C₂²×#43372 #72301360

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

Aliases

direct product

C₂²×#43372

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

79553 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)(7,10)(8,9)(11,12)(13,14,16,17)(15,19,20,18)(21,22,24,26)(23,27,28,25);(6,9)(7,11)(14,17)(15,20)(23,28)(25,27);(14,17)(18,19)(21,23)(22,25)(24,28)(26,27);(5,6)(7,12)(8,9)(10,11)(13,15,16,20)(14,18,17,19)(21,23)(22,25)(24,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,27)(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×#8757652, C₂²×#8598997
maximal normal subgroup of	C₂²×#8757652, C₂²×#8598997
maximal quotient of	C₂²×#540671

GAP

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

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

C22×#43372 #72301360