#72227549

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

derived

#13051525 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, hypocenter	#13051525	C₂
Frattini, socle, cosocle	C₂³	C₂×GL(3,𝔽₂)
Fitting	C₂⁴	GL(3,𝔽₂)
LCS	G ▷ #13051525
UCS	C₁

110 conjugacy classes of subgroups; only the 5 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 5 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) = 28

A minimal faithful permutation representation — 2 generators on 28 points:

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

Conjugacy classes & character table

16 conjugacy classes — view the full character table.

Relationships

maximal quotient of	#72227550
automorphism group of	#13051525

GAP

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

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