Abstract group 26244.hm: $C_3^6.D_{18}$

• Label: 26244.hm
• Order: \( 2^{2} \cdot 3^{8} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{5} \)
• Perm deg.: $27$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.D_{18}$
Order:	\(26244\)\(\medspace = 2^{2} \cdot 3^{8} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$\GL(2,4):C_2^4$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \)
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	1215	2186	14094	4374	4374	26244
Conjugacy classes	1	3	200	27	15	3	249
Divisions	1	3	151	27	4	1	187
Autjugacy classes	1	2	15	4	3	1	26

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	4	12	17	48	0	168	0	249
Irr. rational chars.	4	6	3	50	1	74	49	187

Minimal presentations

Permutation degree:	$27$
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	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{18}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,2,4,8,7,5)(3,6,11,14,9,12)(10,13,16,18,15,17)(19,20,22,24,27,23)(25,26), (2,3,5,9,8,11)(4,7)(6,10,12,15,14,16)(17,18)(20,21,23,25,24,26)\rangle$
Transitive group:	36T12844				more information
Direct product:	not computed
Semidirect product:	$(C_3^7.C_3)$ $\,\rtimes\,$ $C_2^2$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_3^7$ . $D_6$	$C_3^6$ . $D_{18}$	$C_3^6$ . $S_3^2$ (2)	$(C_3^6.S_3)$ . $S_3$ (3)	all 24

Elements of the group are displayed as permutations of degree 27.

Homology

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

Subgroups

There are 68 normal subgroups (14 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 $C_1$
Commutator:	a subgroup isomorphic to 6561.86251
Frattini:	a subgroup isomorphic to $C_3^5$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7.C_3$

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 36 (order at least 729) are shown, as well as all normal subgroups of any index.

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

Order 13122: $C_3^6.C_3.C_6$ x 2, $C_3^7.C_6$

Order 8748: $C_3^5:D_{18}$ x 2, $C_3^6.D_6$

Order 6561: $C_3^7.C_3$

Order 4374: $C_3^7.C_2$ x 3, $C_3^6.C_6$ x 3, $C_3^3.\He_3.C_6$ x 3, $C_3^6.S_3$ x 2, $C_3^4.C_3^3.C_2$

Order 2916: $C_3^5.D_6$ x 27, $C_3^4:D_{18}$ x 6, $C_3^4:D_{18}$

Order 2187: $C_3^6.C_3$ x 6, $C_3^7$

Order 1458: $C_3^5:C_6$ x 75, $C_3^3\wr C_2$ x 54, $C_3^5:C_6$ x 26, $C_3^4:D_9$ x 12, $C_3^4:C_{18}$ x 9, $C_3^4:D_9$ x 6, $C_3^4:D_9$ x 4, $C_3^4:C_{18}$, $C_3^5:S_3$

Order 972: $C_3^3:S_3^2$ x 169, $C_3^3:S_3^2$ x 26, $C_3^3:S_3^2$ x 26, $C_3^3:D_{18}$ x 6, $C_3^3:D_{18}$ x 2

Order 729: $C_3^6$ x 151, $C_3^4:C_9$ x 18, $C_3^4:C_9$ x 4

Order 243: $C_3^5$ x 9

Order 81: $C_3^4$ x 15

Order 27: $C_3^3$ x 15

Order 9: $C_3^2$ x 9

Order 3: $C_3$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 26244: $C_3^6.D_{18}$ ($C_1$)

Order 13122: $C_3^6.C_3.C_6$ ($C_2$) x 2, $C_3^7.C_6$ ($C_2$)

Order 6561: $C_3^7.C_3$ ($C_2^2$)

Order 4374: $C_3^6.S_3$ ($S_3$) x 2, $C_3^7.C_2$ ($S_3$)

Order 2187: $C_3^6.C_3$ ($D_6$) x 2, $C_3^7$ ($D_6$)

Order 1458: $C_3^5:S_3$ ($D_9$)

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

Order 243: $C_3^5$ ($C_3^2:D_6$) x 6, $C_3^5$ ($S_3\times D_9$) x 2, $C_3^5$ ($C_3:S_3^2$)

Order 81: $C_3^4$ ($C_3^3:D_6$) x 6, $C_3^4$ ($C_3^2:D_{18}$) x 6, $C_3^4$ ($C_3^2:S_3^2$) x 2, $C_3^4$ ($C_9:S_3^2$)

Order 27: $C_3^3$ ($C_3^3:S_3^2$) x 6, $C_3^3$ ($C_3^3:D_{18}$) x 6, $C_3^3$ ($C_3^3:D_{18}$) x 2, $C_3^3$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($C_3^4:D_{18}$) x 6, $C_3^2$ ($C_3^5:D_6$) x 2, $C_3^2$ ($C_3^4:D_{18}$)

Order 3: $C_3$ ($C_3^6.D_6$) x 2, $C_3$ ($C_3^6.D_6$)

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

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 7 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $249 \times 249$ 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 $187 \times 187$ rational character table (warning: may be slow to load).