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G = C32×D13order 234 = 2·32·13

Direct product of C32 and D13

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

Aliases: C32×D13, C396C6, (C3×C39)⋊3C2, C133(C3×C6), SmallGroup(234,11)

Series: Derived Chief Lower central Upper central

C1C13 — C32×D13
C1C13C39C3×C39 — C32×D13
C13 — C32×D13
C1C32

Generators and relations for C32×D13
 G = < a,b,c,d | a3=b3=c13=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

13C2
13C6
13C6
13C6
13C6
13C3×C6

Smallest permutation representation of C32×D13
On 117 points
Generators in S117
(1 110 61)(2 111 62)(3 112 63)(4 113 64)(5 114 65)(6 115 53)(7 116 54)(8 117 55)(9 105 56)(10 106 57)(11 107 58)(12 108 59)(13 109 60)(14 88 67)(15 89 68)(16 90 69)(17 91 70)(18 79 71)(19 80 72)(20 81 73)(21 82 74)(22 83 75)(23 84 76)(24 85 77)(25 86 78)(26 87 66)(27 93 52)(28 94 40)(29 95 41)(30 96 42)(31 97 43)(32 98 44)(33 99 45)(34 100 46)(35 101 47)(36 102 48)(37 103 49)(38 104 50)(39 92 51)
(1 27 15)(2 28 16)(3 29 17)(4 30 18)(5 31 19)(6 32 20)(7 33 21)(8 34 22)(9 35 23)(10 36 24)(11 37 25)(12 38 26)(13 39 14)(40 69 62)(41 70 63)(42 71 64)(43 72 65)(44 73 53)(45 74 54)(46 75 55)(47 76 56)(48 77 57)(49 78 58)(50 66 59)(51 67 60)(52 68 61)(79 113 96)(80 114 97)(81 115 98)(82 116 99)(83 117 100)(84 105 101)(85 106 102)(86 107 103)(87 108 104)(88 109 92)(89 110 93)(90 111 94)(91 112 95)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 15)(16 26)(17 25)(18 24)(19 23)(20 22)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 50)(41 49)(42 48)(43 47)(44 46)(51 52)(53 55)(56 65)(57 64)(58 63)(59 62)(60 61)(66 69)(67 68)(70 78)(71 77)(72 76)(73 75)(79 85)(80 84)(81 83)(86 91)(87 90)(88 89)(92 93)(94 104)(95 103)(96 102)(97 101)(98 100)(105 114)(106 113)(107 112)(108 111)(109 110)(115 117)

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

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

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

C32×D13 is a maximal subgroup of   C39⋊Dic3

72 conjugacy classes

class 1  2 3A···3H6A···6H13A···13F39A···39AV
order123···36···613···1339···39
size1131···113···132···22···2

72 irreducible representations

dim111122
type+++
imageC1C2C3C6D13C3×D13
kernelC32×D13C3×C39C3×D13C39C32C3
# reps1188648

Matrix representation of C32×D13 in GL4(𝔽79) generated by

1000
05500
0010
0001
,
23000
0100
0010
0001
,
1000
0100
0001
007818
,
78000
07800
0001
0010
G:=sub<GL(4,GF(79))| [1,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[23,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,78,0,0,1,18],[78,0,0,0,0,78,0,0,0,0,0,1,0,0,1,0] >;

C32×D13 in GAP, Magma, Sage, TeX

C_3^2\times D_{13}
% in TeX

G:=Group("C3^2xD13");
// GroupNames label

G:=SmallGroup(234,11);
// by ID

G=gap.SmallGroup(234,11);
# by ID

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

G:=Group<a,b,c,d|a^3=b^3=c^13=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

Subgroup lattice of C32×D13 in TeX

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