C₂²×#43668 #72296648

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

Aliases

direct product

C₂²×#43668

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

68981 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)(7,11)(8,9)(10,12)(13,14,17,19)(15,16,18,20)(21,22)(23,24)(25,26)(27,28);(6,9)(10,11)(21,23)(22,24)(25,28)(26,27);(22,26)(23,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,27)(24,26)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8197496
maximal normal subgroup of	C₂²×#8197496
maximal quotient of	C₂²×#1427668

GAP

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

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

C22×#43668 #72296648