C₂²×#37848 #72325882

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂²×#37848

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

163121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8108057
maximal normal subgroup of	C₂²×#8108057

GAP

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

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

C22×#37848 #72325882