C₂²×(#4178⋊C₂)⋊C₃ #72229277

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×(#4178⋊C₂)⋊C₃

Splittings

Fitting	C₂²×#6911 ⋊ C₂²
derived	#732 ⋊ C₂⁴×C₃
Sylow	#6911 ⋊ C₂⁴

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	(#4178⋊C₂)⋊C₃
commutator, cosocle, hypocenter	#732	C₂⁴×C₃
Frattini	C₃²	C₂²×C₃×D₃²
Fitting	C₂²×#6911	C₂²
socle	C₂²×C₃	C₃×((C₃⋊D₃)⋊C₃)⋊C₂
LCS	G ▷ #732
UCS	C₁ ◁ C₂²

1685 conjugacy classes of subgroups; only the 223 normal subgroups are shown.

normal subgroups only

This group has 1685 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 31
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 4 generators on 31 points:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#72210336, D₄×(#4178⋊C₂)⋊C₃
maximal normal subgroup of	D₄×(#4178⋊C₂)⋊C₃
maximal quotient of	D₄×(#4178⋊C₂)⋊C₃

GAP

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

gap> G := Group((1,7,23)(2,27,12)(3,25,19)(4,21,26)(5,11,17)(6,10,18)(8,15,9)(13,24,16)(14,22,20)(30,31), (2,6)(3,9)(4,5)(7,17)(8,19)(10,12)(11,23)(13,20)(14,16)(15,24)(18,22)(21,26)(25,27), (2,6)(7,15)(8,16)(11,22)(14,19)(17,24)(18,23)(21,27)(25,26), (28,29));