C₂²×#46387 #72296031

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂²×#46387

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

88973 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8898065
maximal normal subgroup of	C₂²×#8898065
maximal quotient of	C₂²×#4387614

GAP

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

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

C22×#46387 #72296031