C₂×#2109341 #72338046

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

direct product

C₂×#2109341

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

36743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

(16,19)(17,21)(23,24,27,25)(26,29,30,28);(4,5)(9,13)(12,14)(16,19)(17,21)(24,25)(26,30);(4,5)(8,11)(9,13)(17,21)(20,22)(26,30)(28,29);(1,2);(3,4,6,5)(7,8)(9,14)(10,11)(12,13)(15,16)(17,20)(18,19)(21,22)(23,25,27,24)(26,29,30,28);(3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,17,18,21)(16,20,19,22)(23,26,27,30)(24,28,25,29)

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232334, C₂×#72232336, C₂×#72232365
maximal normal subgroup of	C₂×#72232334, C₂×#72232336, C₂×#72232365

GAP

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

gap> G := Group((16,19)(17,21)(23,24,27,25)(26,29,30,28), (4,5)(9,13)(12,14)(16,19)(17,21)(24,25)(26,30), (4,5)(8,11)(9,13)(17,21)(20,22)(26,30)(28,29), (1,2), (3,4,6,5)(7,8)(9,14)(10,11)(12,13)(15,16)(17,20)(18,19)(21,22)(23,25,27,24)(26,29,30,28), (3,5,6,4)(7,9)(8,12)(10,13)(11,14)(15,17,18,21)(16,20,19,22)(23,26,27,30)(24,28,25,29));

C2×#2109341 #72338046