C₂²×#758017 #72293301

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂²×#758017

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

765424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 28 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72210469
maximal normal subgroup of	C₂²×#72210469

GAP

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

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

C22×#758017 #72293301