C₄×AGL(2,𝔽₅) #72278312

order	48000
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

non-solvable

Aliases

direct product

C₄×AGL(2,𝔽₅)

Splittings

Fitting	C₄×C₅² ⋊ GL(2,𝔽₅)
derived	#72209598 ⋊ C₄²

Subgroups

	subgroup	G/H
center, hypercenter	C₄	AGL(2,𝔽₅)
commutator, hypocenter	#72209598	C₄²
Frattini	C₂	C₂×AGL(2,𝔽₅)
Fitting	C₄×C₅²	GL(2,𝔽₅)
socle	C₂×C₅²	C₂×GL(2,𝔽₅)
prime radical, cosocle	C₂×#72210652	C₂²
LCS	G ▷ #72209598
UCS	C₁ ◁ C₄

1020 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 33 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 — 2 generators on 29 points:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

automorphism group of

C₅×C₅⋊D₅

GAP

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

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