C₅×#10615127 #72226508

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₅ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₅×#10615127

Splittings

Fitting	C₅²×#535 ⋊ C₂
Sylow	C₅² ⋊ #3027

Subgroups

	subgroup	G/H
center	C₂×C₅	C₂×#4083
commutator, cosocle	C₂³×C₅	C₂⁴×C₅
Frattini	C₂³	C₂³×C₅×D₅
Fitting	C₅²×#535	C₂
socle	C₂×C₅²	C₂×#120
hypercenter	C₅×#535	D₅
hypocenter	C₅	C₅×#3027
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#535

1878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	#72226514
maximal normal subgroup of	#72226514

GAP

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

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

C5×#10615127 #72226508