Dic₄₉₃₅ #34089464

order	19740
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₉₃₅
unambiguous semidirect product	C₃⋊Dic₁₆₄₅
unambiguous semidirect product	C₇⋊Dic₇₀₅
unambiguous semidirect product	C₄₇⋊Dic₁₀₅

Splittings

derived, Hall	C₃×C₅×C₇×C₄₇ ⋊ C₄
Hall	C₅×C₇×C₄₇ ⋊ Dic₃
Hall	C₃×C₇×C₄₇ ⋊ Dic₅
Hall	C₃×C₅×C₄₇ ⋊ Dic₇
Hall	C₇×C₄₇ ⋊ Dic₁₅
Hall	C₅×C₄₇ ⋊ Dic₂₁
Hall	C₃×C₄₇ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₄₇
Sylow	C₄₇ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₁₄₁
Hall	C₃×C₇ ⋊ Dic₂₃₅
Hall	C₃×C₅ ⋊ Dic₃₂₉
Sylow	C₇ ⋊ Dic₇₀₅
Sylow	C₅ ⋊ Dic₉₈₇
Sylow	C₃ ⋊ Dic₁₆₄₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₉₃₅
commutator, hypocenter	C₃×C₅×C₇×C₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₄₇	C₂
LCS	G ▷ C₃×C₅×C₇×C₄₇
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 66
transitive degree	μ_t(G) = 19740

A minimal faithful permutation representation — 2 generators on 66 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)(21,66)(22,65)(23,64)(24,63)(25,62)(26,61)(27,60)(28,59)(29,58)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)

Polycyclic presentation:

2,3,5,7,47,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^46

Conjugacy classes & character table

4938 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₈₇₀, Dic₁₄₈₀₅
maximal normal subgroup of	Dic₉₈₇₀
maximal quotient of	Dic₁₄₈₀₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19740, 17);

Dic4935 #34089464