D₁₄₆₃₇ #44418083

order	29274
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₄₆₃₇
unambiguous semidirect product	C₃⋊D₄₈₇₉
unambiguous semidirect product	C₇⋊D₂₀₉₁
unambiguous semidirect product	C₁₇⋊D₈₆₁
unambiguous semidirect product	C₄₁⋊D₃₅₇

Splittings

Fitting, socle, derived, Hall	C₃×C₇×C₁₇×C₄₁ ⋊ C₂
Hall	C₇×C₁₇×C₄₁ ⋊ D₃
Hall	C₃×C₁₇×C₄₁ ⋊ D₇
Hall	C₃×C₇×C₄₁ ⋊ D₁₇
Hall	C₁₇×C₄₁ ⋊ D₂₁
Hall	C₃×C₇×C₁₇ ⋊ D₄₁
Hall	C₇×C₄₁ ⋊ D₅₁
Hall	C₃×C₄₁ ⋊ D₁₁₉
Hall	C₇×C₁₇ ⋊ D₁₂₃
Hall	C₃×C₁₇ ⋊ D₂₈₇
Sylow	C₄₁ ⋊ D₃₅₇
Hall	C₃×C₇ ⋊ D₆₉₇
Sylow	C₁₇ ⋊ D₈₆₁
Sylow	C₇ ⋊ D₂₀₉₁
Sylow	C₃ ⋊ D₄₈₇₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₇×C₁₇×C₄₁	C₂
LCS	G ▷ C₃×C₇×C₁₇×C₄₁
UCS	C₁

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 14637

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)(5,10)(6,9)(7,8)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,20)(29,68)(30,67)(31,66)(32,65)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53)(45,52)(46,51)(47,50)(48,49)

Polycyclic presentation:

2,3,7,17,41|b^a=b^2,c^a=c^6,d^a=d^16,e^a=e^40

Conjugacy classes & character table

7320 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₄₆₃₇
maximal normal subgroup of	C₂×D₁₄₆₃₇
maximal quotient of	Dic₁₄₆₃₇, C₂×D₁₄₆₃₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29274, 17);

D14637 #44418083