Abstract group 236196.hp: $C_3^6.(C_9\times D_{18})$

• Label: 236196.hp
• Order: \( 2^{2} \cdot 3^{10} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: 3
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.(C_9\times D_{18})$
Order:	\(236196\)\(\medspace = 2^{2} \cdot 3^{10} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^7.C_3^3.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \)
Composition factors:	$C_2$ x 2, $C_3$ x 10
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	1215	6560	44712	52488	131220	236196
Conjugacy classes	1	3	290	39	405	63	801
Divisions	1	3	168	25	69	11	277
Autjugacy classes	1	2	45	10	21	7	86

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{18}=b^{18}=c^{3}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,24,27,10,13,36,2,23,25,12,15,34,3,22,26,11,14,35)(4,8,29,32,17,19)(5,9,30,31,16,20) \!\cdots\! \rangle$
Transitive group:	36T25004				more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_3^8$ . $(C_6\times S_3)$	$(C_3^7.C_6)$ . $D_9$	$(C_3^8.C_6)$ . $C_6$ (3)	$C_3^7$ . $(S_3\times C_{18})$	all 45

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:	not computed
Commutator length:	$1$

Subgroups

There are 84 normal subgroups (38 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_3$
Commutator:	a subgroup isomorphic to 6561.86251
Frattini:	a subgroup isomorphic to $C_3^6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^8.C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 81 (order at least 2916) are shown, as well as all normal subgroups of any index.

Order 236196: $C_3^6.(C_9\times D_{18})$

Order 118098: $C_3^5.C_3^3.C_3.C_6$ x 2, $C_3^8.C_3.C_6$

Order 78732: $C_3^5.(C_9\times D_{18})$ x 2, $C_3^7.D_{18}$, $C_3^6.(S_3\times C_{18})$

Order 59049: $C_3^8.C_3^2$

Order 39366: $C_3^7.C_3.C_6$ x 5, $C_3^4.C_3^3.C_3.C_6$ x 4, $C_3^5.C_3^3.C_6$ x 2, $C_3^4.C_3^4.C_6$ x 2, $C_3\times C_3^7.C_6$ x 2, $C_3^8.C_6$, $C_3^5.C_3^4.C_2$, $C_3\times C_3^5.C_3^3.C_2$

Order 26244: $C_3^6.D_{18}$ x 2, $C_3^5.(S_3\times C_{18})$ x 2, $C_3^8.C_2^2$, $C_3^4:(C_9\times D_{18})$, $C_3^6.D_{18}$, $C_3^5.(S_3\times C_{18})$

Order 19683: $C_3^7.C_3^2$ x 13, $C_3\times C_3^7.C_3$ x 3

Order 13122: $C_3\times C_3^6.C_6$ x 19, $C_3^6.C_3.C_6$ x 14, $C_3^7.C_6$ x 10, $C_3^4.C_3^2.C_3.C_6$ x 6, $C_3^5.C_3^3.C_2$ x 4, $C_3^3.C_3^3.C_3.C_6$ x 4, $C_3^4.C_3^4.C_2$ x 3, $C_3\times C_3^7.C_2$ x 2, $C_3\times C_3^3.\He_3.C_6$ x 2, $C_3^8.C_2$, $C_3\times C_3^3.\He_3:S_3$

Order 8748: $C_3^7.C_2^2$ x 8, $C_3\times C_3^5.D_6$ x 3, $C_3^3:(C_9\times D_{18})$ x 2, $C_3^5:D_{18}$ x 2, $C_3^5:D_{18}$ x 2, $C_3^4:(S_3\times C_{18})$ x 2, $C_3^5:(C_2\times C_{18})$ x 2, $C_3^6.D_6$, $C_3^5:D_{18}$, $C_3^4:(S_3\times C_{18})$

Order 6561: $C_3^3\times C_3^3:C_9$ x 81, $C_3^7.C_3$ x 18, $C_3^6.C_3^2$ x 12, $C_3\times C_3^6.C_3$ x 12, $C_3^5.C_3^3$ x 10, $C_3^5.C_3^3$ x 9, $C_3^6.C_3^2$ x 4, $C_3^5.C_3^3$ x 3, $C_3\times C_9\times C_3^3:C_9$ x 2, $C_3^4\times C_3^2:C_9$ x 2, $C_3^8$

Order 4374: $C_3^6.C_6$ x 68, $C_3\times (C_3^5.C_3):C_2$ x 48, $C_3\times C_3^6:C_2$ x 41, $C_3^5.C_3.C_6$ x 30, $C_3^2\times C_3^2:(C_3^3:C_2)$ x 21, $C_3^7.C_2$ x 18, $C_3\times (C_3\times C_3\times C_3^2:C_9):C_2$ x 11, $C_3^4.C_3^3.C_2$ x 10, $C_3^3.\He_3.C_6$ x 7, $C_3^4.C_3^2.C_6$ x 6, $C_3^3.C_3^4.C_2$ x 4, $C_3\times (C_3^4.C_3^2):C_2$ x 4, $C_3\times (C_3\times C_3^3:C_9):C_2$ x 4, $C_3^2.C_3^4.C_6$ x 3, $(C_3^2\times C_9^2).C_6$ x 3, $C_3^3.\He_3:S_3$ x 2, $C_3^2\times C_3^2\times C_3^3:C_2$ x 2, $C_3^3.C_3^3.C_6$, $C_3\times (C_3^4\times C_9):C_2$, $(C_3^3\times C_9).C_3.C_6$

Order 2916: $C_3^5:D_6$ x 57, $C_3^6.C_2^2$ x 13, $C_3^5.D_6$ x 11, $C_3\times C_3^5:C_2^2$ x 7, $S_3\times C_3^3:C_{18}$ x 4, $C_3^4:D_{18}$ x 2, $C_3^4:D_{18}$ x 2, $C_3^4:D_{18}$ x 2, $C_3^3:(S_3\times C_{18})$ x 2, $C_9^2:S_3^2$, $C_3^4:D_{18}$, $C_3^4:(C_2\times C_{18})$

Order 2187: $C_3^7$ x 4, $C_3^6.C_3$ x 4, $C_3\times C_3^5.C_3$ x 2

Order 1458: $C_3^5:S_3$

Order 729: $C_3^6$ x 6, $C_3^4:C_9$ x 2

Order 243: $C_3^5$ x 6

Order 81: $C_3^4$ x 6

Order 27: $C_3^3$ x 6

Order 9: $C_3^2$ x 6

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 236196: $C_3^6.(C_9\times D_{18})$ ($C_1$)

Order 118098: $C_3^5.C_3^3.C_3.C_6$ ($C_2$) x 2, $C_3^8.C_3.C_6$ ($C_2$)

Order 78732: $C_3^7.D_{18}$ ($C_3$)

Order 59049: $C_3^8.C_3^2$ ($C_2^2$)

Order 39366: $C_3^4.C_3^4.C_6$ ($S_3$) x 2, $C_3^8.C_6$ ($C_6$), $C_3^5.C_3^4.C_2$ ($C_6$), $C_3\times C_3^7.C_6$ ($S_3$), $C_3\times C_3^5.C_3^3.C_2$ ($C_6$)

Order 26244: $C_3^6.D_{18}$ ($C_9$)

Order 19683: $C_3^7.C_3^2$ ($D_6$) x 2, $C_3\times C_3^7.C_3$ ($C_2\times C_6$), $C_3\times C_3^7.C_3$ ($D_6$)

Order 13122: $C_3^5.C_3^3.C_2$ ($C_{18}$) x 2, $C_3^7.C_6$ ($C_{18}$), $C_3^7.C_6$ ($D_9$), $C_3^4.C_3^4.C_2$ ($C_3\times S_3$), $C_3\times C_3^7.C_2$ ($C_3\times S_3$), $C_3\times C_3^3.\He_3:S_3$ ($C_3\times S_3$)

Order 6561: $C_3\times C_3^6.C_3$ ($C_6\times S_3$) x 2, $C_3\times C_3^6.C_3$ ($S_3^2$) x 2, $C_3^7.C_3$ ($C_2\times C_{18}$), $C_3^7.C_3$ ($D_{18}$), $C_3^6.C_3^2$ ($S_3^2$), $C_3^8$ ($C_6\times S_3$)

Order 4374: $C_3^3.\He_3:S_3$ ($S_3\times C_9$) x 2, $C_3^7.C_2$ ($C_3\times D_9$), $C_3\times C_3^6:C_2$ ($S_3\times C_9$)

Order 2187: $C_3^7$ ($C_3\times S_3^2$) x 2, $C_3^6.C_3$ ($S_3\times C_{18}$) x 2, $C_3^6.C_3$ ($S_3\times D_9$) x 2, $C_3^7$ ($S_3\times C_{18}$), $C_3^7$ ($C_3\times D_{18}$), $C_3\times C_3^5.C_3$ ($C_3:S_3^2$), $C_3\times C_3^5.C_3$ ($C_3\times S_3^2$)

Order 1458: $C_3^5:S_3$ ($C_9\times D_9$)

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

Order 243: $C_3^5$ ($C_3^3:S_3^2$) x 2, $C_3^5$ ($C_9^2:D_6$) x 2, $C_3^5$ ($C_9\times C_3:S_3^2$), $C_3^5$ ($C_3^3:D_{18}$)

Order 81: $C_3^4$ ($C_3^4.C_6.C_6$) x 2, $C_3^4$ ($C_3^3:S_3.C_{18}$) x 2, $C_3^4$ ($C_9^2:S_3^2$), $C_3^4$ ($C_3^4:S_3^2$)

Order 27: $C_3^3$ ($C_3^4.C_6.C_{18}$) x 2, $C_3^3$ ($C_3^5:(C_6\times S_3)$) x 2, $C_3^3$ ($C_3^5.C_6.C_6$), $C_3^3$ ($C_3^4:C_3.C_6.C_6$)

Order 9: $C_3^2$ ($C_3^6.C_6.C_6$) x 2, $C_3^2$ ($C_3^5.C_6.C_{18}$) x 2, $C_3^2$ ($C_3^4:(C_9\times D_{18})$), $C_3^2$ ($\He_3^2:S_3^2$)

Order 3: $C_3$ ($C_3^6.C_6.C_{18}$) x 3, $C_3$ ($C_3^7.C_6.C_6$)

Order 1: $C_1$ ($C_3^6.(C_9\times D_{18})$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

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

The $801 \times 801$ character table is not available for this group.

Rational character table

The $277 \times 277$ rational character table is not available for this group.