C₅×#73041 #14510765

order	1920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

solvable

Aliases

direct product

C₅×#73041

Splittings

Fitting

C₅×#504 ⋊ D₃

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5402
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#504	D₃
prime radical	C₅×#5407	C₂
hypercenter	C₅×C₁₆	S₄
hypocenter	SL(2,𝔽₃)	C₅×D₈
LCS	G ▷ #131⋊C₃ ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×C₈ ◁ C₅×C₁₆

170 conjugacy classes of subgroups; only the 32 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 32 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) = 69
transitive degree	μ_t(G) = 320

A minimal faithful permutation representation — 2 generators on 69 points:

(1,2,3,4,5)(6,42,65,35,48,67,8,43,27,18,36,69,20,38,50,34,28,44,14,58,55,15,51,61,12,16,64,31,24,66,23,32,29,7,46,68,9,56,52,13,10,41,19,62,54,30,21,60)(11,17,37,63,25,39,22,33,26,45,47,59,49,57,53,40);(7,13)(8,19)(9,20)(10,26)(11,28)(14,23)(15,35)(16,40)(17,43)(18,34)(21,37)(22,36)(24,25)(27,29)(30,31)(32,45)(33,42)(38,59)(39,58)(41,61)(44,60)(46,53)(47,51)(48,49)(50,64)(52,65)(56,63)(57,62)(66,68)(67,69)

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236463);