C₂×#72227543 #72227551

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₂×#72227543

Splittings

Fitting, Sylow	C₂×#2867 ⋊ ΓL(1,𝔽₈)
derived	C₄³⋊C₇ ⋊ C₂²×C₃
Hall	C₂×#2867⋊C₇ ⋊ C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#72227543
commutator, cosocle, hypocenter	C₄³⋊C₇	C₂²×C₃
Frattini	C₂³	C₂²×AΓL(1,𝔽₈)
Fitting	C₂×#2867	ΓL(1,𝔽₈)
socle	C₂⁴	C₂×AΓL(1,𝔽₈)
LCS	G ▷ C₄³⋊C₇
UCS	C₁ ◁ C₂

362 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 22 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:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227552, #72227563

GAP

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

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