C₂×#72237112 #72334490

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹×C₄

Aliases

direct product

C₂×#72237112

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

948872 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:

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

Conjugacy classes & character table

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

Relationships

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

GAP

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

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

C2×#72237112 #72334490