C₂×#7642451 #72324366

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

Aliases

direct product

C₂×#7642451

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

93869 conjugacy classes of subgroups.

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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232032
maximal normal subgroup of	C₂×#72232032
maximal quotient of	C₂×#72232056

GAP

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

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

C2×#7642451 #72324366