C₂²×#1420724 #72293312

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

Aliases

direct product

C₂²×#1420724

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

At least 6921 conjugacy classes of subgroups; only the 42696 normal subgroups are shown.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72210469, C₂³×#1420724
maximal normal subgroup of	C₂²×#72210469, C₂³×#1420724
maximal quotient of	C₂³×#1420724

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

C22×#1420724 #72293312