C₂²×AGL(2,𝔽₅) #72275698

order	48000
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

non-solvable

Aliases

direct product

C₂²×AGL(2,𝔽₅)

Splittings

Fitting, socle	C₂²×C₅² ⋊ GL(2,𝔽₅)
derived	#72209598 ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂²	AGL(2,𝔽₅)
commutator, hypocenter	#72209598	C₂²×C₄
Frattini	C₁	G
Fitting, socle	C₂²×C₅²	GL(2,𝔽₅)
prime radical	C₂²×#72210652	C₂
cosocle	#72210652	C₂³
LCS	G ▷ #72209598
UCS	C₁ ◁ C₂²

2044 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 2044 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 29
transitive degree	μ_t(G) = 100

A minimal faithful permutation representation — 3 generators on 29 points:

(3,4)(5,6,8,22)(7,11,19,29)(9,14,25,27)(10,15,16,26)(12,13)(17,23)(18,28,24,21);(1,2)(3,4)(6,10,16,29,12,19,26,21,7,14,25,24,23,18,17,20,9,27,11,13,22,28,8,15);(3,4)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

automorphism group of

#6029, C₄×C₅⋊D₅, C₂²⋊(C₅⋊D₅)

GAP

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

gap> G := Group((3,4)(5,6,8,22)(7,11,19,29)(9,14,25,27)(10,15,16,26)(12,13)(17,23)(18,28,24,21), (1,2)(3,4)(6,10,16,29,12,19,26,21,7,14,25,24,23,18,17,20,9,27,11,13,22,28,8,15), (3,4));