D₈₂₇₅ #30658476

order	16550
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₂₇₅
unambiguous semidirect product	C₂₅⋊D₃₃₁
unambiguous semidirect product	C₃₃₁⋊D₂₅

Splittings

Fitting, derived, Hall	C₂₅×C₃₃₁ ⋊ C₂
Sylow	C₃₃₁ ⋊ D₂₅
Sylow	C₂₅ ⋊ D₃₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₂₅×C₃₃₁	C₂
Frattini	C₅	D₁₆₅₅
socle	C₅×C₃₃₁	D₅
LCS	G ▷ C₂₅×C₃₃₁
UCS	C₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 356
transitive degree	μ_t(G) = 8275

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

Polycyclic presentation:

2,5,331,5|b^5=d^4,b^a=b^4d,c^a=c^330,d^a=d^4

Conjugacy classes & character table

4139 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₂₅×D₃₃₁, C₂×D₈₂₇₅
maximal normal subgroup of	D₂₅×D₃₃₁, C₂×D₈₂₇₅
maximal quotient of	Dic₈₂₇₅, C₂×D₈₂₇₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16550, 6);

D8275 #30658476