C₁₉×D₃₅₃ #27211802

order	13414
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₉ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₉×D₃₅₃
unambiguous semidirect product	C₃₅₃⋊(C₂×C₁₉)

Splittings

Fitting, socle, Hall	C₁₉×C₃₅₃ ⋊ C₂
derived, Sylow	C₃₅₃ ⋊ C₂×C₁₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁₉	D₃₅₃
commutator, cosocle, hypocenter	C₃₅₃	C₂×C₁₉
Frattini	C₁	G
Fitting, socle	C₁₉×C₃₅₃	C₂
LCS	G ▷ C₃₅₃
UCS	C₁ ◁ C₁₉

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 372
transitive degree	μ_t(G) = 6707

A minimal faithful permutation representation — 2 generators on 372 points:

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

Conjugacy classes & character table

3382 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₉×#13083443, C₁₉⋊#13083443, D₁₉×D₃₅₃, …
maximal normal subgroup of	C₁₉×#13083443, C₁₉⋊#13083443, D₁₉×D₃₅₃, …
maximal quotient of	C₁₉×Dic₃₅₃, C₂×C₁₉×D₃₅₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13414, 2);

C19×D353 #27211802