C₂×#72229306 #72276213

order	5184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#72229306

Subgroups

	subgroup	G/H
center	C₂²	C₂×((C₂²⋊C₂₇)⋊C₂)⋊C₃
commutator, cosocle	C₂×C₂²⋊C₂₇	C₂³×C₃
Frattini	C₂×C₉	C₂²×C₃×S₄
Fitting	C₂⁵×C₃×C₉	D₃
socle	C₂⁴×C₃	C₂×C₃×D₉
hypercenter	C₂³	((C₂²⋊C₂₇)⋊C₂)⋊C₃
hypocenter	C₂²⋊C₂₇	C₂×C₃×D₄
LCS	G ▷ C₂×C₂²⋊C₂₇ ▷ C₂²⋊C₂₇
UCS	C₁ ◁ C₂² ◁ C₂³

1702 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 1702 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 37
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 37 points:

(3,5,6)(8,10)(11,36,30,34,19,33,35,18,28,14,27,29,13,16,15,20,32,25,23,31,37,22,12,24,21,26,17);(4,5)(7,8)(9,10)(12,15)(13,20)(14,23)(16,24)(17,27)(18,30)(19,33)(21,34)(22,35)(25,31)(26,29)(28,36)(32,37);(1,2)

Conjugacy classes & character table

276 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)(8,10)(11,36,30,34,19,33,35,18,28,14,27,29,13,16,15,20,32,25,23,31,37,22,12,24,21,26,17), (4,5)(7,8)(9,10)(12,15)(13,20)(14,23)(16,24)(17,27)(18,30)(19,33)(21,34)(22,35)(25,31)(26,29)(28,36)(32,37), (1,2));