C₂²×#55034 #72296203

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

Aliases

direct product

C₂²×#55034

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

80101 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#9228811, C₂²×#8598997
maximal normal subgroup of	C₂²×#9228811, C₂²×#8598997
maximal quotient of	C₂²×#3332795, C₂²×#2108842

GAP

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

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

C22×#55034 #72296203