C₂×#72237103 #72334475

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72237103

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁶
commutator, Frattini, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1172128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 22 points:

(3,4)(5,6)(7,8)(9,10)(12,14)(15,16)(17,22)(18,21)(19,20);(7,8)(9,10)(12,14)(16,20)(17,21);(4,6)(8,10)(12,14)(17,21)(18,22);(4,6)(8,10)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22);(1,2);(3,4)(5,6)(7,8)(9,10);(15,18)(16,21)(17,20)(19,22)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237114, C₂×D₄×#34980, #72339945
maximal normal subgroup of	C₂×#72237114, C₂×D₄×#34980, #72339945

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)(9,10)(12,14)(15,16)(17,22)(18,21)(19,20), (7,8)(9,10)(12,14)(16,20)(17,21), (4,6)(8,10)(12,14)(17,21)(18,22), (4,6)(8,10)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22), (1,2), (3,4)(5,6)(7,8)(9,10), (15,18)(16,21)(17,20)(19,22));

C2×#72237103 #72334475