D₁₃₄₇₅ #41003945

order	26950
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₇₅
unambiguous semidirect product	C₁₁⋊D₁₂₂₅
unambiguous semidirect product	C₂₅⋊D₅₃₉
unambiguous semidirect product	C₄₉⋊D₂₇₅

Splittings

Fitting, derived, Hall	C₁₁×C₂₅×C₄₉ ⋊ C₂
Hall	C₂₅×C₄₉ ⋊ D₁₁
Hall	C₁₁×C₄₉ ⋊ D₂₅
Hall	C₁₁×C₂₅ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₂₇₅
Sylow	C₂₅ ⋊ D₅₃₉
Sylow	C₁₁ ⋊ D₁₂₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₁₁×C₂₅×C₄₉	C₂
Frattini	C₅×C₇	D₃₈₅
socle	C₅×C₇×C₁₁	D₃₅
LCS	G ▷ C₁₁×C₂₅×C₄₉
UCS	C₁

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 13475

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

Polycyclic presentation:

2,5,7,11,5,7|b^5=e^4,c^7=f^6,b^a=b^4e,c^a=c^6f,d^a=d^10,e^a=e^4,f^a=f^6

Conjugacy classes & character table

6739 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₇₅
maximal normal subgroup of	C₂×D₁₃₄₇₅
maximal quotient of	Dic₁₃₄₇₅, C₂×D₁₃₄₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26950, 12);

D13475 #41003945