C₂⁶×C₅⋊D₅ #72258458

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹×C₅

Aliases

direct product

C₂⁶×C₅⋊D₅

Splittings

Fitting, socle	C₂⁶×C₅² ⋊ C₂
derived, Sylow	C₅² ⋊ C₂⁷

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁶	C₅⋊D₅
commutator, cosocle, hypocenter	C₅²	C₂⁷
Frattini	C₁	G
Fitting, socle	C₂⁶×C₅²	C₂
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂⁶

233696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 1600

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂⁶×#1043, C₅⋊D₅×#2846
maximal normal subgroup of	C₂⁶×#1043, C₅⋊D₅×#2846

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

C26×C5⋊D5 #72258458