D₁₃₁₃₅ #39911099

order	26270
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₁₃₅
unambiguous semidirect product	C₅⋊D₂₆₂₇
unambiguous semidirect product	C₃₇⋊D₃₅₅
unambiguous semidirect product	C₇₁⋊D₁₈₅

Splittings

Fitting, socle, derived, Hall	C₅×C₃₇×C₇₁ ⋊ C₂
Hall	C₃₇×C₇₁ ⋊ D₅
Hall	C₅×C₇₁ ⋊ D₃₇
Hall	C₅×C₃₇ ⋊ D₇₁
Sylow	C₇₁ ⋊ D₁₈₅
Sylow	C₃₇ ⋊ D₃₅₅
Sylow	C₅ ⋊ D₂₆₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₃₇×C₇₁	C₂
LCS	G ▷ C₅×C₃₇×C₇₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 113
transitive degree	μ_t(G) = 13135

A minimal faithful permutation representation — 2 generators on 113 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113);(2,5)(3,4)(7,42)(8,41)(9,40)(10,39)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(44,113)(45,112)(46,111)(47,110)(48,109)(49,108)(50,107)(51,106)(52,105)(53,104)(54,103)(55,102)(56,101)(57,100)(58,99)(59,98)(60,97)(61,96)(62,95)(63,94)(64,93)(65,92)(66,91)(67,90)(68,89)(69,88)(70,87)(71,86)(72,85)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)

Polycyclic presentation:

2,5,37,71|b^a=b^4,c^a=c^36,d^a=d^70

Conjugacy classes & character table

6569 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₁₃₅
maximal normal subgroup of	C₂×D₁₃₁₃₅
maximal quotient of	Dic₁₃₁₃₅, C₂×D₁₃₁₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26270, 9);

D13135 #39911099