C₂²×#7985294 #72301656

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

direct product

C₂²×#7985294

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

803628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 8 generators on 28 points:

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

Conjugacy classes & character table

404 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#72236189, C₂³×#7985294
maximal normal subgroup of	C₂²×#72236189, C₂³×#7985294
maximal quotient of	C₂³×#7985294

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

C22×#7985294 #72301656