C₂×#72232275 #72337814

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁵

Aliases

direct product

C₂×#72232275

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁷
commutator, Frattini, cosocle	C₂³	C₂⁸
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 82212 conjugacy classes of subgroups; only the 418576 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 8 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

C₂×#72232371

GAP

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

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

C2×#72232275 #72337814