3.C_2^2$, $C_3\times C_3^3:C_3^2:C_2^2$, $C_3\times (C_3^2\times \He_3):C_2^2$, $C_3^2\times C_3^3:C_2:S_3$, $C_3^5.D_6$, $C_3\times (C_3^2\times \He_3):C_2^2$, $C_3^4:C_3.D_6$, $C_3^4:C_3.D_6$, $(C_3^2\times \He_3).D_6$, $C_3^4.(C_6\times S_3)$, $C_3^5.D_6$, $C_3^3.C_3^3.C_2^2$

Order 2187: $C_3^4.C_3^3$ x 4, $C_3^6.C_3$ x 2, $C_3^5.C_3^2$ x 2, $C_3^3.C_3^4$, $C_3^2\times C_3^4:C_3$, $C_3^4.C_3^3$

Order 1458: $\He_3^2:C_2$ x 7, $C_3^5:S_3$ x 7, $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 6, $C_3^5:S_3$ x 6, $(C_3^2\times \He_3):S_3$ x 5, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^5:S_3$ x 3, $C_3^5:C_6$ x 3, $C_3^3\wr C_2$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^3:(C_3\times C_{18})$, $(C_3^2\times \He_3):S_3$, $(C_3^2\times \He_3):C_6$, $C_3^5:C_6$, $(C_3^2\times \He_3):S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $\He_3^2:C_2$, $\He_3^2:C_2$, $C_3^5.C_6$, $C_3^4.(C_3\times C_6)$, $C_3^4.(C_3\times C_6)$, $C_3^5.C_6$, $C_3^5:S_3$

Order 972: $C_3^3:S_3^2$ x 7, $C_3^3:S_3^2$ x 6, $C_3^3:C_6^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^4:D_6$ x 3, $C_3^3:C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 729: $C_3^5:C_3$ x 16, $C_3^5:C_3$ x 8, $\He_3^2$ x 7, $C_3^4:C_3^2$ x 5, $C_3^3\times \He_3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^3.C_3^3$ x 4, $C_3^5:C_3$ x 3, $C_3^3.C_3^3$ x 3, $C_3^5.C_3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^5.C_3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^6$, $C_3^3:C_3^3$

Order 486: $C_3^4:S_3$ x 53, $C_3^4:S_3$ x 33, $C_3^4:C_6$ x 27, $C_3^4:C_6$ x 25, $C_3^4:C_6$ x 16, $C_3^4:C_6$ x 12, $C_3^4:S_3$ x 12, $C_3^4:C_6$ x 12, $C_3^4:S_3$ x 8, $C_3^4:S_3$ x 6, $C_3^3:(C_3\times C_6)$ x 5, $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 3, $C_3^4.C_6$ x 3, $S_3\times C_3^4$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^3:C_{18}$, $(C_3\times \He_3):C_6$, $(C_3\times \He_3):S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^3:C_{18}$

Order 324: $C_3\wr C_2^2$ x 10, $C_3^2\times S_3^2$ x 9, $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 7, $C_3^3:D_6$ x 5, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^2\times \He_3$ x 82, $C_3^4:C_3$ x 80, $C_3^5$ x 24, $C_3^4:C_3$ x 18, $C_3^4.C_3$ x 15, $C_3^3.C_3^2$ x 10, $C_3^3:C_3^2$ x 6, $C_3^3:C_9$ x 5, $C_3^3:C_9$ x 5, $C_9:C_3^3$ x 2

Order 162: $C_3^2\wr C_2$ x 125, $C_3^3:S_3$ x 64, $C_3^3:C_6$ x 62, $S_3\times C_3^3$ x 44, $C_3^3:S_3$ x 37, $C_3^3:C_6$ x 18, $C_3^3:C_6$ x 18, $S_3\times \He_3$ x 10, $C_3\wr S_3$ x 7, $C_3^3.C_6$ x 3, $C_6\times \He_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_3^2:C_{18}$, $C_3^2:C_{18}$

Order 108: $C_3\times S_3^2$ x 22, $C_3^2\times D_6$ x 6, $C_3:S_3^2$ x 6, $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 5, $C_3^2:D_6$ x 4, $C_3^2:D_6$ x 3, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 291, $C_3^4$ x 250, $C_3^2:C_9$ x 14, $C_9:C_3^2$ x 13, $C_3\wr C_3$ x 12, $C_3^2\times C_9$ x 2

Order 54: $S_3\times C_3^2$ x 169, $C_3^2:C_6$ x 156, $C_3^2:S_3$ x 66, $C_3^2:C_6$ x 31, $C_3^2\times C_6$ x 13, $C_3^2:S_3$ x 6, $C_2\times \He_3$ x 5, $C_9:C_6$ x 3, $C_3\times C_{18}$, $S_3\times C_9$

Order 36: $C_6\times S_3$ x 14, $S_3^2$ x 13, $C_6:S_3$ x 3, $C_6^2$

Order 27: $C_3^3$ x 660, $\He_3$ x 156, $C_9:C_3$ x 11, $C_3\times C_9$ x 5

Order 18: $C_3\times S_3$ x 148, $C_3:S_3$ x 37, $C_3\times C_6$ x 35, $C_{18}$

Order 12: $D_6$ x 8, $C_2\times C_6$ x 2

Order 9: $C_3^2$ x 306, $C_9$ x 2

Order 6: $S_3$ x 28, $C_6$ x 21

Order 4: $C_2^2$

Order 3: $C_3$ x 34

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 26244: $C_3^6.S_3^2$ ($C_1$)

Order 13122: $C_3^3.C_3^4.C_6$ ($C_2$), $C_3^5.C_3^3.C_2$ ($C_2$), $C_3^4.C_3^3.C_6$ ($C_2$)

Order 8748: $C_3.C_3^5.D_6$ ($C_3$)

Order 6561: $C_3^5.C_3^3$ ($C_2^2$)

Order 4374: $C_3^4.\He_3.C_2$ ($S_3$) x 3, $C_3^4.C_3^3.C_2$ ($S_3$), $C_3.C_3^5.C_6$ ($C_6$), $C_3^4.C_3^3.C_2$ ($C_6$), $C_3\times (C_3\times C_3^4:C_3):C_2$ ($S_3$), $C_3.C_3^5.S_3$ ($C_6$)

Order 2187: $C_3^4.C_3^3$ ($D_6$) x 3, $C_3^3.C_3^4$ ($C_2\times C_6$), $C_3^2\times C_3^4:C_3$ ($D_6$), $C_3^6.C_3$ ($D_6$)

Order 1458: $C_3^4:(C_3\times S_3)$ ($C_3\times S_3$) x 3, $C_3^5:C_6$ ($C_3\times S_3$), $C_3^5:S_3$ ($C_3\times S_3$), $C_3^5:S_3$ ($C_3:S_3$)

Order 729: $C_3^4:C_3^2$ ($C_6\times S_3$) x 3, $C_3^5:C_3$ ($S_3^2$) x 3, $C_3^6$ ($S_3^2$), $C_3^3\times \He_3$ ($C_6\times S_3$), $C_3^5:C_3$ ($C_6:S_3$), $C_3^5:C_3$ ($C_6\times S_3$)

Order 486: $C_3^4:C_6$ ($C_3^2:C_6$) x 2, $C_3^4:C_6$ ($C_3^2:C_6$), $C_3^4:S_3$ ($C_3^2:C_6$), $C_3^4:C_6$ ($C_3^2:C_6$)

Order 243: $C_3^2\times \He_3$ ($C_3^2:D_6$) x 3, $C_3^4:C_3$ ($C_3\times S_3^2$) x 3, $C_3^5$ ($C_3:S_3^2$), $C_3^5$ ($C_3\times S_3^2$), $C_3^5$ ($C_3^2:D_6$), $C_3^2\times \He_3$ ($C_3^2:D_6$)

Order 162: $C_3^3:C_6$ ($C_3^3:C_6$), $C_3^3:C_6$ ($C_3\wr S_3$)

Order 81: $C_3^4$ ($C_3^2:S_3^2$) x 4, $C_3\times \He_3$ ($C_3^2:S_3^2$) x 3, $C_3^4$ ($C_3^2:S_3^2$), $C_3^4$ ($C_3^3:D_6$), $C_3\times \He_3$ ($C_3^3:D_6$)

Order 54: $C_3^2:S_3$ ($C_3^4:S_3$)

Order 27: $C_3^3$ ($C_3^3:S_3^2$) x 3, $C_3^3$ ($C_3^3:S_3^2$), $C_3^3$ ($C_3^3:S_3^2$), $C_3^3$ ($C_3^4:D_6$), $C_3^3$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($S_3\times C_3:C_3\wr S_3$), $C_3^2$ ($C_3^4.(C_6\times S_3)$), $C_3^2$ ($C_3.C_3^4:C_6.C_2$), $C_3^2$ ($C_3^5.C_6.C_2$)

Order 3: $C_3$ ($C_3^5.(C_6\times S_3)$), $C_3$ ($C_3.C_3^5.C_6.C_2$)

Order 1: $C_1$ ($C_3^6.S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 26244: $C_3^6.S_3^2$ ($C_1$)

Order 13122: $C_3^3.C_3^4.C_6$ ($C_2$), $C_3^5.C_3^3.C_2$ ($C_2$), $C_3^4.C_3^3.C_6$ ($C_2$)

Order 8748: $C_3.C_3^5.D_6$ ($C_3$)

Order 6561: $C_3^5.C_3^3$ ($C_2^2$)

Order 4374: $C_3^4.C_3^3.C_2$ ($S_3$), $C_3.C_3^5.C_6$ ($C_6$), $C_3^4.C_3^3.C_2$ ($C_6$), $C_3^4.\He_3.C_2$ ($S_3$), $C_3\times (C_3\times C_3^4:C_3):C_2$ ($S_3$), $C_3.C_3^5.S_3$ ($C_6$)

Order 2187: $C_3^3.C_3^4$ ($C_2\times C_6$), $C_3^2\times C_3^4:C_3$ ($D_6$), $C_3^4.C_3^3$ ($D_6$), $C_3^6.C_3$ ($D_6$)

Order 1458: $C_3^5:C_6$ ($C_3\times S_3$), $C_3^4:(C_3\times S_3)$ ($C_3\times S_3$), $C_3^5:S_3$ ($C_3\times S_3$), $C_3^5:S_3$ ($C_3:S_3$)

Order 729: $C_3^6$ ($S_3^2$), $C_3^3\times \He_3$ ($C_6\times S_3$), $C_3^4:C_3^2$ ($C_6\times S_3$), $C_3^5:C_3$ ($C_6:S_3$), $C_3^5:C_3$ ($C_6\times S_3$), $C_3^5:C_3$ ($S_3^2$)

Order 486: $C_3^4:C_6$ ($C_3^2:C_6$) x 2, $C_3^4:C_6$ ($C_3^2:C_6$), $C_3^4:S_3$ ($C_3^2:C_6$), $C_3^4:C_6$ ($C_3^2:C_6$)

Order 243: $C_3^2\times \He_3$ ($C_3^2:D_6$) x 3, $C_3^5$ ($C_3:S_3^2$), $C_3^5$ ($C_3\times S_3^2$), $C_3^5$ ($C_3^2:D_6$), $C_3^2\times \He_3$ ($C_3^2:D_6$), $C_3^4:C_3$ ($C_3\times S_3^2$)

Order 162: $C_3^3:C_6$ ($C_3^3:C_6$), $C_3^3:C_6$ ($C_3\wr S_3$)

Order 81: $C_3^4$ ($C_3^2:S_3^2$) x 4, $C_3^4$ ($C_3^2:S_3^2$), $C_3^4$ ($C_3^3:D_6$), $C_3\times \He_3$ ($C_3^3:D_6$), $C_3\times \He_3$ ($C_3^2:S_3^2$)

Order 54: $C_3^2:S_3$ ($C_3^4:S_3$)

Order 27: $C_3^3$ ($C_3^3:S_3^2$) x 3, $C_3^3$ ($C_3^3:S_3^2$), $C_3^3$ ($C_3^3:S_3^2$), $C_3^3$ ($C_3^4:D_6$), $C_3^3$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($S_3\times C_3:C_3\wr S_3$), $C_3^2$ ($C_3^4.(C_6\times S_3)$), $C_3^2$ ($C_3.C_3^4:C_6.C_2$), $C_3^2$ ($C_3^5.C_6.C_2$)

Order 3: $C_3$ ($C_3^5.(C_6\times S_3)$), $C_3$ ($C_3.C_3^5.C_6.C_2$)

Order 1: $C_1$ ($C_3^6.S_3^2$)

Series

Derived series

$C_3^6.S_3^2$

$\rhd$

$C_3^6.S_3^2$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^6.S_3^2$

$\rhd$

$C_3^6.S_3^2$

$\rhd$

$C_3^4.C_3^3.C_6$

$\rhd$

$C_3^4.C_3^3.C_6$

$\rhd$

$C_3^5.C_3^3$

$\rhd$

$C_3^5.C_3^3$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3\times \He_3$

$\rhd$

$C_3^3\times \He_3$

$\rhd$

$C_3^5:C_3$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^6.S_3^2$

$\rhd$

$C_3^6.S_3^2$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Character theory

Complex character table

See the $165 \times 165$ 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 $105 \times 105$ rational character table.