C₂×#7648691 #72324848

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#7648691

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 6350 conjugacy classes of subgroups; only the 32077 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232067
maximal normal subgroup of	C₂×#72232067
maximal quotient of	C₂×#72232179

GAP

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

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

C2×#7648691 #72324848