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## G = (C3×C39)⋊C4order 468 = 22·32·13

### 1st semidirect product of C3×C39 and C4 acting faithfully

Aliases: (C3×C39)⋊1C4, C13⋊(C32⋊C4), C32⋊(C13⋊C4), C3⋊D39.C2, SmallGroup(468,41)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C39 — (C3×C39)⋊C4
 Chief series C1 — C13 — C3×C39 — C3⋊D39 — (C3×C39)⋊C4
 Lower central C3×C39 — (C3×C39)⋊C4
 Upper central C1

Generators and relations for (C3×C39)⋊C4
G = < a,b,c | a3=b39=c4=1, ab=ba, cac-1=a-1b13, cbc-1=ab31 >

117C2
2C3
2C3
117C4
78S3
78S3
9D13
2C39
2C39
13C3⋊S3
6D39
6D39
13C32⋊C4

Smallest permutation representation of (C3×C39)⋊C4
On 78 points
Generators in S78
(1 37 23)(2 38 24)(3 39 25)(4 27 26)(5 28 14)(6 29 15)(7 30 16)(8 31 17)(9 32 18)(10 33 19)(11 34 20)(12 35 21)(13 36 22)(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)
(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)
(1 40 12 76)(2 61 11 55)(3 43 10 73)(4 64 9 52)(5 46 8 70)(6 67 7 49)(13 58)(14 72 31 44)(15 54 30 62)(16 75 29 41)(17 57 28 59)(18 78 27 77)(19 60 39 56)(20 42 38 74)(21 63 37 53)(22 45 36 71)(23 66 35 50)(24 48 34 68)(25 69 33 47)(26 51 32 65)

G:=sub<Sym(78)| (1,37,23)(2,38,24)(3,39,25)(4,27,26)(5,28,14)(6,29,15)(7,30,16)(8,31,17)(9,32,18)(10,33,19)(11,34,20)(12,35,21)(13,36,22)(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), (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), (1,40,12,76)(2,61,11,55)(3,43,10,73)(4,64,9,52)(5,46,8,70)(6,67,7,49)(13,58)(14,72,31,44)(15,54,30,62)(16,75,29,41)(17,57,28,59)(18,78,27,77)(19,60,39,56)(20,42,38,74)(21,63,37,53)(22,45,36,71)(23,66,35,50)(24,48,34,68)(25,69,33,47)(26,51,32,65)>;

G:=Group( (1,37,23)(2,38,24)(3,39,25)(4,27,26)(5,28,14)(6,29,15)(7,30,16)(8,31,17)(9,32,18)(10,33,19)(11,34,20)(12,35,21)(13,36,22)(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), (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), (1,40,12,76)(2,61,11,55)(3,43,10,73)(4,64,9,52)(5,46,8,70)(6,67,7,49)(13,58)(14,72,31,44)(15,54,30,62)(16,75,29,41)(17,57,28,59)(18,78,27,77)(19,60,39,56)(20,42,38,74)(21,63,37,53)(22,45,36,71)(23,66,35,50)(24,48,34,68)(25,69,33,47)(26,51,32,65) );

G=PermutationGroup([[(1,37,23),(2,38,24),(3,39,25),(4,27,26),(5,28,14),(6,29,15),(7,30,16),(8,31,17),(9,32,18),(10,33,19),(11,34,20),(12,35,21),(13,36,22),(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)], [(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)], [(1,40,12,76),(2,61,11,55),(3,43,10,73),(4,64,9,52),(5,46,8,70),(6,67,7,49),(13,58),(14,72,31,44),(15,54,30,62),(16,75,29,41),(17,57,28,59),(18,78,27,77),(19,60,39,56),(20,42,38,74),(21,63,37,53),(22,45,36,71),(23,66,35,50),(24,48,34,68),(25,69,33,47),(26,51,32,65)]])

33 conjugacy classes

 class 1 2 3A 3B 4A 4B 13A 13B 13C 39A ··· 39X order 1 2 3 3 4 4 13 13 13 39 ··· 39 size 1 117 4 4 117 117 4 4 4 4 ··· 4

33 irreducible representations

 dim 1 1 1 4 4 4 type + + + + + image C1 C2 C4 C32⋊C4 C13⋊C4 (C3×C39)⋊C4 kernel (C3×C39)⋊C4 C3⋊D39 C3×C39 C13 C32 C1 # reps 1 1 2 2 3 24

Matrix representation of (C3×C39)⋊C4 in GL4(𝔽157) generated by

 4 111 0 0 38 152 0 0 44 4 121 111 117 130 24 35
,
 132 103 0 0 147 60 0 0 131 51 48 58 8 63 38 95
,
 133 33 151 31 101 8 81 154 3 31 1 94 56 77 41 15
G:=sub<GL(4,GF(157))| [4,38,44,117,111,152,4,130,0,0,121,24,0,0,111,35],[132,147,131,8,103,60,51,63,0,0,48,38,0,0,58,95],[133,101,3,56,33,8,31,77,151,81,1,41,31,154,94,15] >;

(C3×C39)⋊C4 in GAP, Magma, Sage, TeX

(C_3\times C_{39})\rtimes C_4
% in TeX

G:=Group("(C3xC39):C4");
// GroupNames label

G:=SmallGroup(468,41);
// by ID

G=gap.SmallGroup(468,41);
# by ID

G:=PCGroup([5,-2,-2,-3,3,-13,10,422,67,643,248,7204,5409]);
// Polycyclic

G:=Group<a,b,c|a^3=b^39=c^4=1,a*b=b*a,c*a*c^-1=a^-1*b^13,c*b*c^-1=a*b^31>;
// generators/relations

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