C₂²×#15468 #72296040

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂²×#15468

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

56381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#1082853, C₂²×#2108839, C₂²×#4122655
maximal normal subgroup of	C₂²×#1082853, C₂²×#2108839, C₂²×#4122655

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

C22×#15468 #72296040