C₂×#72334339 #72334573

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

Aliases

direct product

C₂×#72334339

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

932404 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334342, C₂×#72339320
maximal normal subgroup of	C₂×#72334342, C₂×#72339320

GAP

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

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

C2×#72334339 #72334573