C₂²×#56384 #72301654

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#56384

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

80625 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#8698894, C₂²×#8195808, C₂³×#56384, …
maximal normal subgroup of	C₂²×#8698894, C₂²×#8195808, C₂³×#56384, …
maximal quotient of	C₂³×#56384, C₂²×#2814158

GAP

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

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

C22×#56384 #72301654