C₂²×#4312024 #72293294

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

Aliases

direct product

C₂²×#4312024

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

872040 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);(9,10)(11,12)(14,17)(15,19)(21,22,25,26)(23,28,27,24);(14,17)(18,20)(22,26)(23,27);(6,8)(9,10,11,12)(15,19)(18,20)(23,27)(24,28);(13,14,16,17)(15,18,19,20)(21,23,25,27)(22,24,26,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,19)(17,20)(21,24)(22,27)(23,26)(25,28);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72210469, C₂³×#4312024
maximal normal subgroup of	C₂²×#72210469, C₂³×#4312024
maximal quotient of	C₂³×#4312024

GAP

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

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

C22×#4312024 #72293294