C₂×#72335119 #72335188

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

Aliases

direct product

C₂×#72335119

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

1094526 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:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×#72335129, #72337247
maximal normal subgroup of	C₂×#72335129, #72337247

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

C2×#72335119 #72335188