C₂×C₁₉⋊(D₇₁⋊C₅) #41008689

order	26980
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₁₉⋊(D₇₁⋊C₅)
unambiguous semidirect product	(C₂×C₁₉)⋊(D₇₁⋊C₅)
unambiguous semidirect product	(C₂×C₇₁⋊C₅)⋊D₁₉
unambiguous semidirect product	(C₂×D₁₃₄₉)⋊C₅

Splittings

Fitting, socle	C₂×C₁₉×C₇₁ ⋊ C₂×C₅
derived, Hall	C₁₉×C₇₁ ⋊ C₂²×C₅
Hall	C₁₉×C₇₁⋊C₅ ⋊ C₂²
Hall	C₂×D₁₃₄₉ ⋊ C₅
Hall	C₇₁⋊C₅ ⋊ C₂×D₁₉
Sylow	C₇₁ ⋊ C₂×C₅×D₁₉
Sylow	C₁₉ ⋊ C₂×D₇₁⋊C₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₁₉⋊(D₇₁⋊C₅)
commutator, cosocle, hypocenter	C₁₉×C₇₁	C₂²×C₅
Frattini	C₁	G
Fitting, socle	C₂×C₁₉×C₇₁	C₂×C₅
LCS	G ▷ C₁₉×C₇₁
UCS	C₁ ◁ C₂

40 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 92
transitive degree	μ_t(G) = 2698

A minimal faithful permutation representation — 2 generators on 92 points:

(2,6,7,25,30,3,4,12,14,49)(5,19,20,65,51,9,10,40,42,48)(8,35,36,37,54,16,17,61,43,63)(11,47,38,39,24,22,23,18,64,26)(13,50,29,46,70,27,28,52,53,69)(15,57,58,55,56,32,33,59,31,62)(21,34,68,41,71,44,45,60,66,67)(73,74)(75,76)(77,78)(79,80)(81,82)(83,84)(85,86)(87,88)(89,90);(1,2,5,11,24,7,15,34,56,61,20,43,35,55,25,38,16,6,13,29,23,48,50,60,70,30,54,71,65,64,58,57,53,44,19,41,66,10,21,46,33,59,39,42,67,63,49,69,68,28,51,47,52,27,4,8,18,14,31,17,37,40,36,62,45,32,12,26,22,9,3)(72,73,75,77,79,81,83,85,87,89,90,88,86,84,82,80,78,76,74)(91,92)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26980, 16);

C2×C19⋊(D71⋊C5) #41008689