C₅²×#1313 #72242111

order	3200
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₄×C₅²×C₈ · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product

C₅²×#1313

Subgroups

	subgroup	G/H
center	C₂×C₄×C₅²	C₄²
commutator	C₄	C₄×C₅²×C₈
Frattini, cosocle	C₄×C₈	C₂²×C₅²
socle, prime radical	C₂²×C₅²	#97
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅² ◁ G

320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 34
transitive degree	μ_t(G) = 3200

A minimal faithful permutation representation — 2 generators on 34 points:

(1,2,3,5,4,6,7,8)(10,14,15,20)(11,17)(16,23,22,21)(18,24)(25,26,27,28,29);(9,10,11,16,12,14,18,21,13,15,17,22,19,20,24,23)(30,31,32,33,34)

Conjugacy classes & character table

1100 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₅⋊D₅×#1313, #1043×#1313, #72253106, …
maximal subgroup of	C₅⋊D₅×#1313, C₅²×#12067, #72253106, …
maximal normal subgroup of	C₅⋊D₅×#1313, C₅²×#12067, #72253106, …

GAP

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

gap> G := Group((1,2,3,5,4,6,7,8)(10,14,15,20)(11,17)(16,23,22,21)(18,24)(25,26,27,28,29), (9,10,11,16,12,14,18,21,13,15,17,22,19,20,24,23)(30,31,32,33,34));

C52×#1313 #72242111