C₂×#72286443 #72291057

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂×#72286443

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator	C₂⁴	C₂⁵×C₄
Frattini, cosocle	C₂⁵	C₂⁶
prime radical	C₂³×#2846	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2799572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

(15,16,18,17)(20,22);(4,6)(7,8,9,10)(12,14)(16,17)(19,20)(21,22);(1,2);(7,8)(9,10);(4,6)(8,10)(11,12,13,14)(15,17,18,16);(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×#72296159, C₂×#72286712, C₂×#72296169, …
maximal normal subgroup of	C₂×#72296159, C₂×#72286712, C₂×#72296169, …

GAP

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

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

C2×#72286443 #72291057