C₂×#72232049 #72324463

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#72232049

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

657112 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232064
maximal normal subgroup of	C₂×#72232064

GAP

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

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

C2×#72232049 #72324463