C₅×#73138 #14510862

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

solvable

Aliases

direct product

C₅×#73138

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	#5415
commutator, cosocle	#131⋊C₃	C₂²×C₅
Frattini	C₈	C₂×C₅×S₄
Fitting	C₅×#576	D₃
prime radical	C₅×#576⋊C₃	C₂
hypercenter	C₅×Q₁₆	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×Q₁₆

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

normal subgroups only

This group has 184 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) = 101
transitive degree	μ_t(G) = 480

A minimal faithful permutation representation — 2 generators on 101 points:

(1,2,3,4,5)(6,33,90,25,57,50,13,38,59,8,36,35)(7,88,74,42,85,101,20,44,70,14,51,95)(9,68,83,63,62,86,31,29,93,27,67,97)(10,11,79,23,60,71,12,37,92,32,24,78)(15,53,65,49,41,75,46,17,94,43,84,58)(16,61,77,40,54,72,18,87,34,47,55,99)(19,82,80,45,52,73,21,89,76,26,48,98)(22,56,66,28,64,100,39,81,91,30,69,96);(6,7,13,20)(8,21,25,19)(9,26,31,45)(10,18,12,16)(11,34,37,77)(14,39,42,22)(15,30,46,28)(17,50,53,35)(23,49,32,43)(24,58,60,75)(27,47,63,40)(29,65,68,94)(33,70,38,74)(36,80,57,76)(41,83,84,93)(44,86,88,97)(48,71,52,78)(51,79,85,92)(54,59,55,90)(56,72,81,99)(61,91,87,66)(62,95,67,101)(64,73,69,98)(82,96,89,100)

Conjugacy classes & character table

125 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1920, 236560);