C₂×#687337 #72294693

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

Aliases

direct product

C₂×#687337

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

33725 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293944
maximal normal subgroup of	C₂×#72293944
maximal quotient of	C₂×#72293952

GAP

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

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

C2×#687337 #72294693