C₂×#7644775 #72330609

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

Aliases

direct product

C₂×#7644775

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

111649 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232294
maximal normal subgroup of	C₂×#72232294
maximal quotient of	C₂×#72323281, C₂×#72323279, C₂×#72323283

GAP

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

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

C2×#7644775 #72330609