Copied to
clipboard

## G = C32×C13⋊C3order 351 = 33·13

### Direct product of C32 and C13⋊C3

Aliases: C32×C13⋊C3, C13⋊C33, C39⋊C32, (C3×C39)⋊3C3, SmallGroup(351,13)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C32×C13⋊C3
G = < a,b,c,d | a3=b3=c13=d3=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c9 >

Subgroups: 320 in 56 conjugacy classes, 34 normal (5 characteristic)
C1, C3, C3, C32, C32, C13, C33, C13⋊C3, C39, C3×C13⋊C3, C3×C39, C32×C13⋊C3
Quotients: C1, C3, C32, C33, C13⋊C3, C3×C13⋊C3, C32×C13⋊C3

Smallest permutation representation of C32×C13⋊C3
On 117 points
Generators in S117
(1 105 53)(2 106 54)(3 107 55)(4 108 56)(5 109 57)(6 110 58)(7 111 59)(8 112 60)(9 113 61)(10 114 62)(11 115 63)(12 116 64)(13 117 65)(14 79 66)(15 80 67)(16 81 68)(17 82 69)(18 83 70)(19 84 71)(20 85 72)(21 86 73)(22 87 74)(23 88 75)(24 89 76)(25 90 77)(26 91 78)(27 92 40)(28 93 41)(29 94 42)(30 95 43)(31 96 44)(32 97 45)(33 98 46)(34 99 47)(35 100 48)(36 101 49)(37 102 50)(38 103 51)(39 104 52)
(1 27 14)(2 28 15)(3 29 16)(4 30 17)(5 31 18)(6 32 19)(7 33 20)(8 34 21)(9 35 22)(10 36 23)(11 37 24)(12 38 25)(13 39 26)(40 66 53)(41 67 54)(42 68 55)(43 69 56)(44 70 57)(45 71 58)(46 72 59)(47 73 60)(48 74 61)(49 75 62)(50 76 63)(51 77 64)(52 78 65)(79 105 92)(80 106 93)(81 107 94)(82 108 95)(83 109 96)(84 110 97)(85 111 98)(86 112 99)(87 113 100)(88 114 101)(89 115 102)(90 116 103)(91 117 104)
(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 14 27)(2 17 36)(3 20 32)(4 23 28)(5 26 37)(6 16 33)(7 19 29)(8 22 38)(9 25 34)(10 15 30)(11 18 39)(12 21 35)(13 24 31)(40 53 66)(41 56 75)(42 59 71)(43 62 67)(44 65 76)(45 55 72)(46 58 68)(47 61 77)(48 64 73)(49 54 69)(50 57 78)(51 60 74)(52 63 70)(79 92 105)(80 95 114)(81 98 110)(82 101 106)(83 104 115)(84 94 111)(85 97 107)(86 100 116)(87 103 112)(88 93 108)(89 96 117)(90 99 113)(91 102 109)

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

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

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

63 conjugacy classes

 class 1 3A ··· 3H 3I ··· 3Z 13A 13B 13C 13D 39A ··· 39AF order 1 3 ··· 3 3 ··· 3 13 13 13 13 39 ··· 39 size 1 1 ··· 1 13 ··· 13 3 3 3 3 3 ··· 3

63 irreducible representations

 dim 1 1 1 3 3 type + image C1 C3 C3 C13⋊C3 C3×C13⋊C3 kernel C32×C13⋊C3 C3×C13⋊C3 C3×C39 C32 C3 # reps 1 24 2 4 32

Matrix representation of C32×C13⋊C3 in GL4(𝔽79) generated by

 1 0 0 0 0 55 0 0 0 0 55 0 0 0 0 55
,
 23 0 0 0 0 23 0 0 0 0 23 0 0 0 0 23
,
 1 0 0 0 0 29 41 29 0 1 0 67 0 0 1 40
,
 23 0 0 0 0 52 15 73 0 61 25 31 0 7 6 2
G:=sub<GL(4,GF(79))| [1,0,0,0,0,55,0,0,0,0,55,0,0,0,0,55],[23,0,0,0,0,23,0,0,0,0,23,0,0,0,0,23],[1,0,0,0,0,29,1,0,0,41,0,1,0,29,67,40],[23,0,0,0,0,52,61,7,0,15,25,6,0,73,31,2] >;

C32×C13⋊C3 in GAP, Magma, Sage, TeX

C_3^2\times C_{13}\rtimes C_3
% in TeX

G:=Group("C3^2xC13:C3");
// GroupNames label

G:=SmallGroup(351,13);
// by ID

G=gap.SmallGroup(351,13);
# by ID

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

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

׿
×
𝔽