Copied to
clipboard

G = S3xC7xD5order 420 = 22·3·5·7

Direct product of C7, S3 and D5

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3xC7xD5, C35:6D6, D15:C14, C21:6D10, C105:7C22, C15:(C2xC14), (C5xS3):C14, C5:1(S3xC14), (C3xD5):C14, C3:1(D5xC14), (S3xC35):3C2, (D5xC21):3C2, (C7xD15):3C2, SmallGroup(420,27)

Series: Derived Chief Lower central Upper central

C1C15 — S3xC7xD5
C1C5C15C105D5xC21 — S3xC7xD5
C15 — S3xC7xD5
C1C7

Generators and relations for S3xC7xD5
 G = < a,b,c,d,e | a7=b3=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 144 in 40 conjugacy classes, 20 normal (all characteristic)
Quotients: C1, C2, C22, S3, C7, D5, D6, C14, D10, C2xC14, S3xC7, S3xD5, C7xD5, S3xC14, D5xC14, S3xC7xD5
3C2
5C2
15C2
15C22
5S3
5C6
3D5
3C10
3C14
5C14
15C14
5D6
3D10
15C2xC14
5S3xC7
5C42
3C7xD5
3C70
5S3xC14
3D5xC14

Smallest permutation representation of S3xC7xD5
On 105 points
Generators in S105
(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)
(1 54 94)(2 55 95)(3 56 96)(4 50 97)(5 51 98)(6 52 92)(7 53 93)(8 39 58)(9 40 59)(10 41 60)(11 42 61)(12 36 62)(13 37 63)(14 38 57)(15 85 65)(16 86 66)(17 87 67)(18 88 68)(19 89 69)(20 90 70)(21 91 64)(22 100 43)(23 101 44)(24 102 45)(25 103 46)(26 104 47)(27 105 48)(28 99 49)(29 75 79)(30 76 80)(31 77 81)(32 71 82)(33 72 83)(34 73 84)(35 74 78)
(8 39)(9 40)(10 41)(11 42)(12 36)(13 37)(14 38)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 75)(30 76)(31 77)(32 71)(33 72)(34 73)(35 74)(50 97)(51 98)(52 92)(53 93)(54 94)(55 95)(56 96)
(1 70 79 102 59)(2 64 80 103 60)(3 65 81 104 61)(4 66 82 105 62)(5 67 83 99 63)(6 68 84 100 57)(7 69 78 101 58)(8 53 19 35 44)(9 54 20 29 45)(10 55 21 30 46)(11 56 15 31 47)(12 50 16 32 48)(13 51 17 33 49)(14 52 18 34 43)(22 38 92 88 73)(23 39 93 89 74)(24 40 94 90 75)(25 41 95 91 76)(26 42 96 85 77)(27 36 97 86 71)(28 37 98 87 72)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 53)(9 54)(10 55)(11 56)(12 50)(13 51)(14 52)(15 47)(16 48)(17 49)(18 43)(19 44)(20 45)(21 46)(22 88)(23 89)(24 90)(25 91)(26 85)(27 86)(28 87)(36 97)(37 98)(38 92)(39 93)(40 94)(41 95)(42 96)(64 103)(65 104)(66 105)(67 99)(68 100)(69 101)(70 102)

G:=sub<Sym(105)| (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), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,75)(30,76)(31,77)(32,71)(33,72)(34,73)(35,74)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,53)(9,54)(10,55)(11,56)(12,50)(13,51)(14,52)(15,47)(16,48)(17,49)(18,43)(19,44)(20,45)(21,46)(22,88)(23,89)(24,90)(25,91)(26,85)(27,86)(28,87)(36,97)(37,98)(38,92)(39,93)(40,94)(41,95)(42,96)(64,103)(65,104)(66,105)(67,99)(68,100)(69,101)(70,102)>;

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)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105), (1,54,94)(2,55,95)(3,56,96)(4,50,97)(5,51,98)(6,52,92)(7,53,93)(8,39,58)(9,40,59)(10,41,60)(11,42,61)(12,36,62)(13,37,63)(14,38,57)(15,85,65)(16,86,66)(17,87,67)(18,88,68)(19,89,69)(20,90,70)(21,91,64)(22,100,43)(23,101,44)(24,102,45)(25,103,46)(26,104,47)(27,105,48)(28,99,49)(29,75,79)(30,76,80)(31,77,81)(32,71,82)(33,72,83)(34,73,84)(35,74,78), (8,39)(9,40)(10,41)(11,42)(12,36)(13,37)(14,38)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,75)(30,76)(31,77)(32,71)(33,72)(34,73)(35,74)(50,97)(51,98)(52,92)(53,93)(54,94)(55,95)(56,96), (1,70,79,102,59)(2,64,80,103,60)(3,65,81,104,61)(4,66,82,105,62)(5,67,83,99,63)(6,68,84,100,57)(7,69,78,101,58)(8,53,19,35,44)(9,54,20,29,45)(10,55,21,30,46)(11,56,15,31,47)(12,50,16,32,48)(13,51,17,33,49)(14,52,18,34,43)(22,38,92,88,73)(23,39,93,89,74)(24,40,94,90,75)(25,41,95,91,76)(26,42,96,85,77)(27,36,97,86,71)(28,37,98,87,72), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,53)(9,54)(10,55)(11,56)(12,50)(13,51)(14,52)(15,47)(16,48)(17,49)(18,43)(19,44)(20,45)(21,46)(22,88)(23,89)(24,90)(25,91)(26,85)(27,86)(28,87)(36,97)(37,98)(38,92)(39,93)(40,94)(41,95)(42,96)(64,103)(65,104)(66,105)(67,99)(68,100)(69,101)(70,102) );

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),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105)], [(1,54,94),(2,55,95),(3,56,96),(4,50,97),(5,51,98),(6,52,92),(7,53,93),(8,39,58),(9,40,59),(10,41,60),(11,42,61),(12,36,62),(13,37,63),(14,38,57),(15,85,65),(16,86,66),(17,87,67),(18,88,68),(19,89,69),(20,90,70),(21,91,64),(22,100,43),(23,101,44),(24,102,45),(25,103,46),(26,104,47),(27,105,48),(28,99,49),(29,75,79),(30,76,80),(31,77,81),(32,71,82),(33,72,83),(34,73,84),(35,74,78)], [(8,39),(9,40),(10,41),(11,42),(12,36),(13,37),(14,38),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,75),(30,76),(31,77),(32,71),(33,72),(34,73),(35,74),(50,97),(51,98),(52,92),(53,93),(54,94),(55,95),(56,96)], [(1,70,79,102,59),(2,64,80,103,60),(3,65,81,104,61),(4,66,82,105,62),(5,67,83,99,63),(6,68,84,100,57),(7,69,78,101,58),(8,53,19,35,44),(9,54,20,29,45),(10,55,21,30,46),(11,56,15,31,47),(12,50,16,32,48),(13,51,17,33,49),(14,52,18,34,43),(22,38,92,88,73),(23,39,93,89,74),(24,40,94,90,75),(25,41,95,91,76),(26,42,96,85,77),(27,36,97,86,71),(28,37,98,87,72)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,53),(9,54),(10,55),(11,56),(12,50),(13,51),(14,52),(15,47),(16,48),(17,49),(18,43),(19,44),(20,45),(21,46),(22,88),(23,89),(24,90),(25,91),(26,85),(27,86),(28,87),(36,97),(37,98),(38,92),(39,93),(40,94),(41,95),(42,96),(64,103),(65,104),(66,105),(67,99),(68,100),(69,101),(70,102)]])

84 conjugacy classes

class 1 2A2B2C 3 5A5B 6 7A···7F10A10B14A···14F14G···14L14M···14R15A15B21A···21F35A···35L42A···42F70A···70L105A···105L
order122235567···7101014···1414···1414···14151521···2135···3542···4270···70105···105
size13515222101···1663···35···515···15442···22···210···106···64···4

84 irreducible representations

dim111111112222222244
type+++++++++
imageC1C2C2C2C7C14C14C14S3D5D6D10S3xC7C7xD5S3xC14D5xC14S3xD5S3xC7xD5
kernelS3xC7xD5D5xC21S3xC35C7xD15S3xD5C5xS3C3xD5D15C7xD5S3xC7C35C21D5S3C5C3C7C1
# reps111166661212612612212

Matrix representation of S3xC7xD5 in GL4(F211) generated by

123000
012300
00580
00058
,
1000
0100
00209176
001931
,
1000
0100
0010
0018210
,
0100
21017800
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(211))| [123,0,0,0,0,123,0,0,0,0,58,0,0,0,0,58],[1,0,0,0,0,1,0,0,0,0,209,193,0,0,176,1],[1,0,0,0,0,1,0,0,0,0,1,18,0,0,0,210],[0,210,0,0,1,178,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

S3xC7xD5 in GAP, Magma, Sage, TeX

S_3\times C_7\times D_5
% in TeX

G:=Group("S3xC7xD5");
// GroupNames label

G:=SmallGroup(420,27);
// by ID

G=gap.SmallGroup(420,27);
# by ID

G:=PCGroup([5,-2,-2,-7,-3,-5,568,8404]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^3=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

Export

Subgroup lattice of S3xC7xD5 in TeX

׿
x
:
Z
F
o
wr
Q
<