C₂×#72334343 #72334695

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

Aliases

direct product

C₂×#72334343

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

917854 conjugacy classes of subgroups.

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334357
maximal normal subgroup of	C₂×#72334357

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

C2×#72334343 #72334695