D₉₆₅₃ #34064487

order	19306
automorphism	Aut(G) order 79463496
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₆₅₃
unambiguous semidirect product	C₄₉⋊D₁₉₇
unambiguous semidirect product	C₁₉₇⋊D₄₉

Splittings

Fitting, derived, Hall	C₄₉×C₁₉₇ ⋊ C₂
Sylow	C₁₉₇ ⋊ D₄₉
Sylow	C₄₉ ⋊ D₁₉₇

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₁₉₇	C₂
Frattini	C₇	D₁₃₇₉
socle	C₇×C₁₉₇	D₇
LCS	G ▷ C₄₉×C₁₉₇
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 246
transitive degree	μ_t(G) = 9653

A minimal faithful permutation representation — 2 generators on 246 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)(50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246);(2,49)(3,48)(4,47)(5,46)(6,45)(7,44)(8,43)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(51,246)(52,245)(53,244)(54,243)(55,242)(56,241)(57,240)(58,239)(59,238)(60,237)(61,236)(62,235)(63,234)(64,233)(65,232)(66,231)(67,230)(68,229)(69,228)(70,227)(71,226)(72,225)(73,224)(74,223)(75,222)(76,221)(77,220)(78,219)(79,218)(80,217)(81,216)(82,215)(83,214)(84,213)(85,212)(86,211)(87,210)(88,209)(89,208)(90,207)(91,206)(92,205)(93,204)(94,203)(95,202)(96,201)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,192)(106,191)(107,190)(108,189)(109,188)(110,187)(111,186)(112,185)(113,184)(114,183)(115,182)(116,181)(117,180)(118,179)(119,178)(120,177)(121,176)(122,175)(123,174)(124,173)(125,172)(126,171)(127,170)(128,169)(129,168)(130,167)(131,166)(132,165)(133,164)(134,163)(135,162)(136,161)(137,160)(138,159)(139,158)(140,157)(141,156)(142,155)(143,154)(144,153)(145,152)(146,151)(147,150)(148,149)

Polycyclic presentation:

2,7,197,7|b^7=d^6,b^a=b^6d,c^a=c^196,d^a=d^6

Conjugacy classes & character table

4828 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₄₉×D₁₉₇, C₂×D₉₆₅₃, D₂₈₉₅₉
maximal normal subgroup of	D₄₉×D₁₉₇, C₂×D₉₆₅₃
maximal quotient of	Dic₉₆₅₃, C₂×D₉₆₅₃, D₂₈₉₅₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(19306, 6);

D9653 #34064487