#120×#1035 #72275581

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

#120×#1035

Splittings

Fitting	C₂₅×#120 ⋊ C₄
Sylow	C₂₅ ⋊ C₄×#120

Subgroups

	subgroup	G/H
center	C₂²	C₂³×#1035
commutator	C₂²×C₂₅	C₂³×C₄
Frattini, socle	C₂²×C₅	C₂³×AGL(1,𝔽₅)
Fitting	C₂₅×#120	C₄
prime radical	#120×D₂₅	C₂
cosocle	C₂²×D₂₅	C₂⁴
hypercenter	#120	#1035
hypocenter	C₂₅	C₄×#120
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂² ◁ #120

1836 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 1836 conjugacy classes of subgroups — too many to draw. Only its 236 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) = 33
transitive degree	μ_t(G) = 200

A minimal faithful permutation representation — 4 generators on 33 points:

(6,8)(9,27,29,33,32,11,10,31,12,30,18,15,14,21,20,16,23,19,24,13,17,26,28,25,22);(10,12,13,19)(11,16,17,18)(14,23,21,30)(15,25,20,29)(22,31,27,24)(26,33,32,28);(2,4);(1,2)(3,4)(5,6)(7,8)

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×AGL(1,ℤ/25), B(4,𝔽₂)×#1035
maximal normal subgroup of	#120×AGL(1,ℤ/25), B(4,𝔽₂)×#1035
maximal quotient of	B(4,𝔽₂)×#1035, #453×#1035

GAP

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

gap> G := Group((6,8)(9,27,29,33,32,11,10,31,12,30,18,15,14,21,20,16,23,19,24,13,17,26,28,25,22), (10,12,13,19)(11,16,17,18)(14,23,21,30)(15,25,20,29)(22,31,27,24)(26,33,32,28), (2,4), (1,2)(3,4)(5,6)(7,8));