C₂×#6342374 #72286858

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#6342374

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

425483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72256576, C₂×#72233691, #72287899, …
maximal normal subgroup of	C₂×#72256576, C₂×#72233691, #72287899, …
maximal quotient of	#72287899, C₂×#72286433, C₂²×#6342374, …

GAP

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

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

C2×#6342374 #72286858