Dic₈₅₃₃ #46802419

order	34132
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₅₃₃
unambiguous semidirect product	C₇⋊Dic₁₂₁₉
unambiguous semidirect product	C₂₃⋊Dic₃₇₁

Splittings

derived, Hall	C₇×C₂₃×C₅₃ ⋊ C₄
Hall	C₂₃×C₅₃ ⋊ Dic₇
Hall	C₇×C₅₃ ⋊ Dic₂₃
Hall	C₇×C₂₃ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₆₁
Sylow	C₂₃ ⋊ Dic₃₇₁
Sylow	C₇ ⋊ Dic₁₂₁₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₅₃₃
commutator, hypocenter	C₇×C₂₃×C₅₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₇×C₂₃×C₅₃	C₂
LCS	G ▷ C₇×C₂₃×C₅₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 34132

A minimal faithful permutation representation — 2 generators on 87 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);(1,3,2,4)(6,11)(7,10)(8,9)(13,34)(14,33)(15,32)(16,31)(17,30)(18,29)(19,28)(20,27)(21,26)(22,25)(23,24)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2,7,23,53,2|a^2=e,b^a=b^6,c^a=c^22,d^a=d^52

Conjugacy classes & character table

8536 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(34132, 8);

Dic8533 #46802419