C₂×#72298091 #72299573

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72298091

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 22074 conjugacy classes of subgroups; only the 38120 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298104
maximal normal subgroup of	C₂×#72298104

GAP

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

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

C2×#72298091 #72299573