C₂²×#62652 #72299726

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷×C₄ · Schur multiplier (H₂) = C₂²⁷

Aliases

direct product

C₂²×#62652

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂	C₂⁷×C₄
Frattini, cosocle	C₂²	C₂⁸
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂² ▷ C₁

843517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 24 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂²×#10586066
maximal normal subgroup of	C₂²×#10586066
maximal quotient of	C₂²×D₄×#518, C₂²×C₂²⋊C₄×H(5,𝔽₂), C₂²×#7626568

GAP

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

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

C22×#62652 #72299726