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## G = C13⋊He3order 351 = 33·13

### The semidirect product of C13 and He3 acting via He3/C32=C3

Aliases: C13⋊He3, C39.6C32, (C3×C39)⋊2C3, C32⋊(C13⋊C3), (C3×C13⋊C3)⋊C3, C3.6(C3×C13⋊C3), SmallGroup(351,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — C13⋊He3
 Chief series C1 — C13 — C39 — C3×C13⋊C3 — C13⋊He3
 Lower central C13 — C39 — C13⋊He3
 Upper central C1 — C3 — C32

Generators and relations for C13⋊He3
G = < a,b,c,d | a13=b3=c3=d3=1, ab=ba, ac=ca, dad-1=a9, bc=cb, dbd-1=bc-1, cd=dc >

Smallest permutation representation of C13⋊He3
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 88 46)(2 89 47)(3 90 48)(4 91 49)(5 79 50)(6 80 51)(7 81 52)(8 82 40)(9 83 41)(10 84 42)(11 85 43)(12 86 44)(13 87 45)(14 99 61)(15 100 62)(16 101 63)(17 102 64)(18 103 65)(19 104 53)(20 92 54)(21 93 55)(22 94 56)(23 95 57)(24 96 58)(25 97 59)(26 98 60)(27 105 68)(28 106 69)(29 107 70)(30 108 71)(31 109 72)(32 110 73)(33 111 74)(34 112 75)(35 113 76)(36 114 77)(37 115 78)(38 116 66)(39 117 67)
(1 35 17)(2 36 18)(3 37 19)(4 38 20)(5 39 21)(6 27 22)(7 28 23)(8 29 24)(9 30 25)(10 31 26)(11 32 14)(12 33 15)(13 34 16)(40 70 58)(41 71 59)(42 72 60)(43 73 61)(44 74 62)(45 75 63)(46 76 64)(47 77 65)(48 78 53)(49 66 54)(50 67 55)(51 68 56)(52 69 57)(79 117 93)(80 105 94)(81 106 95)(82 107 96)(83 108 97)(84 109 98)(85 110 99)(86 111 100)(87 112 101)(88 113 102)(89 114 103)(90 115 104)(91 116 92)
(2 4 10)(3 7 6)(5 13 11)(8 9 12)(14 21 16)(15 24 25)(18 20 26)(19 23 22)(27 37 28)(29 30 33)(31 36 38)(32 39 34)(40 59 74)(41 62 70)(42 65 66)(43 55 75)(44 58 71)(45 61 67)(46 64 76)(47 54 72)(48 57 68)(49 60 77)(50 63 73)(51 53 69)(52 56 78)(79 112 99)(80 115 95)(81 105 104)(82 108 100)(83 111 96)(84 114 92)(85 117 101)(86 107 97)(87 110 93)(88 113 102)(89 116 98)(90 106 94)(91 109 103)

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

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

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

47 conjugacy classes

 class 1 3A 3B 3C 3D 3E ··· 3J 13A 13B 13C 13D 39A ··· 39AF order 1 3 3 3 3 3 ··· 3 13 13 13 13 39 ··· 39 size 1 1 1 3 3 39 ··· 39 3 3 3 3 3 ··· 3

47 irreducible representations

 dim 1 1 1 3 3 3 3 type + image C1 C3 C3 He3 C13⋊C3 C3×C13⋊C3 C13⋊He3 kernel C13⋊He3 C3×C13⋊C3 C3×C39 C13 C32 C3 C1 # reps 1 6 2 2 4 8 24

Matrix representation of C13⋊He3 in GL3(𝔽79) generated by

 0 1 0 0 0 1 1 10 25
,
 55 8 57 57 72 11 11 9 31
,
 23 0 0 0 23 0 0 0 23
,
 1 0 0 24 68 54 3 55 10
G:=sub<GL(3,GF(79))| [0,0,1,1,0,10,0,1,25],[55,57,11,8,72,9,57,11,31],[23,0,0,0,23,0,0,0,23],[1,24,3,0,68,55,0,54,10] >;

C13⋊He3 in GAP, Magma, Sage, TeX

C_{13}\rtimes {\rm He}_3
% in TeX

G:=Group("C13:He3");
// GroupNames label

G:=SmallGroup(351,8);
// by ID

G=gap.SmallGroup(351,8);
# by ID

G:=PCGroup([4,-3,-3,-3,-13,97,1299]);
// Polycyclic

G:=Group<a,b,c,d|a^13=b^3=c^3=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^9,b*c=c*b,d*b*d^-1=b*c^-1,c*d=d*c>;
// generators/relations

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