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## G = C39⋊Dic3order 468 = 22·32·13

### 1st semidirect product of C39 and Dic3 acting via Dic3/C3=C4

Aliases: C391Dic3, (C3×C39)⋊2C4, C3⋊(C39⋊C4), C13⋊(C3⋊Dic3), D13.(C3⋊S3), C323(C13⋊C4), (C3×D13).3S3, (C32×D13).1C2, SmallGroup(468,38)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C39 — C39⋊Dic3
 Chief series C1 — C13 — C39 — C3×C39 — C32×D13 — C39⋊Dic3
 Lower central C3×C39 — C39⋊Dic3
 Upper central C1

Generators and relations for C39⋊Dic3
G = < a,b,c | a39=b6=1, c2=b3, bab-1=a25, cac-1=a8, cbc-1=b-1 >

Smallest permutation representation of C39⋊Dic3
On 117 points
Generators in S117
(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)
(1 112 73)(2 98 74 26 113 59)(3 84 75 12 114 45)(4 109 76 37 115 70)(5 95 77 23 116 56)(6 81 78 9 117 42)(7 106 40 34 79 67)(8 92 41 20 80 53)(10 103 43 31 82 64)(11 89 44 17 83 50)(13 100 46 28 85 61)(14 86 47)(15 111 48 39 87 72)(16 97 49 25 88 58)(18 108 51 36 90 69)(19 94 52 22 91 55)(21 105 54 33 93 66)(24 102 57 30 96 63)(27 99 60)(29 110 62 38 101 71)(32 107 65 35 104 68)
(2 6 26 9)(3 11 12 17)(4 16 37 25)(5 21 23 33)(7 31 34 10)(8 36 20 18)(13 22 28 19)(14 27)(15 32 39 35)(24 38 30 29)(40 103 67 82)(41 108 53 90)(42 113 78 98)(43 79 64 106)(44 84 50 114)(45 89 75 83)(46 94 61 91)(47 99)(48 104 72 107)(49 109 58 115)(51 80 69 92)(52 85 55 100)(54 95 66 116)(56 105 77 93)(57 110 63 101)(59 81 74 117)(60 86)(62 96 71 102)(65 111 68 87)(70 97 76 88)(73 112)

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

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

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

39 conjugacy classes

 class 1 2 3A 3B 3C 3D 4A 4B 6A 6B 6C 6D 13A 13B 13C 39A ··· 39X order 1 2 3 3 3 3 4 4 6 6 6 6 13 13 13 39 ··· 39 size 1 13 2 2 2 2 117 117 26 26 26 26 4 4 4 4 ··· 4

39 irreducible representations

 dim 1 1 1 2 2 4 4 type + + + - + image C1 C2 C4 S3 Dic3 C13⋊C4 C39⋊C4 kernel C39⋊Dic3 C32×D13 C3×C39 C3×D13 C39 C32 C3 # reps 1 1 2 4 4 3 24

Matrix representation of C39⋊Dic3 in GL6(𝔽157)

 0 1 0 0 0 0 156 156 0 0 0 0 0 0 138 155 100 115 0 0 49 73 23 26 0 0 6 106 139 47 0 0 92 140 54 114
,
 0 156 0 0 0 0 1 1 0 0 0 0 0 0 1 18 150 87 0 0 151 64 131 12 0 0 40 98 145 95 0 0 120 17 9 104
,
 28 0 0 0 0 0 129 129 0 0 0 0 0 0 25 146 45 13 0 0 83 105 101 9 0 0 40 119 72 84 0 0 27 24 132 112

G:=sub<GL(6,GF(157))| [0,156,0,0,0,0,1,156,0,0,0,0,0,0,138,49,6,92,0,0,155,73,106,140,0,0,100,23,139,54,0,0,115,26,47,114],[0,1,0,0,0,0,156,1,0,0,0,0,0,0,1,151,40,120,0,0,18,64,98,17,0,0,150,131,145,9,0,0,87,12,95,104],[28,129,0,0,0,0,0,129,0,0,0,0,0,0,25,83,40,27,0,0,146,105,119,24,0,0,45,101,72,132,0,0,13,9,84,112] >;

C39⋊Dic3 in GAP, Magma, Sage, TeX

C_{39}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C39:Dic3");
// GroupNames label

G:=SmallGroup(468,38);
// by ID

G=gap.SmallGroup(468,38);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-13,10,122,483,4504,5409]);
// Polycyclic

G:=Group<a,b,c|a^39=b^6=1,c^2=b^3,b*a*b^-1=a^25,c*a*c^-1=a^8,c*b*c^-1=b^-1>;
// generators/relations

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