C₂×#72334389 #72334752

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁷×C₄

Aliases

direct product

C₂×#72334389

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁶×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

837784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

416 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×#72334421
maximal normal subgroup of	C₂×#72334421

GAP

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

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

C2×#72334389 #72334752