C₂×#5021145 #72295178

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

Aliases

direct product

C₂×#5021145

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

124825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293925
maximal normal subgroup of	C₂×#72293925
maximal quotient of	C₂×#72293979

GAP

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

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

C2×#5021145 #72295178