Abstract group 26244.ki: $C_3^4:(C_9\times D_{18})$

• Label: 26244.ki
• Order: \( 2^{2} \cdot 3^{8} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{3} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^4:(C_9\times D_{18})$
Order:	\(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3^3.C_6^2.C_2^2$, of order \(944784\)\(\medspace = 2^{4} \cdot 3^{10} \)
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	243	728	4860	5832	14580	26244
Conjugacy classes	1	3	65	21	243	63	396
Divisions	1	3	39	13	42	11	109
Autjugacy classes	1	2	21	8	19	7	58

Dimension	1	2	4	6	8	12	18	24	36	72
Irr. complex chars.	36	108	153	0	0	81	12	0	6	0	396
Irr. rational chars.	4	10	9	6	6	19	4	34	6	11	109

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$72$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{18}=b^{18}=c^{3}=d^{3}=e^{3}=f^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,30,25,16,13,6,3,29,27,17,14,5,2,28,26,18,15,4)(7,35,32,24,21,12,9,34,33,22,20,10,8,36,31,23,19,11) \!\cdots\! \rangle$
Transitive group:	36T12926				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:D_{18})$ $\,\rtimes\,$ $C_9$	$(C_3^4:D_9)$ $\,\rtimes\,$ $C_{18}$	$(C_3^4:C_{18})$ $\,\rtimes\,$ $D_9$	$(C_3^4:C_9)$ $\,\rtimes\,$ $D_{18}$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^6$ . $(C_6\times S_3)$	$C_3^5$ . $(S_3\times C_{18})$	$(C_3^5:D_{18})$ . $C_3$	$C_3^5$ . $(C_3\times D_{18})$	all 32

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

Homology

Abelianization:	$C_{2} \times C_{18} \simeq C_{2}^{2} \times C_{9}$
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 159745 subgroups in 5401 conjugacy classes, 72 normal (34 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^5.C_6.C_6$
Commutator:	$G' \simeq$ $C_3^4:C_9$	$G/G' \simeq$ $C_2\times C_{18}$
Frattini:	$\Phi \simeq$ $C_3^4$	$G/\Phi \simeq$ $C_3\wr C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6.C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4:(C_9\times D_{18})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6.C_3^2$

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: $C_3^4:(C_9\times D_{18})$

Order 13122: $C_3^3.C_3^3.C_3.C_6$ x 2, $C_3^6.C_3.C_6$

Order 8748: $C_3^3:(C_9\times D_{18})$ x 2, $C_3^5:D_{18}$, $C_3^4:(S_3\times C_{18})$

Order 6561: $C_3^6.C_3^2$

Order 4374: $C_3^5.C_3.C_6$ x 2, $C_3\times (C_3^4.C_3^2):C_2$ x 2, $(C_3^3\times C_9).C_3.C_6$ x 2, $(C_3^2\times C_9^2).C_6$ x 2, $C_3^3.C_3^3.C_6$ x 2, $C_3^2.C_3^4.C_6$ x 2, $C_3^5.C_3.C_6$ x 2, $C_3\times (C_3^5.C_3):C_2$ x 2, $C_3^5.C_3.C_6$, $C_3\times (C_3\times C_3\times C_3^2:C_9):C_2$

Order 2916: $C_3^4:D_{18}$ x 2, $C_3^3:(S_3\times C_{18})$ x 2, $C_9^2:S_3^2$, $C_3^5:D_6$, $C_3^4:D_{18}$, $C_3^4:(C_2\times C_{18})$

Order 2187: $C_3^4.C_3^3$ x 8, $C_3^4.C_3^3$ x 3, $C_3\times C_3^5.C_3$ x 2, $C_3\times C_9\times C_3^2:C_9$ x 2, $C_3^3\times C_3^2:C_9$

Order 1458: $C_3^4:C_{18}$ x 16, $(C_3^2\times C_9):C_{18}$ x 6, $C_3^4:D_9$ x 4, $C_3^4:C_{18}$ x 4, $C_3^4:C_{18}$ x 4, $C_3^4:D_9$ x 4, $(C_3\times C_9)\wr C_2$ x 4, $C_9\times C_3^2:C_{18}$ x 3, $C_3^4:C_{18}$ x 3, $C_3^3\wr C_2$ x 2, $C_3^4:D_9$ x 2, $C_3^4:C_{18}$ x 2, $C_3^5:C_6$, $C_3^4:C_{18}$, $C_9\times C_3^2:C_{18}$

Order 972: $C_3^3:S_3^2$ x 4, $S_3\times C_3^2:C_{18}$ x 4, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_9^2:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_9\times C_3:S_3^2$, $C_3^3:D_{18}$

Order 729: $C_3^4:C_9$ x 48, $C_3^2.C_9^2$ x 21, $C_3^4:C_9$ x 18, $C_3^2:C_9^2$ x 15, $C_3^2\times C_9^2$ x 4, $C_3^4\times C_9$ x 2, $C_3^6$

Order 486: $C_3^3:C_{18}$ x 36, $C_3^3:C_{18}$ x 25, $C_3^3:D_9$ x 20, $C_3^4:C_6$ x 18, $C_3^3:C_{18}$ x 13, $C_3^4:C_6$ x 10, $C_3^3:C_{18}$ x 6, $(C_3\times C_9):C_{18}$ x 6, $C_9^2:C_6$ x 4, $C_3^3:D_9$ x 4, $C_9^2:S_3$ x 4, $S_3\times C_9^2$ x 3, $C_3^4:C_6$ x 3, $C_3^3:C_{18}$ x 3, $C_3^3:D_9$ x 2, $C_3^3:D_9$ x 2

Order 324: $C_3\wr C_2^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 4, $C_9\times S_3^2$ x 3, $C_2\times C_3^2:C_{18}$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:D_{18}$ x 2, $C_9\times D_{18}$, $C_3^2:S_3^2$, $C_9:S_3^2$

Order 243: $C_3^3:C_9$ x 341, $C_3^3\times C_9$ x 67, $C_3^5$ x 39, $C_3.C_9^2$ x 39, $C_3\times C_9^2$ x 22

Order 162: $C_3^2\wr C_2$ x 97, $C_3^2:C_{18}$ x 51, $C_3^3:C_6$ x 38, $C_3^2:C_{18}$ x 29, $C_3^2:C_{18}$ x 27, $C_3^2:D_9$ x 14, $S_3\times C_3^3$ x 10, $C_3^2:D_9$ x 10, $C_9\times D_9$ x 10, $C_3^2\times D_9$ x 4, $C_3^2:C_{18}$ x 2, $C_3^2:D_9$ x 2, $C_3^3:S_3$, $C_9\times C_{18}$

Order 108: $C_3\times S_3^2$ x 10, $C_3:S_3^2$ x 6, $C_3:S_3^2$ x 6, $S_3\times C_{18}$ x 3, $C_3^2:D_6$ x 2, $S_3\times D_9$ x 2, $C_3\times D_{18}$

Order 81: $C_3^4$ x 536, $C_3^2\times C_9$ x 406, $C_3^2:C_9$ x 286, $C_9^2$ x 25

Order 54: $C_3^2:C_6$ x 146, $S_3\times C_3^2$ x 86, $S_3\times C_9$ x 49, $C_3\times D_9$ x 14, $C_3^2:S_3$ x 10, $C_3\times C_{18}$ x 7, $C_3:D_9$ x 4, $C_3^2\times C_6$ x 2

Order 36: $S_3^2$ x 10, $C_6\times S_3$ x 5, $C_6:S_3$ x 2, $C_2\times C_{18}$, $D_{18}$

Order 27: $C_3^3$ x 1356, $C_3\times C_9$ x 379

Order 18: $C_3\times S_3$ x 106, $C_3:S_3$ x 29, $C_3\times C_6$ x 11, $C_{18}$ x 11, $D_9$ x 4

Order 12: $D_6$ x 5, $C_2\times C_6$

Order 9: $C_3^2$ x 536, $C_9$ x 42

Order 6: $S_3$ x 18, $C_6$ x 13

Order 4: $C_2^2$

Order 3: $C_3$ x 39

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 26244: $C_3^4:(C_9\times D_{18})$

Order 13122: $C_3^6.C_3.C_6$, $C_3^3.C_3^3.C_3.C_6$

Order 8748: $C_3^3:(C_9\times D_{18})$, $C_3^5:D_{18}$, $C_3^4:(S_3\times C_{18})$

Order 6561: $C_3^6.C_3^2$

Order 4374: $C_3\times (C_3^5.C_3):C_2$ x 2, $C_3^5.C_3.C_6$, $C_3^5.C_3.C_6$, $C_3\times (C_3^4.C_3^2):C_2$, $(C_3^3\times C_9).C_3.C_6$, $(C_3^2\times C_9^2).C_6$, $C_3^3.C_3^3.C_6$, $C_3^2.C_3^4.C_6$, $C_3^5.C_3.C_6$, $C_3\times (C_3\times C_3\times C_3^2:C_9):C_2$

Order 2916: $C_9^2:S_3^2$, $C_3^5:D_6$, $C_3^4:D_{18}$, $C_3^4:D_{18}$, $C_3^4:(C_2\times C_{18})$, $C_3^3:(S_3\times C_{18})$

Order 2187: $C_3^4.C_3^3$ x 4, $C_3^4.C_3^3$ x 3, $C_3\times C_3^5.C_3$ x 2, $C_3\times C_9\times C_3^2:C_9$ x 2, $C_3^3\times C_3^2:C_9$

Order 1458: $C_3^4:C_{18}$ x 9, $C_3^4:C_{18}$ x 4, $(C_3^2\times C_9):C_{18}$ x 3, $C_3^4:D_9$ x 2, $C_3^4:C_{18}$ x 2, $C_3^4:D_9$ x 2, $(C_3\times C_9)\wr C_2$ x 2, $C_9\times C_3^2:C_{18}$, $C_3^3\wr C_2$, $C_3^5:C_6$, $C_3^4:C_{18}$, $C_3^4:D_9$, $C_3^4:C_{18}$, $C_3^4:C_{18}$, $C_9\times C_3^2:C_{18}$

Order 972: $C_3^3:S_3^2$ x 2, $S_3\times C_3^2:C_{18}$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_9\times C_3:S_3^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_9^2:D_6$

Order 729: $C_3^4:C_9$ x 19, $C_3^4:C_9$ x 14, $C_3^2.C_9^2$ x 12, $C_3^2:C_9^2$ x 4, $C_3^2\times C_9^2$ x 3, $C_3^4\times C_9$ x 2, $C_3^6$

Order 486: $C_3^3:C_{18}$ x 15, $C_3^3:C_{18}$ x 11, $C_3^3:D_9$ x 9, $C_3^4:C_6$ x 9, $C_3^3:C_{18}$ x 8, $C_3^4:C_6$ x 5, $C_3^3:C_{18}$ x 3, $(C_3\times C_9):C_{18}$ x 3, $C_3^4:C_6$ x 3, $S_3\times C_9^2$ x 2, $C_9^2:C_6$ x 2, $C_3^3:D_9$ x 2, $C_9^2:S_3$ x 2, $C_3^3:C_{18}$, $C_3^3:D_9$, $C_3^3:D_9$

Order 324: $C_3\wr C_2^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_2\times C_3^2:C_{18}$, $C_9\times D_{18}$, $C_3^2:D_{18}$, $C_3^2:S_3^2$, $C_9:S_3^2$, $C_3^2:D_{18}$

Order 243: $C_3^3:C_9$ x 87, $C_3^3\times C_9$ x 27, $C_3^5$ x 21, $C_3.C_9^2$ x 14, $C_3\times C_9^2$ x 10

Order 162: $C_3^2\wr C_2$ x 36, $C_3^3:C_6$ x 19, $C_3^2:C_{18}$ x 17, $C_3^2:C_{18}$ x 14, $C_3^2:C_{18}$ x 10, $C_3^2:D_9$ x 6, $S_3\times C_3^3$ x 5, $C_3^2:D_9$ x 5, $C_9\times D_9$ x 3, $C_3^2\times D_9$ x 2, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_9\times C_{18}$, $C_3^2:D_9$

Order 108: $C_3\times S_3^2$ x 5, $C_3:S_3^2$ x 3, $C_3:S_3^2$ x 3, $S_3\times C_{18}$ x 2, $C_3^2:D_6$, $C_3\times D_{18}$, $S_3\times D_9$

Order 81: $C_3^4$ x 146, $C_3^2\times C_9$ x 104, $C_3^2:C_9$ x 57, $C_9^2$ x 7

Order 54: $C_3^2:C_6$ x 58, $S_3\times C_3^2$ x 31, $S_3\times C_9$ x 20, $C_3\times D_9$ x 6, $C_3\times C_{18}$ x 5, $C_3^2:S_3$ x 5, $C_3:D_9$ x 2, $C_3^2\times C_6$

Order 36: $S_3^2$ x 5, $C_6\times S_3$ x 3, $C_2\times C_{18}$, $D_{18}$, $C_6:S_3$

Order 27: $C_3^3$ x 303, $C_3\times C_9$ x 105

Order 18: $C_3\times S_3$ x 41, $C_3:S_3$ x 14, $C_{18}$ x 7, $C_3\times C_6$ x 6, $D_9$ x 2

Order 12: $D_6$ x 3, $C_2\times C_6$

Order 9: $C_3^2$ x 146, $C_9$ x 19

Order 6: $S_3$ x 9, $C_6$ x 8

Order 4: $C_2^2$

Order 3: $C_3$ x 21

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 26244: $C_3^4:(C_9\times D_{18})$ ($C_1$)

Order 13122: $C_3^3.C_3^3.C_3.C_6$ ($C_2$) x 2, $C_3^6.C_3.C_6$ ($C_2$)

Order 8748: $C_3^5:D_{18}$ ($C_3$)

Order 6561: $C_3^6.C_3^2$ ($C_2^2$)

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

Order 2916: $C_3^4:D_{18}$ ($C_9$)

Order 2187: $C_3^4.C_3^3$ ($D_6$) x 2, $C_3\times C_3^5.C_3$ ($C_2\times C_6$), $C_3\times C_3^5.C_3$ ($D_6$)

Order 1458: $C_3^4:D_9$ ($C_{18}$) x 2, $C_3^4:D_9$ ($C_3\times S_3$) x 2, $C_3^4:C_{18}$ ($C_{18}$), $C_3^4:C_{18}$ ($D_9$), $C_3^5:C_6$ ($C_3\times S_3$)

Order 729: $C_3^4:C_9$ ($C_6\times S_3$) x 2, $C_3^4:C_9$ ($S_3^2$) x 2, $C_3^6$ ($C_6\times S_3$), $C_3^2\times C_9^2$ ($S_3^2$), $C_3^4:C_9$ ($C_2\times C_{18}$), $C_3^4:C_9$ ($D_{18}$)

Order 486: $C_3^3:D_9$ ($S_3\times C_9$) x 2, $C_3^4:C_6$ ($S_3\times C_9$), $C_3^4:C_6$ ($C_3\times D_9$)

Order 243: $C_3^5$ ($C_3\times S_3^2$) x 2, $C_3^3:C_9$ ($S_3\times C_{18}$) x 2, $C_3^3:C_9$ ($S_3\times D_9$) x 2, $C_3^5$ ($S_3\times C_{18}$), $C_3^5$ ($C_3\times D_{18}$), $C_3^3\times C_9$ ($C_3:S_3^2$), $C_3^3\times C_9$ ($C_3\times S_3^2$)

Order 162: $C_3^3:S_3$ ($C_9\times D_9$)

Order 81: $C_3^4$ ($C_9\times S_3^2$) x 2, $C_3^4$ ($C_3^2:D_{18}$) x 2, $C_3^4$ ($C_9\times D_{18}$), $C_3^4$ ($C_3\wr C_2^2$), $C_3^2\times C_9$ ($C_9:S_3^2$), $C_3^2\times C_9$ ($C_9\times S_3^2$)

Order 27: $C_3^3$ ($C_3^3:S_3^2$) x 2, $C_3^3$ ($C_9^2:D_6$) x 2, $C_3^3$ ($C_9\times C_3:S_3^2$), $C_3^3$ ($C_3^3:D_{18}$)

Order 9: $C_3^2$ ($C_3^3:S_3.C_{18}$) x 2, $C_3^2$ ($C_3^4.C_6.C_6$) x 2, $C_3^2$ ($C_9^2:S_3^2$), $C_3^2$ ($C_3^4:S_3^2$)

Order 3: $C_3$ ($C_3^4.C_6.C_{18}$) x 2, $C_3$ ($C_3^5.C_6.C_6$), $C_3$ ($C_3^4:C_3.C_6.C_6$)

Order 1: $C_1$ ($C_3^4:(C_9\times D_{18})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 26244: $C_3^4:(C_9\times D_{18})$ ($C_1$)

Order 13122: $C_3^6.C_3.C_6$ ($C_2$), $C_3^3.C_3^3.C_3.C_6$ ($C_2$)

Order 8748: $C_3^5:D_{18}$ ($C_3$)

Order 6561: $C_3^6.C_3^2$ ($C_2^2$)

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

Order 2916: $C_3^4:D_{18}$ ($C_9$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_2\times C_6$), $C_3\times C_3^5.C_3$ ($D_6$), $C_3^4.C_3^3$ ($D_6$)

Order 1458: $C_3^4:D_9$ ($C_{18}$), $C_3^4:C_{18}$ ($C_{18}$), $C_3^4:C_{18}$ ($D_9$), $C_3^5:C_6$ ($C_3\times S_3$), $C_3^4:D_9$ ($C_3\times S_3$)

Order 729: $C_3^6$ ($C_6\times S_3$), $C_3^4:C_9$ ($C_6\times S_3$), $C_3^4:C_9$ ($S_3^2$), $C_3^2\times C_9^2$ ($S_3^2$), $C_3^4:C_9$ ($C_2\times C_{18}$), $C_3^4:C_9$ ($D_{18}$)

Order 486: $C_3^4:C_6$ ($S_3\times C_9$), $C_3^4:C_6$ ($C_3\times D_9$), $C_3^3:D_9$ ($S_3\times C_9$)

Order 243: $C_3^5$ ($C_3\times S_3^2$), $C_3^5$ ($S_3\times C_{18}$), $C_3^5$ ($C_3\times D_{18}$), $C_3^3\times C_9$ ($C_3:S_3^2$), $C_3^3\times C_9$ ($C_3\times S_3^2$), $C_3^3:C_9$ ($S_3\times C_{18}$), $C_3^3:C_9$ ($S_3\times D_9$)

Order 162: $C_3^3:S_3$ ($C_9\times D_9$)

Order 81: $C_3^4$ ($C_9\times D_{18}$), $C_3^4$ ($C_3\wr C_2^2$), $C_3^4$ ($C_9\times S_3^2$), $C_3^4$ ($C_3^2:D_{18}$), $C_3^2\times C_9$ ($C_9:S_3^2$), $C_3^2\times C_9$ ($C_9\times S_3^2$)

Order 27: $C_3^3$ ($C_3^3:S_3^2$), $C_3^3$ ($C_9\times C_3:S_3^2$), $C_3^3$ ($C_3^3:D_{18}$), $C_3^3$ ($C_9^2:D_6$)

Order 9: $C_3^2$ ($C_3^3:S_3.C_{18}$), $C_3^2$ ($C_3^4.C_6.C_6$), $C_3^2$ ($C_9^2:S_3^2$), $C_3^2$ ($C_3^4:S_3^2$)

Order 3: $C_3$ ($C_3^5.C_6.C_6$), $C_3$ ($C_3^4.C_6.C_{18}$), $C_3$ ($C_3^4:C_3.C_6.C_6$)

Order 1: $C_1$ ($C_3^4:(C_9\times D_{18})$)

Series

Derived series

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^4:C_9$

$\rhd$

$C_3^4:C_9$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^6.C_3.C_6$

$\rhd$

$C_3^6.C_3.C_6$

$\rhd$

$C_3\times (C_3^5.C_3):C_2$

$\rhd$

$C_3\times (C_3^5.C_3):C_2$

$\rhd$

$C_3^4:C_{18}$

$\rhd$

$C_3^4:C_{18}$

$\rhd$

$C_3^4:C_9$

$\rhd$

$C_3^4:C_9$

$\rhd$

$C_3^5$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^4:(C_9\times D_{18})$

$\rhd$

$C_3^4:C_9$

$\rhd$

$C_3^4:C_9$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 6 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 $396 \times 396$ 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 $109 \times 109$ rational character table.