C₂×#7641850 #72334332

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

Aliases

direct product

C₂×#7641850

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

114501 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72235095, C₂²×#7641850
maximal normal subgroup of	C₂×#72235095, C₂²×#7641850
maximal quotient of	C₂×#72256575, C₂×#72237102, C₂×#72237112, …

GAP

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

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

C2×#7641850 #72334332