C₂²×#2108839 #72296051

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

Aliases

direct product

C₂²×#2108839

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

595872 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231761
maximal normal subgroup of	C₂²×#72231761

GAP

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

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

C22×#2108839 #72296051