C₂×#7648649 #72324854

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

Aliases

direct product

C₂×#7648649

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 6382 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232067, C₂×#72232285
maximal normal subgroup of	C₂×#72232067, C₂×#72232285
maximal quotient of	C₂×#72232164

GAP

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

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

C2×#7648649 #72324854