C₂²×#4418479 #72296985

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄²

Aliases

direct product

C₂²×#4418479

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

1019616 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:

(5,6)(7,8)(15,19)(18,20);(6,7)(9,10,12,11)(15,19)(18,20)(21,22)(23,27)(24,25)(26,28);(10,11)(14,17)(18,20)(22,25)(23,28);(10,11)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,7,8,6)(9,11,12,10)(13,15,16,19)(14,18,17,20)(21,23,24,28)(22,26,25,27);(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×#72231349, C₂²×#72233705
maximal normal subgroup of	C₂²×#72231349, C₂²×#72233705

GAP

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

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

C22×#4418479 #72296985