Dic₇₇₅₁ #44526505

order	31004
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₇₇₅₁
unambiguous semidirect product	C₂₃⋊Dic₃₃₇

Splittings

derived, Hall	C₂₃×C₃₃₇ ⋊ C₄
Sylow	C₃₃₇ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₃₃₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₇₇₅₁
commutator, hypocenter	C₂₃×C₃₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₂₃×C₃₃₇	C₂
LCS	G ▷ C₂₃×C₃₃₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 364
transitive degree	μ_t(G) = 31004

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

Polycyclic presentation:

2,23,337,2|a^2=d,b^a=b^22,c^a=c^336

Conjugacy classes & character table

7754 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₅₅₀₂
maximal normal subgroup of	Dic₁₅₅₀₂

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31004, 4);

Dic7751 #44526505