C₂²×#520464 #72340087

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²³

Aliases

direct product

C₂²×#520464

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁶×#41	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4702488 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337391
maximal normal subgroup of	C₂×#72337391

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

C22×#520464 #72340087