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## G = C37⋊Dic3order 444 = 22·3·37

### The semidirect product of C37 and Dic3 acting via Dic3/C3=C4

Aliases: C37⋊Dic3, C1111C4, D37.S3, C3⋊(C37⋊C4), (C3×D37).1C2, SmallGroup(444,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C111 — C37⋊Dic3
 Chief series C1 — C37 — C111 — C3×D37 — C37⋊Dic3
 Lower central C111 — C37⋊Dic3
 Upper central C1

Generators and relations for C37⋊Dic3
G = < a,b,c | a37=b6=1, c2=b3, bab-1=a-1, cac-1=a31, cbc-1=b-1 >

Smallest permutation representation of C37⋊Dic3
On 111 points
Generators in S111
(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)
(1 101 69)(2 100 70 37 102 68)(3 99 71 36 103 67)(4 98 72 35 104 66)(5 97 73 34 105 65)(6 96 74 33 106 64)(7 95 38 32 107 63)(8 94 39 31 108 62)(9 93 40 30 109 61)(10 92 41 29 110 60)(11 91 42 28 111 59)(12 90 43 27 75 58)(13 89 44 26 76 57)(14 88 45 25 77 56)(15 87 46 24 78 55)(16 86 47 23 79 54)(17 85 48 22 80 53)(18 84 49 21 81 52)(19 83 50 20 82 51)
(2 7 37 32)(3 13 36 26)(4 19 35 20)(5 25 34 14)(6 31 33 8)(9 12 30 27)(10 18 29 21)(11 24 28 15)(16 17 23 22)(38 100 63 102)(39 106 62 96)(40 75 61 90)(41 81 60 84)(42 87 59 78)(43 93 58 109)(44 99 57 103)(45 105 56 97)(46 111 55 91)(47 80 54 85)(48 86 53 79)(49 92 52 110)(50 98 51 104)(64 108 74 94)(65 77 73 88)(66 83 72 82)(67 89 71 76)(68 95 70 107)(69 101)

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

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

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

33 conjugacy classes

 class 1 2 3 4A 4B 6 37A ··· 37I 111A ··· 111R order 1 2 3 4 4 6 37 ··· 37 111 ··· 111 size 1 37 2 111 111 74 4 ··· 4 4 ··· 4

33 irreducible representations

 dim 1 1 1 2 2 4 4 type + + + - + image C1 C2 C4 S3 Dic3 C37⋊C4 C37⋊Dic3 kernel C37⋊Dic3 C3×D37 C111 D37 C37 C3 C1 # reps 1 1 2 1 1 9 18

Matrix representation of C37⋊Dic3 in GL4(𝔽1777) generated by

 0 1 0 0 0 0 1 0 0 0 0 1 1776 1020 1014 1020
,
 527 623 1055 269 1509 556 1155 1250 854 1284 1548 268 1634 328 618 923
,
 1 0 0 0 92 346 796 1033 686 1180 1538 1073 1578 930 94 1669
G:=sub<GL(4,GF(1777))| [0,0,0,1776,1,0,0,1020,0,1,0,1014,0,0,1,1020],[527,1509,854,1634,623,556,1284,328,1055,1155,1548,618,269,1250,268,923],[1,92,686,1578,0,346,1180,930,0,796,1538,94,0,1033,1073,1669] >;

C37⋊Dic3 in GAP, Magma, Sage, TeX

C_{37}\rtimes {\rm Dic}_3
% in TeX

G:=Group("C37:Dic3");
// GroupNames label

G:=SmallGroup(444,10);
// by ID

G=gap.SmallGroup(444,10);
# by ID

G:=PCGroup([4,-2,-2,-3,-37,8,98,1155,3463]);
// Polycyclic

G:=Group<a,b,c|a^37=b^6=1,c^2=b^3,b*a*b^-1=a^-1,c*a*c^-1=a^31,c*b*c^-1=b^-1>;
// generators/relations

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