Dic₅₈₉₉ #37556209

order	23596
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₈₉₉
unambiguous semidirect product	C₃₄₇⋊Dic₁₇

Splittings

derived, Hall	C₁₇×C₃₄₇ ⋊ C₄
Sylow	C₃₄₇ ⋊ Dic₁₇
Sylow	C₁₇ ⋊ Dic₃₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₈₉₉
commutator, hypocenter	C₁₇×C₃₄₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₇×C₃₄₇	C₂
LCS	G ▷ C₁₇×C₃₄₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 368
transitive degree	μ_t(G) = 23596

A minimal faithful permutation representation — 2 generators on 368 points:

(1,2)(3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21)(22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368);(1,3,2,4)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(23,368)(24,367)(25,366)(26,365)(27,364)(28,363)(29,362)(30,361)(31,360)(32,359)(33,358)(34,357)(35,356)(36,355)(37,354)(38,353)(39,352)(40,351)(41,350)(42,349)(43,348)(44,347)(45,346)(46,345)(47,344)(48,343)(49,342)(50,341)(51,340)(52,339)(53,338)(54,337)(55,336)(56,335)(57,334)(58,333)(59,332)(60,331)(61,330)(62,329)(63,328)(64,327)(65,326)(66,325)(67,324)(68,323)(69,322)(70,321)(71,320)(72,319)(73,318)(74,317)(75,316)(76,315)(77,314)(78,313)(79,312)(80,311)(81,310)(82,309)(83,308)(84,307)(85,306)(86,305)(87,304)(88,303)(89,302)(90,301)(91,300)(92,299)(93,298)(94,297)(95,296)(96,295)(97,294)(98,293)(99,292)(100,291)(101,290)(102,289)(103,288)(104,287)(105,286)(106,285)(107,284)(108,283)(109,282)(110,281)(111,280)(112,279)(113,278)(114,277)(115,276)(116,275)(117,274)(118,273)(119,272)(120,271)(121,270)(122,269)(123,268)(124,267)(125,266)(126,265)(127,264)(128,263)(129,262)(130,261)(131,260)(132,259)(133,258)(134,257)(135,256)(136,255)(137,254)(138,253)(139,252)(140,251)(141,250)(142,249)(143,248)(144,247)(145,246)(146,245)(147,244)(148,243)(149,242)(150,241)(151,240)(152,239)(153,238)(154,237)(155,236)(156,235)(157,234)(158,233)(159,232)(160,231)(161,230)(162,229)(163,228)(164,227)(165,226)(166,225)(167,224)(168,223)(169,222)(170,221)(171,220)(172,219)(173,218)(174,217)(175,216)(176,215)(177,214)(178,213)(179,212)(180,211)(181,210)(182,209)(183,208)(184,207)(185,206)(186,205)(187,204)(188,203)(189,202)(190,201)(191,200)(192,199)(193,198)(194,197)(195,196)

Polycyclic presentation:

2,17,347,2|a^2=d,b^a=b^16,c^a=c^346

Conjugacy classes & character table

5902 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₇₉₈
maximal normal subgroup of	Dic₁₁₇₉₈

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23596, 4);

Dic5899 #37556209