C₂×#72334345 #72334701

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

Aliases

direct product

C₂×#72334345

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 7234 conjugacy classes of subgroups; only the 39864 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:

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

Conjugacy classes & character table

296 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((8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,25)(22,23)(24,26), (11,13)(12,16)(14,17)(15,18)(19,21)(20,24)(22,25)(23,26), (3,4,5,6)(7,8,10,9)(13,18)(14,16)(19,20,23,22)(21,24,26,25), (4,6)(12,17)(14,16)(19,21)(20,25)(22,24)(23,26), (3,4)(5,6)(7,9,10,8)(11,14,15,16)(12,13,17,18)(19,22,23,20)(21,25,26,24), (1,2), (11,13)(12,14)(15,18)(16,17));

C2×#72334345 #72334701