#6953

order	250
isoclinism	isoclinic class of #6953
automorphism	Aut(G) = AGL(2,𝔽₅) (order 12000) · Inn(G) = C₅⋊D₅ · Out(G) = A₅⋊C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₅) ⋊ C₂

Subgroups

	subgroup	G/H
center, Frattini, socle, hypercenter	C₅	C₅⋊D₅
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₅)	C₂
LCS	G ▷ H(3,𝔽₅)
UCS	C₁ ◁ C₅

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 25
transitive degree	μ_t(G) = 25

A minimal faithful permutation representation — 3 generators on 25 points:

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

Polycyclic presentation:

2,5^3|b^a=b^4,c^a=c^4,c^b=cd

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

derived subgroup of	(((C₅⋊D₅)⋊C₅)⋊C₂)⋊C₂, #11758708, #14522308, …
maximal subgroup of	#92537, #92539, ((C₅⋊D₅)⋊C₅)⋊C₂, …
maximal normal subgroup of	#92537, #92539, ((C₅⋊D₅)⋊C₅)⋊C₂, …
maximal quotient of	#92525, C₂×#6953, C₃×#6953, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(250, 8);