C₂×#72298481 #72334591

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#72298481

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂³×#2403	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6088 conjugacy classes of subgroups; only the 25384 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72298627
maximal normal subgroup of	C₂×#72298627

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

C2×#72298481 #72334591