Dic₆₀₉₇ #37567217

order	24388
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₉₇
unambiguous semidirect product	C₇⋊Dic₈₇₁
unambiguous semidirect product	C₆₇⋊Dic₉₁

Splittings

derived, Hall	C₇×C₁₃×C₆₇ ⋊ C₄
Hall	C₁₃×C₆₇ ⋊ Dic₇
Hall	C₇×C₆₇ ⋊ Dic₁₃
Hall	C₇×C₁₃ ⋊ Dic₆₇
Sylow	C₆₇ ⋊ Dic₉₁
Sylow	C₁₃ ⋊ Dic₄₆₉
Sylow	C₇ ⋊ Dic₈₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₀₉₇
commutator, hypocenter	C₇×C₁₃×C₆₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₁₃×C₆₇	C₂
LCS	G ▷ C₇×C₁₃×C₆₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 24388

A minimal faithful permutation representation — 2 generators on 91 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(26,91)(27,90)(28,89)(29,88)(30,87)(31,86)(32,85)(33,84)(34,83)(35,82)(36,81)(37,80)(38,79)(39,78)(40,77)(41,76)(42,75)(43,74)(44,73)(45,72)(46,71)(47,70)(48,69)(49,68)(50,67)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,60)(58,59)

Polycyclic presentation:

2,7,13,67,2|a^2=e,b^a=b^6,c^a=c^12,d^a=d^66

Conjugacy classes & character table

6100 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₁₉₄
maximal normal subgroup of	Dic₁₂₁₉₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24388, 8);

Dic6097 #37567217