C₂×#13047416 #72268813

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

Aliases

direct product

C₂×#13047416

Splittings

Fitting	C₂²×C₄×C₉² ⋊ C₂
Sylow	C₉² ⋊ C₂×#41

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₉⋊D₉
commutator, cosocle	C₂×C₉²	C₂⁴
Frattini	C₂×C₃²	C₂³×C₃⋊D₃
Fitting	C₂²×C₄×C₉²	C₂
socle	C₂²×C₃²	C₂²×C₃⋊D₃
hypercenter	C₂²×C₄	C₉⋊D₉
hypocenter	C₉²	C₂×#41
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

1886 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1886 conjugacy classes of subgroups — too many to draw. Only its 488 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 1296

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#41×C₉⋊D₉, C₂×#72244240, C₂×#72244713
maximal normal subgroup of	C₂×#41×C₉⋊D₉, C₂×#72244240, C₂×#72244713

GAP

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

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