C₂²×#1268761 #72296941

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂²×#1268761

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 7257 conjugacy classes of subgroups; only the 38784 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#72231349
maximal normal subgroup of	C₂²×#72231349

GAP

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

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

C22×#1268761 #72296941