C₂×#1099965 #72338038

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

Aliases

direct product

C₂×#1099965

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

34669 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232291, C₂×#72232350
maximal normal subgroup of	C₂×#72232318, C₂×#72232291, C₂×#72232350

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

C2×#1099965 #72338038