C₂²×#847053 #72334501

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂²×#847053

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

979716 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)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(22,25)(26,28);(6,8)(10,12)(13,15)(14,17)(16,20)(18,19)(23,27)(26,28);(5,6,7,8)(10,12)(14,18)(15,20)(21,22)(23,28)(24,25)(26,27);(9,10)(11,12)(13,15)(14,19)(16,20)(17,18);(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,27)(25,28);(3,4);(1,2)(3,4)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72233705
maximal normal subgroup of	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)(9,10)(11,12)(13,14)(15,17)(16,18)(19,20)(22,25)(26,28), (6,8)(10,12)(13,15)(14,17)(16,20)(18,19)(23,27)(26,28), (5,6,7,8)(10,12)(14,18)(15,20)(21,22)(23,28)(24,25)(26,27), (9,10)(11,12)(13,15)(14,19)(16,20)(17,18), (5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,26)(24,27)(25,28), (3,4), (1,2)(3,4));

C22×#847053 #72334501