D₁₅₉₅₃ #44542403

order	31906
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₅₉₅₃
unambiguous semidirect product	C₇⋊D₂₂₇₉
unambiguous semidirect product	C₄₃⋊D₃₇₁
unambiguous semidirect product	C₅₃⋊D₃₀₁

Splittings

Fitting, socle, derived, Hall	C₇×C₄₃×C₅₃ ⋊ C₂
Hall	C₄₃×C₅₃ ⋊ D₇
Hall	C₇×C₅₃ ⋊ D₄₃
Hall	C₇×C₄₃ ⋊ D₅₃
Sylow	C₅₃ ⋊ D₃₀₁
Sylow	C₄₃ ⋊ D₃₇₁
Sylow	C₇ ⋊ D₂₂₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₄₃×C₅₃	C₂
LCS	G ▷ C₇×C₄₃×C₅₃
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 15953

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

Polycyclic presentation:

2,7,43,53|b^a=b^6,c^a=c^42,d^a=d^52

Conjugacy classes & character table

7978 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₅₉₅₃
maximal normal subgroup of	C₂×D₁₅₉₅₃
maximal quotient of	Dic₁₅₉₅₃, C₂×D₁₅₉₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31906, 9);

D15953 #44542403