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## G = C32×C13⋊C4order 468 = 22·32·13

### Direct product of C32 and C13⋊C4

Aliases: C32×C13⋊C4, C397C12, (C3×C39)⋊4C4, C133(C3×C12), D13.2(C3×C6), (C3×D13).7C6, (C32×D13).3C2, SmallGroup(468,36)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C32×C13⋊C4
 Chief series C1 — C13 — D13 — C3×D13 — C32×D13 — C32×C13⋊C4
 Lower central C13 — C32×C13⋊C4
 Upper central C1 — C32

Generators and relations for C32×C13⋊C4
G = < a,b,c,d | a3=b3=c13=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Smallest permutation representation of C32×C13⋊C4
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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)(67 74 78 71)(68 69 77 76)(70 72 75 73)(80 87 91 84)(81 82 90 89)(83 85 88 86)(93 100 104 97)(94 95 103 102)(96 98 101 99)(106 113 117 110)(107 108 116 115)(109 111 114 112)

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

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

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

63 conjugacy classes

 class 1 2 3A ··· 3H 4A 4B 6A ··· 6H 12A ··· 12P 13A 13B 13C 39A ··· 39X order 1 2 3 ··· 3 4 4 6 ··· 6 12 ··· 12 13 13 13 39 ··· 39 size 1 13 1 ··· 1 13 13 13 ··· 13 13 ··· 13 4 4 4 4 ··· 4

63 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C3 C4 C6 C12 C13⋊C4 C3×C13⋊C4 kernel C32×C13⋊C4 C32×D13 C3×C13⋊C4 C3×C39 C3×D13 C39 C32 C3 # reps 1 1 8 2 8 16 3 24

Matrix representation of C32×C13⋊C4 in GL5(𝔽157)

 144 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 103 122 103 156 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
,
 28 0 0 0 0 0 1 0 0 0 0 54 89 109 55 0 122 48 67 102 0 0 0 1 0

G:=sub<GL(5,GF(157))| [144,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,103,1,0,0,0,122,0,1,0,0,103,0,0,1,0,156,0,0,0],[28,0,0,0,0,0,1,54,122,0,0,0,89,48,0,0,0,109,67,1,0,0,55,102,0] >;

C32×C13⋊C4 in GAP, Magma, Sage, TeX

C_3^2\times C_{13}\rtimes C_4
% in TeX

G:=Group("C3^2xC13:C4");
// GroupNames label

G:=SmallGroup(468,36);
// by ID

G=gap.SmallGroup(468,36);
# by ID

G:=PCGroup([5,-2,-3,-3,-2,-13,90,7204,619]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^13=d^4=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^5>;
// generators/relations

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