C₂×#72323285 #72331265

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰×C₄

Aliases

direct product

C₂×#72323285

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 10788 conjugacy classes of subgroups; only the 42696 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72323292
maximal normal subgroup of	C₂×#72323292

GAP

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

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

C2×#72323285 #72331265