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## G = C3×C39⋊C4order 468 = 22·32·13

### Direct product of C3 and C39⋊C4

Aliases: C3×C39⋊C4, C396C12, C392Dic3, (C3×C39)⋊3C4, C322(C13⋊C4), C133(C3×Dic3), (C3×D13).4S3, (C3×D13).5C6, D13.2(C3×S3), (C32×D13).2C2, C3⋊(C3×C13⋊C4), SmallGroup(468,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — C3×C39⋊C4
 Chief series C1 — C13 — C39 — C3×D13 — C32×D13 — C3×C39⋊C4
 Lower central C39 — C3×C39⋊C4
 Upper central C1 — C3

Generators and relations for C3×C39⋊C4
G = < a,b,c | a3=b39=c4=1, ab=ba, ac=ca, cbc-1=b8 >

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

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

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

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

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 12A 12B 12C 12D 13A 13B 13C 39A ··· 39X order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 12 12 12 12 13 13 13 39 ··· 39 size 1 13 1 1 2 2 2 39 39 13 13 26 26 26 39 39 39 39 4 4 4 4 ··· 4

45 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + image C1 C2 C3 C4 C6 C12 S3 Dic3 C3×S3 C3×Dic3 C13⋊C4 C3×C13⋊C4 C39⋊C4 C3×C39⋊C4 kernel C3×C39⋊C4 C32×D13 C39⋊C4 C3×C39 C3×D13 C39 C3×D13 C39 D13 C13 C32 C3 C3 C1 # reps 1 1 2 2 2 4 1 1 2 2 3 6 6 12

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

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 138 52 0 0 105 143 0 0 0 0 145 145 0 0 12 11
,
 0 0 1 0 0 0 0 1 1 0 0 0 142 156 0 0
G:=sub<GL(4,GF(157))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[138,105,0,0,52,143,0,0,0,0,145,12,0,0,145,11],[0,0,1,142,0,0,0,156,1,0,0,0,0,1,0,0] >;

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(468,37);
// by ID

G=gap.SmallGroup(468,37);
# by ID

G:=PCGroup([5,-2,-3,-2,-3,-13,30,483,4504,1814]);
// Polycyclic

G:=Group<a,b,c|a^3=b^39=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^8>;
// generators/relations

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