C₂×#72232148 #72325729

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

Aliases

direct product

C₂×#72232148

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

At least 25824 conjugacy classes of subgroups; only the 42584 normal subgroups are shown.

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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232191
maximal normal subgroup of	C₂×#72232191

GAP

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

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

C2×#72232148 #72325729