Dic₇₄₉₇ #44472371

order	29988
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₄₉₇
unambiguous semidirect product	C₉⋊Dic₈₃₃
unambiguous semidirect product	C₄₉⋊Dic₁₅₃

Splittings

derived, Hall	C₉×C₁₇×C₄₉ ⋊ C₄
Hall	C₁₇×C₄₉ ⋊ Dic₉
Hall	C₉×C₄₉ ⋊ Dic₁₇
Hall	C₉×C₁₇ ⋊ Dic₄₉
Sylow	C₄₉ ⋊ Dic₁₅₃
Sylow	C₁₇ ⋊ Dic₄₄₁
Sylow	C₉ ⋊ Dic₈₃₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₄₉₇
commutator, hypocenter	C₉×C₁₇×C₄₉	C₄
Frattini	C₂×C₃×C₇	D₃₅₇
Fitting, cosocle	C₂×C₉×C₁₇×C₄₉	C₂
socle, prime radical	C₂×C₃×C₇×C₁₇	D₂₁
LCS	G ▷ C₉×C₁₇×C₄₉
UCS	C₁ ◁ C₂

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 29988

A minimal faithful permutation representation — 2 generators on 79 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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,3,7,17,2,3,7|a^2=e,b^3=f^2,c^7=g^6,b^a=b^2f,c^a=c^6g,d^a=d^16,f^a=f^2,g^a=g^6

Conjugacy classes & character table

7500 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₄₉₉₄
maximal normal subgroup of	Dic₁₄₉₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29988, 12);

Dic7497 #44472371