C₂×C₅×5₋¹⁺² #11929362

order	1250
nilpotency class	2
isoclinism	isoclinic class of H(3,𝔽₅)
automorphism	?
homology	Abelianization (H₁) = C₂×C₅³ · Schur multiplier (H₂) = C₅²

Aliases

direct product	C₂×C₅×5₋¹⁺²
unambiguous semidirect product	(C₂×C₂₅)⋊C₅²

Subgroups

	subgroup	G/H
center, socle	C₂×C₅²	C₅²
commutator, Frattini, cosocle	C₅	C₂×C₅³
prime radical	C₂×C₅³	C₅
LCS	G ▷ C₅ ▷ C₁
UCS	C₁ ◁ C₂×C₅² ◁ G

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 250

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

290 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₅×D₂₅⋊C₅, C₄×C₅×5₋¹⁺², C₅×Dic₂₅⋊C₅
maximal normal subgroup of	C₂×C₅×D₂₅⋊C₅, C₄×C₅×5₋¹⁺², C₅×Dic₂₅⋊C₅
maximal quotient of	C₄×C₅×5₋¹⁺²

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1250, 13);

C2×C5×5−1+2 #11929362