C₂×#72252963 #72269045

order	7776
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

solvable

Aliases

direct product

C₂×#72252963

Splittings

derived

C₉×Q₈⋊C₉ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	#14518082
commutator, cosocle, hypocenter	C₉×Q₈⋊C₉	C₂²×C₃
Frattini	C₂×C₃²	C₂×C₃×C₃⋊S₄
Fitting	C₂×C₃×Q₈×3₋¹⁺²	D₃
socle	C₂²×C₃²	C₃×C₃⋊S₄
LCS	G ▷ C₉×Q₈⋊C₉
UCS	C₁ ◁ C₂²

1266 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 124 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 — 3 generators on 28 points:

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

Conjugacy classes & character table

162 conjugacy classes — view the full character table.

GAP

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

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