Dic₆₀₂₃ #37562821

order	24092
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₀₂₃
unambiguous semidirect product	C₁₉⋊Dic₃₁₇

Splittings

derived, Hall	C₁₉×C₃₁₇ ⋊ C₄
Sylow	C₃₁₇ ⋊ Dic₁₉
Sylow	C₁₉ ⋊ Dic₃₁₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₀₂₃
commutator, hypocenter	C₁₉×C₃₁₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₉×C₃₁₇	C₂
LCS	G ▷ C₁₉×C₃₁₇
UCS	C₁ ◁ C₂

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 340
transitive degree	μ_t(G) = 24092

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

Polycyclic presentation:

2,19,317,2|a^2=d,b^a=b^18,c^a=c^316

Conjugacy classes & character table

6026 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₂₀₄₆
maximal normal subgroup of	Dic₁₂₀₄₆

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(24092, 4);

Dic6023 #37562821