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G = C3⋊D39order 234 = 2·32·13

The semidirect product of C3 and D39 acting via D39/C39=C2

metabelian, supersoluble, monomial, A-group

Aliases: C3⋊D39, C391S3, C322D13, C13⋊(C3⋊S3), (C3×C39)⋊1C2, SmallGroup(234,15)

Series: Derived Chief Lower central Upper central

C1C3×C39 — C3⋊D39
C1C13C39C3×C39 — C3⋊D39
C3×C39 — C3⋊D39
C1

Generators and relations for C3⋊D39
 G = < a,b,c | a3=b39=c2=1, ab=ba, cac=a-1, cbc=b-1 >

117C2
39S3
39S3
39S3
39S3
9D13
13C3⋊S3
3D39
3D39
3D39
3D39

Smallest permutation representation of C3⋊D39
On 117 points
Generators in S117
(1 92 51)(2 93 52)(3 94 53)(4 95 54)(5 96 55)(6 97 56)(7 98 57)(8 99 58)(9 100 59)(10 101 60)(11 102 61)(12 103 62)(13 104 63)(14 105 64)(15 106 65)(16 107 66)(17 108 67)(18 109 68)(19 110 69)(20 111 70)(21 112 71)(22 113 72)(23 114 73)(24 115 74)(25 116 75)(26 117 76)(27 79 77)(28 80 78)(29 81 40)(30 82 41)(31 83 42)(32 84 43)(33 85 44)(34 86 45)(35 87 46)(36 88 47)(37 89 48)(38 90 49)(39 91 50)
(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)
(1 39)(2 38)(3 37)(4 36)(5 35)(6 34)(7 33)(8 32)(9 31)(10 30)(11 29)(12 28)(13 27)(14 26)(15 25)(16 24)(17 23)(18 22)(19 21)(40 102)(41 101)(42 100)(43 99)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 85)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 117)(65 116)(66 115)(67 114)(68 113)(69 112)(70 111)(71 110)(72 109)(73 108)(74 107)(75 106)(76 105)(77 104)(78 103)

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

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

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

C3⋊D39 is a maximal subgroup of   (C3×C39)⋊C4  C3⋊S3×D13  S3×D39
C3⋊D39 is a maximal quotient of   C3⋊Dic39

60 conjugacy classes

class 1  2 3A3B3C3D13A···13F39A···39AV
order12333313···1339···39
size111722222···22···2

60 irreducible representations

dim11222
type+++++
imageC1C2S3D13D39
kernelC3⋊D39C3×C39C39C32C3
# reps114648

Matrix representation of C3⋊D39 in GL4(𝔽79) generated by

497100
202900
0010
0001
,
347000
625100
001919
007723
,
112500
116800
00653
001414
G:=sub<GL(4,GF(79))| [49,20,0,0,71,29,0,0,0,0,1,0,0,0,0,1],[34,62,0,0,70,51,0,0,0,0,19,77,0,0,19,23],[11,11,0,0,25,68,0,0,0,0,65,14,0,0,3,14] >;

C3⋊D39 in GAP, Magma, Sage, TeX

C_3\rtimes D_{39}
% in TeX

G:=Group("C3:D39");
// GroupNames label

G:=SmallGroup(234,15);
// by ID

G=gap.SmallGroup(234,15);
# by ID

G:=PCGroup([4,-2,-3,-3,-13,33,146,3459]);
// Polycyclic

G:=Group<a,b,c|a^3=b^39=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C3⋊D39 in TeX

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