C₂×#72334337 #72334571

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

Aliases

direct product

C₂×#72334337

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

935480 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)(13,18)(14,16)(15,17)(20,23)(24,25);(11,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26);(3,4)(5,6)(11,14)(12,16)(13,17)(15,18)(19,21)(20,24)(22,26)(23,25);(7,8)(9,10)(12,17)(14,18);(3,4,5,6)(7,8,9,10)(11,14)(12,13)(15,18)(16,17)(19,20)(21,25)(22,23)(24,26);(1,2);(3,4)(5,6)(7,8)(9,10)(11,13)(12,18)(14,17)(15,16)(19,21)(20,24)(22,26)(23,25)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72334342, C₂×#72334357
maximal normal subgroup of	C₂×#72334342, C₂×#72334357

GAP

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

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

C2×#72334337 #72334571