C₂×#612803 #72334534

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#612803

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2847	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

76089 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×D₄×#2849, C₂×#72334290, C₂×#72334291, …
maximal normal subgroup of	C₂×D₄×#2849, C₂×#72334290, C₂×#72334291, …
maximal quotient of	C₂×#72334310, C₂×#72334426

GAP

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

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

C2×#612803 #72334534