C₂×#814132 #72337915

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

direct product

C₂×#814132

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2410	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37957 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:

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

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232291, C₂×#72232339, C₂×#72232314
maximal normal subgroup of	C₂×#72232291, C₂×#72232339, C₂×#72232314

GAP

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

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

C2×#814132 #72337915