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G = C3×C43⋊C3order 387 = 32·43

Direct product of C3 and C43⋊C3

direct product, metacyclic, supersoluble, monomial, A-group, 3-hyperelementary

Aliases: C3×C43⋊C3, C129⋊C3, C43⋊C32, SmallGroup(387,3)

Series: Derived Chief Lower central Upper central

C1C43 — C3×C43⋊C3
C1C43C43⋊C3 — C3×C43⋊C3
C43 — C3×C43⋊C3
C1C3

Generators and relations for C3×C43⋊C3
 G = < a,b,c | a3=b43=c3=1, ab=ba, ac=ca, cbc-1=b6 >

43C3
43C3
43C3
43C32

Smallest permutation representation of C3×C43⋊C3
On 129 points
Generators in S129
(1 87 44)(2 88 45)(3 89 46)(4 90 47)(5 91 48)(6 92 49)(7 93 50)(8 94 51)(9 95 52)(10 96 53)(11 97 54)(12 98 55)(13 99 56)(14 100 57)(15 101 58)(16 102 59)(17 103 60)(18 104 61)(19 105 62)(20 106 63)(21 107 64)(22 108 65)(23 109 66)(24 110 67)(25 111 68)(26 112 69)(27 113 70)(28 114 71)(29 115 72)(30 116 73)(31 117 74)(32 118 75)(33 119 76)(34 120 77)(35 121 78)(36 122 79)(37 123 80)(38 124 81)(39 125 82)(40 126 83)(41 127 84)(42 128 85)(43 129 86)
(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)
(1 44 87)(2 80 93)(3 73 99)(4 66 105)(5 59 111)(6 52 117)(7 45 123)(8 81 129)(9 74 92)(10 67 98)(11 60 104)(12 53 110)(13 46 116)(14 82 122)(15 75 128)(16 68 91)(17 61 97)(18 54 103)(19 47 109)(20 83 115)(21 76 121)(22 69 127)(23 62 90)(24 55 96)(25 48 102)(26 84 108)(27 77 114)(28 70 120)(29 63 126)(30 56 89)(31 49 95)(32 85 101)(33 78 107)(34 71 113)(35 64 119)(36 57 125)(37 50 88)(38 86 94)(39 79 100)(40 72 106)(41 65 112)(42 58 118)(43 51 124)

G:=sub<Sym(129)| (1,87,44)(2,88,45)(3,89,46)(4,90,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,102,59)(17,103,60)(18,104,61)(19,105,62)(20,106,63)(21,107,64)(22,108,65)(23,109,66)(24,110,67)(25,111,68)(26,112,69)(27,113,70)(28,114,71)(29,115,72)(30,116,73)(31,117,74)(32,118,75)(33,119,76)(34,120,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86), (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), (1,44,87)(2,80,93)(3,73,99)(4,66,105)(5,59,111)(6,52,117)(7,45,123)(8,81,129)(9,74,92)(10,67,98)(11,60,104)(12,53,110)(13,46,116)(14,82,122)(15,75,128)(16,68,91)(17,61,97)(18,54,103)(19,47,109)(20,83,115)(21,76,121)(22,69,127)(23,62,90)(24,55,96)(25,48,102)(26,84,108)(27,77,114)(28,70,120)(29,63,126)(30,56,89)(31,49,95)(32,85,101)(33,78,107)(34,71,113)(35,64,119)(36,57,125)(37,50,88)(38,86,94)(39,79,100)(40,72,106)(41,65,112)(42,58,118)(43,51,124)>;

G:=Group( (1,87,44)(2,88,45)(3,89,46)(4,90,47)(5,91,48)(6,92,49)(7,93,50)(8,94,51)(9,95,52)(10,96,53)(11,97,54)(12,98,55)(13,99,56)(14,100,57)(15,101,58)(16,102,59)(17,103,60)(18,104,61)(19,105,62)(20,106,63)(21,107,64)(22,108,65)(23,109,66)(24,110,67)(25,111,68)(26,112,69)(27,113,70)(28,114,71)(29,115,72)(30,116,73)(31,117,74)(32,118,75)(33,119,76)(34,120,77)(35,121,78)(36,122,79)(37,123,80)(38,124,81)(39,125,82)(40,126,83)(41,127,84)(42,128,85)(43,129,86), (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), (1,44,87)(2,80,93)(3,73,99)(4,66,105)(5,59,111)(6,52,117)(7,45,123)(8,81,129)(9,74,92)(10,67,98)(11,60,104)(12,53,110)(13,46,116)(14,82,122)(15,75,128)(16,68,91)(17,61,97)(18,54,103)(19,47,109)(20,83,115)(21,76,121)(22,69,127)(23,62,90)(24,55,96)(25,48,102)(26,84,108)(27,77,114)(28,70,120)(29,63,126)(30,56,89)(31,49,95)(32,85,101)(33,78,107)(34,71,113)(35,64,119)(36,57,125)(37,50,88)(38,86,94)(39,79,100)(40,72,106)(41,65,112)(42,58,118)(43,51,124) );

G=PermutationGroup([[(1,87,44),(2,88,45),(3,89,46),(4,90,47),(5,91,48),(6,92,49),(7,93,50),(8,94,51),(9,95,52),(10,96,53),(11,97,54),(12,98,55),(13,99,56),(14,100,57),(15,101,58),(16,102,59),(17,103,60),(18,104,61),(19,105,62),(20,106,63),(21,107,64),(22,108,65),(23,109,66),(24,110,67),(25,111,68),(26,112,69),(27,113,70),(28,114,71),(29,115,72),(30,116,73),(31,117,74),(32,118,75),(33,119,76),(34,120,77),(35,121,78),(36,122,79),(37,123,80),(38,124,81),(39,125,82),(40,126,83),(41,127,84),(42,128,85),(43,129,86)], [(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)], [(1,44,87),(2,80,93),(3,73,99),(4,66,105),(5,59,111),(6,52,117),(7,45,123),(8,81,129),(9,74,92),(10,67,98),(11,60,104),(12,53,110),(13,46,116),(14,82,122),(15,75,128),(16,68,91),(17,61,97),(18,54,103),(19,47,109),(20,83,115),(21,76,121),(22,69,127),(23,62,90),(24,55,96),(25,48,102),(26,84,108),(27,77,114),(28,70,120),(29,63,126),(30,56,89),(31,49,95),(32,85,101),(33,78,107),(34,71,113),(35,64,119),(36,57,125),(37,50,88),(38,86,94),(39,79,100),(40,72,106),(41,65,112),(42,58,118),(43,51,124)]])

51 conjugacy classes

class 1 3A3B3C···3H43A···43N129A···129AB
order1333···343···43129···129
size11143···433···33···3

51 irreducible representations

dim11133
type+
imageC1C3C3C43⋊C3C3×C43⋊C3
kernelC3×C43⋊C3C43⋊C3C129C3C1
# reps1621428

Matrix representation of C3×C43⋊C3 in GL3(𝔽1033) generated by

83700
08370
00837
,
001
10831
01476
,
195702259
0471385
098367
G:=sub<GL(3,GF(1033))| [837,0,0,0,837,0,0,0,837],[0,1,0,0,0,1,1,831,476],[195,0,0,702,471,98,259,385,367] >;

C3×C43⋊C3 in GAP, Magma, Sage, TeX

C_3\times C_{43}\rtimes C_3
% in TeX

G:=Group("C3xC43:C3");
// GroupNames label

G:=SmallGroup(387,3);
// by ID

G=gap.SmallGroup(387,3);
# by ID

G:=PCGroup([3,-3,-3,-43,2918]);
// Polycyclic

G:=Group<a,b,c|a^3=b^43=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^6>;
// generators/relations

Export

Subgroup lattice of C3×C43⋊C3 in TeX

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