C₂×#72232044 #72324469

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

Aliases

direct product

C₂×#72232044

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 15340 conjugacy classes of subgroups; only the 40112 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72232064
maximal normal subgroup of	C₂×#72232064

GAP

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

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

C2×#72232044 #72324469