D₁₃₄₈₉ #41008670

order	26978
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₈₉
unambiguous semidirect product	C₇⋊D₁₉₂₇
unambiguous semidirect product	C₄₁⋊D₃₂₉
unambiguous semidirect product	C₄₇⋊D₂₈₇

Splittings

Fitting, socle, derived, Hall	C₇×C₄₁×C₄₇ ⋊ C₂
Hall	C₄₁×C₄₇ ⋊ D₇
Hall	C₇×C₄₇ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₃₂₉
Sylow	C₇ ⋊ D₁₉₂₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₄₁×C₄₇	C₂
LCS	G ▷ C₇×C₄₁×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 95
transitive degree	μ_t(G) = 13489

A minimal faithful permutation representation — 2 generators on 95 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);(2,7)(3,6)(4,5)(9,48)(10,47)(11,46)(12,45)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,36)(22,35)(23,34)(24,33)(25,32)(26,31)(27,30)(28,29)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,7,41,47|b^a=b^6,c^a=c^40,d^a=d^46

Conjugacy classes & character table

6746 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₈₉
maximal normal subgroup of	C₂×D₁₃₄₈₉
maximal quotient of	Dic₁₃₄₈₉, C₂×D₁₃₄₈₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26978, 8);

D13489 #41008670