C₂×#3942518 #72337901

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

Aliases

direct product

C₂×#3942518

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

34749 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232318, C₂×#72232336, C₂×#72232339
maximal normal subgroup of	C₂×#72232318, C₂×#72232336, C₂×#72232339

GAP

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

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

C2×#3942518 #72337901