C₂×#72237108 #72334485

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

Aliases

direct product

C₂×#72237108

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 11294 conjugacy classes of subgroups; only the 41024 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 26 points:

(3,4)(5,6)(7,8,9,10)(11,12,15,16)(13,17,18,14)(19,20,22,23)(21,24,25,26);(4,6)(8,10)(13,18)(14,17)(21,25)(24,26);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18)(16,17)(20,23)(21,25);(19,20)(21,24)(22,23)(25,26);(1,2);(8,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,26)(22,23)(24,25);(3,4)(5,6)(7,8)(9,10)(11,14,15,17)(12,13,16,18)(19,21,22,25)(20,24,23,26)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72237114, C₂×#72334382
maximal normal subgroup of	C₂×#72237114, C₂×#72334382

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)(11,12,15,16)(13,17,18,14)(19,20,22,23)(21,24,25,26), (4,6)(8,10)(13,18)(14,17)(21,25)(24,26), (3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18)(16,17)(20,23)(21,25), (19,20)(21,24)(22,23)(25,26), (1,2), (8,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,26)(22,23)(24,25), (3,4)(5,6)(7,8)(9,10)(11,14,15,17)(12,13,16,18)(19,21,22,25)(20,24,23,26));

C2×#72237108 #72334485