Dic₉₂₇₅ #49150646

order	37100
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₉₂₇₅
unambiguous semidirect product	C₇⋊Dic₁₃₂₅

Splittings

derived, Hall	C₇×C₂₅×C₅₃ ⋊ C₄
Hall	C₂₅×C₅₃ ⋊ Dic₇
Hall	C₇×C₅₃ ⋊ Dic₂₅
Hall	C₇×C₂₅ ⋊ Dic₅₃
Sylow	C₅₃ ⋊ Dic₁₇₅
Sylow	C₂₅ ⋊ Dic₃₇₁
Sylow	C₇ ⋊ Dic₁₃₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₉₂₇₅
commutator, hypocenter	C₇×C₂₅×C₅₃	C₄
Frattini	C₂×C₅	D₁₈₅₅
Fitting, cosocle	C₂×C₇×C₂₅×C₅₃	C₂
socle, prime radical	C₂×C₅×C₇×C₅₃	D₅
LCS	G ▷ C₇×C₂₅×C₅₃
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 37100

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)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)(31,36)(32,35)(33,34)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2,5,7,53,2,5|a^2=e,b^5=f^4,b^a=b^4f,c^a=c^6,d^a=d^52,f^a=f^4

Conjugacy classes & character table

9278 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37100, 8);

Dic9275 #49150646