C₂×#72232050 #72324482

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

Aliases

direct product

C₂×#72232050

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 6214 conjugacy classes of subgroups; only the 39848 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:

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

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

C2×#72232050 #72324482