C₂×#613453 #72341651

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#613453

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

60541 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72303276, C₂×#72303265, C₂×#72260984
maximal normal subgroup of	C₂×#72303276, C₂×#72303265, C₂×#72260984

GAP

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

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

C2×#613453 #72341651