C₂×#72334925 #72339339

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

Aliases

direct product

C₂×#72334925

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

929876 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)(8,10)(11,12)(13,17)(14,18)(15,16)(19,20,22,24)(21,25,26,23);(4,6)(8,10)(13,18)(14,17)(19,21,22,26)(20,23,24,25);(4,6)(7,8,9,10)(12,16)(14,17)(19,21)(20,23)(22,26)(24,25);(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,18)(16,17);(1,2);(19,20)(21,25)(22,24)(23,26);(11,14)(12,13)(15,17)(16,18)(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×#72339320
maximal normal subgroup of	C₂×#72339320

GAP

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

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

C2×#72334925 #72339339