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G = C19×F5order 380 = 22·5·19

Direct product of C19 and F5

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C19×F5, C5⋊C76, C952C4, D5.C38, (D5×C19).2C2, SmallGroup(380,5)

Series: Derived Chief Lower central Upper central

C1C5 — C19×F5
C1C5D5D5×C19 — C19×F5
C5 — C19×F5
C1C19

Generators and relations for C19×F5
 G = < a,b,c | a19=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

5C2
5C4
5C38
5C76

Smallest permutation representation of C19×F5
On 95 points
Generators in S95
(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)
(1 62 34 95 50)(2 63 35 77 51)(3 64 36 78 52)(4 65 37 79 53)(5 66 38 80 54)(6 67 20 81 55)(7 68 21 82 56)(8 69 22 83 57)(9 70 23 84 39)(10 71 24 85 40)(11 72 25 86 41)(12 73 26 87 42)(13 74 27 88 43)(14 75 28 89 44)(15 76 29 90 45)(16 58 30 91 46)(17 59 31 92 47)(18 60 32 93 48)(19 61 33 94 49)
(20 55 81 67)(21 56 82 68)(22 57 83 69)(23 39 84 70)(24 40 85 71)(25 41 86 72)(26 42 87 73)(27 43 88 74)(28 44 89 75)(29 45 90 76)(30 46 91 58)(31 47 92 59)(32 48 93 60)(33 49 94 61)(34 50 95 62)(35 51 77 63)(36 52 78 64)(37 53 79 65)(38 54 80 66)

G:=sub<Sym(95)| (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), (1,62,34,95,50)(2,63,35,77,51)(3,64,36,78,52)(4,65,37,79,53)(5,66,38,80,54)(6,67,20,81,55)(7,68,21,82,56)(8,69,22,83,57)(9,70,23,84,39)(10,71,24,85,40)(11,72,25,86,41)(12,73,26,87,42)(13,74,27,88,43)(14,75,28,89,44)(15,76,29,90,45)(16,58,30,91,46)(17,59,31,92,47)(18,60,32,93,48)(19,61,33,94,49), (20,55,81,67)(21,56,82,68)(22,57,83,69)(23,39,84,70)(24,40,85,71)(25,41,86,72)(26,42,87,73)(27,43,88,74)(28,44,89,75)(29,45,90,76)(30,46,91,58)(31,47,92,59)(32,48,93,60)(33,49,94,61)(34,50,95,62)(35,51,77,63)(36,52,78,64)(37,53,79,65)(38,54,80,66)>;

G:=Group( (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), (1,62,34,95,50)(2,63,35,77,51)(3,64,36,78,52)(4,65,37,79,53)(5,66,38,80,54)(6,67,20,81,55)(7,68,21,82,56)(8,69,22,83,57)(9,70,23,84,39)(10,71,24,85,40)(11,72,25,86,41)(12,73,26,87,42)(13,74,27,88,43)(14,75,28,89,44)(15,76,29,90,45)(16,58,30,91,46)(17,59,31,92,47)(18,60,32,93,48)(19,61,33,94,49), (20,55,81,67)(21,56,82,68)(22,57,83,69)(23,39,84,70)(24,40,85,71)(25,41,86,72)(26,42,87,73)(27,43,88,74)(28,44,89,75)(29,45,90,76)(30,46,91,58)(31,47,92,59)(32,48,93,60)(33,49,94,61)(34,50,95,62)(35,51,77,63)(36,52,78,64)(37,53,79,65)(38,54,80,66) );

G=PermutationGroup([(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)], [(1,62,34,95,50),(2,63,35,77,51),(3,64,36,78,52),(4,65,37,79,53),(5,66,38,80,54),(6,67,20,81,55),(7,68,21,82,56),(8,69,22,83,57),(9,70,23,84,39),(10,71,24,85,40),(11,72,25,86,41),(12,73,26,87,42),(13,74,27,88,43),(14,75,28,89,44),(15,76,29,90,45),(16,58,30,91,46),(17,59,31,92,47),(18,60,32,93,48),(19,61,33,94,49)], [(20,55,81,67),(21,56,82,68),(22,57,83,69),(23,39,84,70),(24,40,85,71),(25,41,86,72),(26,42,87,73),(27,43,88,74),(28,44,89,75),(29,45,90,76),(30,46,91,58),(31,47,92,59),(32,48,93,60),(33,49,94,61),(34,50,95,62),(35,51,77,63),(36,52,78,64),(37,53,79,65),(38,54,80,66)])

95 conjugacy classes

class 1  2 4A4B 5 19A···19R38A···38R76A···76AJ95A···95R
order1244519···1938···3876···7695···95
size155541···15···55···54···4

95 irreducible representations

dim11111144
type+++
imageC1C2C4C19C38C76F5C19×F5
kernelC19×F5D5×C19C95F5D5C5C19C1
# reps112181836118

Matrix representation of C19×F5 in GL4(𝔽761) generated by

680000
068000
006800
000680
,
760760760760
1000
0100
0010
,
1000
0001
0100
760760760760
G:=sub<GL(4,GF(761))| [680,0,0,0,0,680,0,0,0,0,680,0,0,0,0,680],[760,1,0,0,760,0,1,0,760,0,0,1,760,0,0,0],[1,0,0,760,0,0,1,760,0,0,0,760,0,1,0,760] >;

C19×F5 in GAP, Magma, Sage, TeX

C_{19}\times F_5
% in TeX

G:=Group("C19xF5");
// GroupNames label

G:=SmallGroup(380,5);
// by ID

G=gap.SmallGroup(380,5);
# by ID

G:=PCGroup([4,-2,-19,-2,-5,152,2435,139]);
// Polycyclic

G:=Group<a,b,c|a^19=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

Export

Subgroup lattice of C19×F5 in TeX

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