D₁₃₄₈₅ #41004200

order	26970
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₃₄₈₅
unambiguous semidirect product	C₃⋊D₄₄₉₅
unambiguous semidirect product	C₅⋊D₂₆₉₇
unambiguous semidirect product	C₂₉⋊D₄₆₅
unambiguous semidirect product	C₃₁⋊D₄₃₅

Splittings

Fitting, socle, derived, Hall	C₃×C₅×C₂₉×C₃₁ ⋊ C₂
Hall	C₅×C₂₉×C₃₁ ⋊ D₃
Hall	C₃×C₂₉×C₃₁ ⋊ D₅
Hall	C₂₉×C₃₁ ⋊ D₁₅
Hall	C₃×C₅×C₃₁ ⋊ D₂₉
Hall	C₃×C₅×C₂₉ ⋊ D₃₁
Hall	C₅×C₃₁ ⋊ D₈₇
Hall	C₅×C₂₉ ⋊ D₉₃
Hall	C₃×C₃₁ ⋊ D₁₄₅
Hall	C₃×C₂₉ ⋊ D₁₅₅
Sylow	C₃₁ ⋊ D₄₃₅
Sylow	C₂₉ ⋊ D₄₆₅
Hall	C₃×C₅ ⋊ D₈₉₉
Sylow	C₅ ⋊ D₂₆₉₇
Sylow	C₃ ⋊ D₄₄₉₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₅×C₂₉×C₃₁	C₂
LCS	G ▷ C₃×C₅×C₂₉×C₃₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 13485

A minimal faithful permutation representation — 2 generators on 68 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);(2,3)(5,8)(6,7)(10,37)(11,36)(12,35)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(39,68)(40,67)(41,66)(42,65)(43,64)(44,63)(45,62)(46,61)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)

Polycyclic presentation:

2,3,5,29,31|b^a=b^2,c^a=c^4,d^a=d^28,e^a=e^30

Conjugacy classes & character table

6744 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₃₄₈₅
maximal normal subgroup of	C₂×D₁₃₄₈₅
maximal quotient of	Dic₁₃₄₈₅, C₂×D₁₃₄₈₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26970, 19);

D13485 #41004200