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G = C19⋊F5order 380 = 22·5·19

The semidirect product of C19 and F5 acting via F5/D5=C2

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

Aliases: C19⋊F5, C951C4, C5⋊Dic19, D5.D19, (D5×C19).1C2, SmallGroup(380,6)

Series: Derived Chief Lower central Upper central

C1C95 — C19⋊F5
C1C19C95D5×C19 — C19⋊F5
C95 — C19⋊F5
C1

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

5C2
95C4
5C38
19F5
5Dic19

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

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

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

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

41 conjugacy classes

class 1  2 4A4B 5 19A···19I38A···38I95A···95R
order1244519···1938···3895···95
size15959542···210···104···4

41 irreducible representations

dim1112244
type+++-+
imageC1C2C4D19Dic19F5C19⋊F5
kernelC19⋊F5D5×C19C95D5C5C19C1
# reps11299118

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

5032800
76010000
0050328
00760100
,
007600
000760
10712545
0140948
,
323127438634
335438426323
0089382
00553672
G:=sub<GL(4,GF(761))| [50,760,0,0,328,100,0,0,0,0,50,760,0,0,328,100],[0,0,1,0,0,0,0,1,760,0,712,409,0,760,545,48],[323,335,0,0,127,438,0,0,438,426,89,553,634,323,382,672] >;

C19⋊F5 in GAP, Magma, Sage, TeX

C_{19}\rtimes F_5
% in TeX

G:=Group("C19:F5");
// GroupNames label

G:=SmallGroup(380,6);
// by ID

G=gap.SmallGroup(380,6);
# by ID

G:=PCGroup([4,-2,-2,-5,-19,8,146,102,5763]);
// Polycyclic

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

Export

Subgroup lattice of C19⋊F5 in TeX

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