C₂²×#517592 #72340083

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

Aliases

direct product

C₂²×#517592

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂³	C₂⁶×C₄
Frattini, cosocle	C₂⁴	C₂⁷
prime radical	C₂⁵×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4740088 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72337391
maximal normal subgroup of	C₂×#72337391
maximal quotient of	C₂×#72337391, C₂×#72337376

GAP

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

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

C22×#517592 #72340083