C₂×#7904581 #72299997

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

Aliases

direct product

C₂×#7904581

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 6838 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:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72238233, C₂×#72297795, C₂×#72304329, …
maximal normal subgroup of	C₂×#72238233, C₂×#72297795, C₂×#72304329, …
maximal quotient of	C₂×#72232055, C₂×#72232074

GAP

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

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

C2×#7904581 #72299997