C₂×#72278106 #72295261

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

direct product

C₂×#72278106

Subgroups

	subgroup	G/H
center, socle	C₂⁶	C₂⁵
commutator, Frattini, cosocle	C₂⁵	C₂⁶
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 17730 conjugacy classes of subgroups; only the 9328 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72293994, C₂×#72278177
maximal normal subgroup of	C₂×#72293994, C₂×#72278177

GAP

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

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

C2×#72278106 #72295261