D₉₁₉₂ #72214031

order	18384
automorphism	Aut(G) order 28090752
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₁₉₂

Splittings

Fitting	C₃×C₈×C₃₈₃ ⋊ C₂
Hall	C₃×C₃₈₃ ⋊ D₈
Sylow	C₃₈₃ ⋊ D₂₄
Sylow	C₃ ⋊ D₃₀₆₄

Subgroups

	subgroup	G/H
center	C₂	D₄₅₉₆
commutator, cosocle	C₃×C₄×C₃₈₃	C₂²
Frattini	C₄	C₂×D₁₁₄₉
Fitting	C₃×C₈×C₃₈₃	C₂
socle	C₂×C₃×C₃₈₃	D₄
hypercenter	C₈	D₁₁₄₉
hypocenter	C₃×C₃₈₃	D₈
LCS	G ▷ C₃×C₄×C₃₈₃ ▷ C₂×C₃×C₃₈₃ ▷ C₃×C₃₈₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 394
transitive degree	μ_t(G) = 9192

A minimal faithful permutation representation — 2 generators on 394 points:

(1,2,3,4,5,6,7,8)(9,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394);(2,8)(3,7)(4,6)(10,11)(13,394)(14,393)(15,392)(16,391)(17,390)(18,389)(19,388)(20,387)(21,386)(22,385)(23,384)(24,383)(25,382)(26,381)(27,380)(28,379)(29,378)(30,377)(31,376)(32,375)(33,374)(34,373)(35,372)(36,371)(37,370)(38,369)(39,368)(40,367)(41,366)(42,365)(43,364)(44,363)(45,362)(46,361)(47,360)(48,359)(49,358)(50,357)(51,356)(52,355)(53,354)(54,353)(55,352)(56,351)(57,350)(58,349)(59,348)(60,347)(61,346)(62,345)(63,344)(64,343)(65,342)(66,341)(67,340)(68,339)(69,338)(70,337)(71,336)(72,335)(73,334)(74,333)(75,332)(76,331)(77,330)(78,329)(79,328)(80,327)(81,326)(82,325)(83,324)(84,323)(85,322)(86,321)(87,320)(88,319)(89,318)(90,317)(91,316)(92,315)(93,314)(94,313)(95,312)(96,311)(97,310)(98,309)(99,308)(100,307)(101,306)(102,305)(103,304)(104,303)(105,302)(106,301)(107,300)(108,299)(109,298)(110,297)(111,296)(112,295)(113,294)(114,293)(115,292)(116,291)(117,290)(118,289)(119,288)(120,287)(121,286)(122,285)(123,284)(124,283)(125,282)(126,281)(127,280)(128,279)(129,278)(130,277)(131,276)(132,275)(133,274)(134,273)(135,272)(136,271)(137,270)(138,269)(139,268)(140,267)(141,266)(142,265)(143,264)(144,263)(145,262)(146,261)(147,260)(148,259)(149,258)(150,257)(151,256)(152,255)(153,254)(154,253)(155,252)(156,251)(157,250)(158,249)(159,248)(160,247)(161,246)(162,245)(163,244)(164,243)(165,242)(166,241)(167,240)(168,239)(169,238)(170,237)(171,236)(172,235)(173,234)(174,233)(175,232)(176,231)(177,230)(178,229)(179,228)(180,227)(181,226)(182,225)(183,224)(184,223)(185,222)(186,221)(187,220)(188,219)(189,218)(190,217)(191,216)(192,215)(193,214)(194,213)(195,212)(196,211)(197,210)(198,209)(199,208)(200,207)(201,206)(202,205)(203,204)

Polycyclic presentation:

2^4,3,383|b^2=cd,c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^382

Conjugacy classes & character table

4599 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₈₃₈₄, D₂₇₅₇₆
maximal normal subgroup of	D₁₈₃₈₄
maximal quotient of	Dic₉₁₉₂, D₁₈₃₈₄, D₂₇₅₇₆

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 383]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.6^382);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9192 #72214031