C₂⁶×#6923 #30626753

order	15616
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₄ · Schur multiplier (H₂) = C₂²¹

Aliases

direct product	C₂⁶×#6923
unambiguous semidirect product	D₆₁⋊(C₂⁵×C₄)

Splittings

Fitting, socle	C₂⁶×C₆₁ ⋊ C₄
derived, Sylow	C₆₁ ⋊ C₂⁶×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂⁶	#6923
commutator, hypocenter	C₆₁	C₂⁶×C₄
Frattini	C₁	G
Fitting, socle	C₂⁶×C₆₁	C₄
prime radical	C₂⁶×D₆₁	C₂
cosocle	D₆₁	C₂⁷
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₂⁶

111198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 73
transitive degree	μ_t(G) = 3904

A minimal faithful permutation representation — 7 generators on 73 points:

(14,15,16,18)(17,20,21,27)(19,24,25,23)(22,29,30,40)(26,33,34,36)(28,37,38,32)(31,42,43,57)(35,46,47,53)(39,50,51,49)(41,54,55,45)(44,59,60,61)(48,63,64,72)(52,67,68,70)(56,71,69,66)(58,73,65,62);(11,12)(13,14,17,22,31,44,61,73,54,37,24,15,19,26,35,48,65,57,72,71,50,33,20,28,39,52,69,55,40,53,70,67,46,29,41,56,68,51,38,27,36,49,66,63,42,58,64,47,34,25,18,23,32,45,62,59,60,43,30,21,16);(9,10);(7,8);(5,6);(3,4);(1,2)

Conjugacy classes & character table

1216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15616, 1116308);

C26×#6923 #30626753