D₁₆₉₇₅ #46800470

order	33950
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₉₇₅
unambiguous semidirect product	C₇⋊D₂₄₂₅
unambiguous semidirect product	C₂₅⋊D₆₇₉
unambiguous semidirect product	C₉₇⋊D₁₇₅

Splittings

Fitting, derived, Hall	C₇×C₂₅×C₉₇ ⋊ C₂
Hall	C₂₅×C₉₇ ⋊ D₇
Hall	C₇×C₉₇ ⋊ D₂₅
Hall	C₇×C₂₅ ⋊ D₉₇
Sylow	C₉₇ ⋊ D₁₇₅
Sylow	C₂₅ ⋊ D₆₇₉
Sylow	C₇ ⋊ D₂₄₂₅

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₇×C₂₅×C₉₇	C₂
Frattini	C₅	D₃₃₉₅
socle	C₅×C₇×C₉₇	D₅
LCS	G ▷ C₇×C₂₅×C₉₇
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 129
transitive degree	μ_t(G) = 16975

A minimal faithful permutation representation — 2 generators on 129 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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,25)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(27,32)(28,31)(29,30)(34,129)(35,128)(36,127)(37,126)(38,125)(39,124)(40,123)(41,122)(42,121)(43,120)(44,119)(45,118)(46,117)(47,116)(48,115)(49,114)(50,113)(51,112)(52,111)(53,110)(54,109)(55,108)(56,107)(57,106)(58,105)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85)(79,84)(80,83)(81,82)

Polycyclic presentation:

2,5,7,97,5|b^5=e^4,b^a=b^4e,c^a=c^6,d^a=d^96,e^a=e^4

Conjugacy classes & character table

8489 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33950, 8);

D16975 #46800470