C₂×C₉×#63710 #72268896

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₉ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂×C₉×#63710

Splittings

Fitting	C₂²×D₄×C₉² ⋊ C₂
Sylow	C₉² ⋊ C₂×H(5,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂²×C₉	C₂³×D₉
commutator	C₂×C₉	C₂⁵×C₉
Frattini	C₂×C₃²	C₂⁴×C₃×D₃
Fitting	C₂²×D₄×C₉²	C₂
socle	C₂²×C₃²	C₂³×C₃×D₃
prime radical	C₂×C₃×#63710	C₃
cosocle	C₂×C₃×C₉	C₂⁵×C₃
hypercenter	C₂²×D₄×C₉	D₉
hypocenter	C₉	C₂×C₉×H(5,𝔽₂)
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₉ ◁ C₂²×D₄×C₉

4867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 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);(4,7)(8,10);(1,2);(5,9)(8,10);(3,4)(5,8)(6,7)(9,10)(21,23)(22,25)(24,26)(27,28)

Conjugacy classes & character table

1026 conjugacy classes. The character table is too large to display.

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), (4,7)(8,10), (1,2), (5,9)(8,10), (3,4)(5,8)(6,7)(9,10)(21,23)(22,25)(24,26)(27,28));

C2×C9×#63710 #72268896