Dic₇₈₄₇ #44535999

order	31388
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₈₄₇
unambiguous semidirect product	C₇⋊Dic₁₁₂₁
unambiguous semidirect product	C₁₉⋊Dic₄₁₃
unambiguous semidirect product	C₅₉⋊Dic₁₃₃

Splittings

derived, Hall	C₇×C₁₉×C₅₉ ⋊ C₄
Hall	C₁₉×C₅₉ ⋊ Dic₇
Hall	C₇×C₅₉ ⋊ Dic₁₉
Hall	C₇×C₁₉ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₁₃₃
Sylow	C₁₉ ⋊ Dic₄₁₃
Sylow	C₇ ⋊ Dic₁₁₂₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₈₄₇
commutator, hypocenter	C₇×C₁₉×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₉×C₅₉	C₂
LCS	G ▷ C₇×C₁₉×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 31388

A minimal faithful permutation representation — 2 generators on 89 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,30)(14,29)(15,28)(16,27)(17,26)(18,25)(19,24)(20,23)(21,22)(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:

2,7,19,59,2|a^2=e,b^a=b^6,c^a=c^18,d^a=d^58

Conjugacy classes & character table

7850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₆₉₄
maximal normal subgroup of	Dic₁₅₆₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31388, 8);

Dic7847 #44535999