C₂×#72233526 #72298790

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

Aliases

direct product

C₂×#72233526

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

992406 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)(15,16,18,19)(17,21,22,20);(4,6)(8,10)(11,12)(13,14)(17,22)(20,21);(7,8)(9,10)(16,19)(20,21);(7,8)(9,10)(15,16)(17,20)(18,19)(21,22);(7,8)(9,10);(3,4)(5,6)(7,8)(9,10)(15,17)(16,20)(18,22)(19,21);(1,2)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72233580
maximal normal subgroup of	C₂×#72233580

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

C2×#72233526 #72298790