Dic₆₁₉₅ #37613355

order	24780
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₁₉₅
unambiguous semidirect product	C₃⋊Dic₂₀₆₅
unambiguous semidirect product	C₇⋊Dic₈₈₅
unambiguous semidirect product	C₅₉⋊Dic₁₀₅

Splittings

derived, Hall	C₃×C₅×C₇×C₅₉ ⋊ C₄
Hall	C₅×C₇×C₅₉ ⋊ Dic₃
Hall	C₃×C₇×C₅₉ ⋊ Dic₅
Hall	C₃×C₅×C₅₉ ⋊ Dic₇
Hall	C₇×C₅₉ ⋊ Dic₁₅
Hall	C₅×C₅₉ ⋊ Dic₂₁
Hall	C₃×C₅₉ ⋊ Dic₃₅
Hall	C₃×C₅×C₇ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₀₅
Hall	C₅×C₇ ⋊ Dic₁₇₇
Hall	C₃×C₇ ⋊ Dic₂₉₅
Hall	C₃×C₅ ⋊ Dic₄₁₃
Sylow	C₇ ⋊ Dic₈₈₅
Sylow	C₅ ⋊ Dic₁₂₃₉
Sylow	C₃ ⋊ Dic₂₀₆₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₁₉₅
commutator, hypocenter	C₃×C₅×C₇×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₃×C₅×C₇×C₅₉	C₂
LCS	G ▷ C₃×C₅×C₇×C₅₉
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 78
transitive degree	μ_t(G) = 24780

A minimal faithful permutation representation — 2 generators on 78 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);(1,3,2,4)(6,7)(9,12)(10,11)(14,19)(15,18)(16,17)(21,78)(22,77)(23,76)(24,75)(25,74)(26,73)(27,72)(28,71)(29,70)(30,69)(31,68)(32,67)(33,66)(34,65)(35,64)(36,63)(37,62)(38,61)(39,60)(40,59)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)

Polycyclic presentation:

2,3,5,7,59,2|a^2=f,b^a=b^2,c^a=c^4,d^a=d^6,e^a=e^58

Conjugacy classes & character table

6198 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₃₉₀
maximal normal subgroup of	Dic₁₂₃₉₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24780, 17);

Dic6195 #37613355