Dic₈₀₂₇ #44545747

order	32108
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₈₀₂₇
unambiguous semidirect product	C₂₃⋊Dic₃₄₉

Splittings

derived, Hall	C₂₃×C₃₄₉ ⋊ C₄
Sylow	C₃₄₉ ⋊ Dic₂₃
Sylow	C₂₃ ⋊ Dic₃₄₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₈₀₂₇
commutator, hypocenter	C₂₃×C₃₄₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₂₃×C₃₄₉	C₂
LCS	G ▷ C₂₃×C₃₄₉
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 376
transitive degree	μ_t(G) = 32108

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

Polycyclic presentation:

2,23,349,2|a^2=d,b^a=b^22,c^a=c^348

Conjugacy classes & character table

8030 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₆₀₅₄
maximal normal subgroup of	Dic₁₆₀₅₄

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32108, 4);

Dic8027 #44545747