Dic₃₈₄₃ #29506702

order	15372
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₃₈₄₃
unambiguous semidirect product	C₇⋊Dic₅₄₉
unambiguous semidirect product	C₉⋊Dic₄₂₇

Splittings

derived, Hall	C₇×C₉×C₆₁ ⋊ C₄
Hall	C₉×C₆₁ ⋊ Dic₇
Hall	C₇×C₆₁ ⋊ Dic₉
Hall	C₇×C₉ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₆₃
Sylow	C₉ ⋊ Dic₄₂₇
Sylow	C₇ ⋊ Dic₅₄₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₃₈₄₃
commutator, hypocenter	C₇×C₉×C₆₁	C₄
Frattini	C₂×C₃	D₁₂₈₁
Fitting, cosocle	C₂×C₇×C₉×C₆₁	C₂
socle, prime radical	C₂×C₃×C₇×C₆₁	D₃
LCS	G ▷ C₇×C₉×C₆₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 15372

A minimal faithful permutation representation — 2 generators on 81 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,20)(16,19)(17,18)(22,81)(23,80)(24,79)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,72)(32,71)(33,70)(34,69)(35,68)(36,67)(37,66)(38,65)(39,64)(40,63)(41,62)(42,61)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(51,52)

Polycyclic presentation:

2,3,7,61,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^6,d^a=d^60,f^a=f^2

Conjugacy classes & character table

3846 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₇₆₈₆, Dic₁₁₅₂₉
maximal normal subgroup of	Dic₇₆₈₆
maximal quotient of	Dic₁₁₅₂₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15372, 12);

Dic3843 #29506702