C₂²×#521530 #72339143

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²²

Aliases

direct product

C₂²×#521530

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1005688 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)(14,17)(15,19)(21,22,24,26)(23,27,28,25);(6,7)(10,12)(14,17)(18,20);(5,7,8,6)(9,10)(11,12)(15,19)(18,20)(23,28)(25,27);(13,14)(15,18)(16,17)(19,20)(21,22)(23,25)(24,26)(27,28);(3,4);(1,2)(3,4);(13,15)(14,18)(16,19)(17,20)(21,23)(22,25)(24,28)(26,27)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337376
maximal normal subgroup of	C₂×#72337376

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

C22×#521530 #72339143