C₂×#7625692 #72340006

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product

C₂×#7625692

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator	C₂²	C₂⁶×C₄
Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

312045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

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

Relationships

maximal quotient of

#72340011, C₂×#72337292, C₂×#72337324

GAP

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

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

C2×#7625692 #72340006