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#67×C₂₅ #10597043

order	600
isoclinism	isoclinic class of Dic₆
automorphism	?
homology	Abelianization (H₁) = C₂²×C₂₅ · Schur multiplier (H₂) = C₂

Aliases

#67×C₂₅

Splittings

Fitting	C₂²×C₃×C₂₅ ⋊ C₂
Hall	C₃×C₂₅ ⋊ D₄
Sylow	C₃ ⋊ D₄×C₂₅

Subgroups

	subgroup	G/H
center	C₂×C₂₅	C₂×D₃
commutator	C₂×C₃	C₂²×C₂₅
Frattini	C₂×C₅	C₂×C₅×D₃
Fitting	C₂²×C₃×C₂₅	C₂
socle, cosocle	C₂×C₃×C₅	C₂²×C₅
prime radical	C₅×#67	C₅
hypercenter	C₂²×C₂₅	D₃
hypocenter	C₃	D₄×C₂₅
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₂₅ ◁ C₂²×C₂₅

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 300

A minimal faithful permutation representation — 2 generators on 32 points:

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

Conjugacy classes & character table

225 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#11924926, #11924930, #67×D₂₅, …
maximal normal subgroup of	#11924926, #11924930, #67×D₂₅, …
maximal quotient of	C₂₅×Dic₃⋊C₄, C₂₅×(C₂×D₃)⋊C₄, C₂₅×#213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(600, 24);