A₄×D₂₅⋊C₅ #72228840

order	3000
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product

A₄×D₂₅⋊C₅

Splittings

Fitting	C₂²×5₋¹⁺² ⋊ C₂×C₃
derived	C₂²×C₂₅ ⋊ C₂×C₃×C₅
Hall	C₂²×D₂₅⋊C₅ ⋊ C₃
Sylow	5₋¹⁺² ⋊ C₂×A₄

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₂²×C₂₅	C₂×C₃×C₅
Frattini	C₅	C₅×D₅×A₄
Fitting	C₂²×5₋¹⁺²	C₂×C₃
socle	C₂²×C₅	C₃×C₅×D₅
LCS	G ▷ C₂²×C₂₅
UCS	C₁

82 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×AGL(1,ℤ/25), S₄×D₂₅⋊C₅, #72228881
maximal normal subgroup of	A₄×AGL(1,ℤ/25), S₄×D₂₅⋊C₅, #72228881

GAP

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

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

A4×D25⋊C5 #72228840