Dic₇₈₃₉ #44535576

order	31356
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₃₉
unambiguous semidirect product	C₉⋊Dic₈₇₁
unambiguous semidirect product	C₆₇⋊Dic₁₁₇

Splittings

derived, Hall	C₉×C₁₃×C₆₇ ⋊ C₄
Hall	C₁₃×C₆₇ ⋊ Dic₉
Hall	C₉×C₆₇ ⋊ Dic₁₃
Hall	C₉×C₁₃ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₁₁₇
Sylow	C₁₃ ⋊ Dic₆₀₃
Sylow	C₉ ⋊ Dic₈₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₇₈₃₉
commutator, hypocenter	C₉×C₁₃×C₆₇	C₄
Frattini	C₂×C₃	D₂₆₁₃
Fitting, cosocle	C₂×C₉×C₁₃×C₆₇	C₂
socle, prime radical	C₂×C₃×C₁₃×C₆₇	D₃
LCS	G ▷ C₉×C₁₃×C₆₇
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 93
transitive degree	μ_t(G) = 31356

A minimal faithful permutation representation — 2 generators on 93 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)(27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)(28,93)(29,92)(30,91)(31,90)(32,89)(33,88)(34,87)(35,86)(36,85)(37,84)(38,83)(39,82)(40,81)(41,80)(42,79)(43,78)(44,77)(45,76)(46,75)(47,74)(48,73)(49,72)(50,71)(51,70)(52,69)(53,68)(54,67)(55,66)(56,65)(57,64)(58,63)(59,62)(60,61)

Polycyclic presentation:

2,3,13,67,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^12,d^a=d^66,f^a=f^2

Conjugacy classes & character table

7842 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₆₇₈
maximal normal subgroup of	Dic₁₅₆₇₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31356, 12);

Dic7839 #44535576