#72227889

order	7680
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

non-solvable

Aliases

No aliases for this group.

Splittings

derived

#72209606 ⋊ C₂

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄	#11757163
commutator, hypocenter	#72209606	C₂
Frattini, socle	C₂²	C₂×#11757163
Fitting	C₂×#585	A₅
cosocle	C₂×2₋¹⁺⁴	C₂×A₅
LCS	G ▷ #72209606
UCS	C₁ ◁ C₂×C₄

597 conjugacy classes of subgroups; only the 15 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 15 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) = 88
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 88 points:

(1,2,6,15,7)(3,10,20,11,14)(4,8,17,9,19)(5,12,22,13,24)(27,37,45,43,66)(28,29,42,72,69)(30,48,78,40,38)(31,51,52,83,80)(32,46,39,54,79)(33,34,49,68,75)(35,60,65,56,55)(36,62,63,73,67)(41,50,58,71,81)(44,77,76,57,61)(53,84,86,74,85)(64,70,87,82,88);(1,5,4,14)(2,9,8,15)(3,13,12,20)(6,18,17,16)(7,11,19,24)(10,23,22,21)(25,31,54,51)(26,36,43,62)(27,41,75,63)(28,44,29,47)(30,50,66,53)(32,57,69,52)(33,58,34,59)(35,61,79,64)(37,67,42,76)(38,70,39,71)(40,73,60,74)(45,77,55,84)(46,80,49,81)(48,82,56,83)(65,85,72,87)(68,86,78,88)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72227900

GAP

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

gap> G := Group((1,2,6,15,7)(3,10,20,11,14)(4,8,17,9,19)(5,12,22,13,24)(27,37,45,43,66)(28,29,42,72,69)(30,48,78,40,38)(31,51,52,83,80)(32,46,39,54,79)(33,34,49,68,75)(35,60,65,56,55)(36,62,63,73,67)(41,50,58,71,81)(44,77,76,57,61)(53,84,86,74,85)(64,70,87,82,88), (1,5,4,14)(2,9,8,15)(3,13,12,20)(6,18,17,16)(7,11,19,24)(10,23,22,21)(25,31,54,51)(26,36,43,62)(27,41,75,63)(28,44,29,47)(30,50,66,53)(32,57,69,52)(33,58,34,59)(35,61,79,64)(37,67,42,76)(38,70,39,71)(40,73,60,74)(45,77,55,84)(46,80,49,81)(48,82,56,83)(65,85,72,87)(68,86,78,88));