C₂×C₉×#3699 #72268837

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₉×#3699

Splittings

Fitting	C₂×D₄×C₉² ⋊ C₂
Sylow	C₉² ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂²×D₉
commutator	C₂×C₉	C₂⁴×C₉
Frattini	C₂×C₃²	C₂³×C₃×D₃
Fitting	C₂×D₄×C₉²	C₂
socle	C₂²×C₃²	C₂²×C₃×D₃
prime radical	C₂×C₃×#3699	C₃
cosocle	C₂×C₃×C₉	C₂⁴×C₃
hypercenter	C₂×D₄×C₉	D₉
hypocenter	C₉	C₂×C₉×#41
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₉ ◁ C₂×D₄×C₉

927 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 144

A minimal faithful permutation representation — 4 generators on 28 points:

(3,5,6,9)(4,8,7,10)(11,15,19,13,18,14,16,12,17)(20,27,25,23,24,26,21,22,28);(1,2);(4,7)(5,9);(3,4)(5,8)(6,7)(9,10)(21,23)(22,25)(24,26)(27,28)

Conjugacy classes & character table

540 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72219961, C₂×C₉×#63710, C₂×#72219950
maximal normal subgroup of	C₂×#72219961, C₂×C₉×#63710, C₂×#72219950

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((3,5,6,9)(4,8,7,10)(11,15,19,13,18,14,16,12,17)(20,27,25,23,24,26,21,22,28), (1,2), (4,7)(5,9), (3,4)(5,8)(6,7)(9,10)(21,23)(22,25)(24,26)(27,28));

C2×C9×#3699 #72268837