C₂×#7916677 #72299873

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7916677

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6920 conjugacy classes of subgroups; only the 31893 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 22 points:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297207, C₂×#72233749, C₂×#72232070
maximal normal subgroup of	C₂×#72297207, C₂×#72233749, C₂×#72232070
maximal quotient of	C₂×#72232171

GAP

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

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

C2×#7916677 #72299873