direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×S3×D13, C39⋊C23, C6⋊1D26, C26⋊1D6, C78⋊C22, D78⋊5C2, D39⋊C22, (S3×C26)⋊3C2, (C6×D13)⋊3C2, (S3×C13)⋊C22, C13⋊1(C22×S3), (C3×D13)⋊C22, C3⋊1(C22×D13), SmallGroup(312,54)
Series: Derived ►Chief ►Lower central ►Upper central
| C39 — C2×S3×D13 |
Generators and relations for C2×S3×D13
G = < a,b,c,d,e | a2=b3=c2=d13=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 564 in 64 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C3, C22, S3, S3, C6, C6, C23, D6, D6, C2×C6, C13, C22×S3, D13, D13, C26, C26, C39, D26, D26, C2×C26, S3×C13, C3×D13, D39, C78, C22×D13, S3×D13, C6×D13, S3×C26, D78, C2×S3×D13
Quotients: C1, C2, C22, S3, C23, D6, C22×S3, D13, D26, C22×D13, S3×D13, C2×S3×D13
►On 78 points
Generators in S78
(1 51)(2 52)(3 40)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 53)(27 76)(28 77)(29 78)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)
(1 23 27)(2 24 28)(3 25 29)(4 26 30)(5 14 31)(6 15 32)(7 16 33)(8 17 34)(9 18 35)(10 19 36)(11 20 37)(12 21 38)(13 22 39)(40 65 78)(41 53 66)(42 54 67)(43 55 68)(44 56 69)(45 57 70)(46 58 71)(47 59 72)(48 60 73)(49 61 74)(50 62 75)(51 63 76)(52 64 77)
(14 31)(15 32)(16 33)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 27)(24 28)(25 29)(26 30)(53 66)(54 67)(55 68)(56 69)(57 70)(58 71)(59 72)(60 73)(61 74)(62 75)(63 76)(64 77)(65 78)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)(40 48)(41 47)(42 46)(43 45)(49 52)(50 51)(53 59)(54 58)(55 57)(60 65)(61 64)(62 63)(66 72)(67 71)(68 70)(73 78)(74 77)(75 76)
G:=sub<Sym(78)| (1,51)(2,52)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,53)(27,76)(28,77)(29,78)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75), (1,23,27)(2,24,28)(3,25,29)(4,26,30)(5,14,31)(6,15,32)(7,16,33)(8,17,34)(9,18,35)(10,19,36)(11,20,37)(12,21,38)(13,22,39)(40,65,78)(41,53,66)(42,54,67)(43,55,68)(44,56,69)(45,57,70)(46,58,71)(47,59,72)(48,60,73)(49,61,74)(50,62,75)(51,63,76)(52,64,77), (14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,27)(24,28)(25,29)(26,30)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76)>;
G:=Group( (1,51)(2,52)(3,40)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,53)(27,76)(28,77)(29,78)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75), (1,23,27)(2,24,28)(3,25,29)(4,26,30)(5,14,31)(6,15,32)(7,16,33)(8,17,34)(9,18,35)(10,19,36)(11,20,37)(12,21,38)(13,22,39)(40,65,78)(41,53,66)(42,54,67)(43,55,68)(44,56,69)(45,57,70)(46,58,71)(47,59,72)(48,60,73)(49,61,74)(50,62,75)(51,63,76)(52,64,77), (14,31)(15,32)(16,33)(17,34)(18,35)(19,36)(20,37)(21,38)(22,39)(23,27)(24,28)(25,29)(26,30)(53,66)(54,67)(55,68)(56,69)(57,70)(58,71)(59,72)(60,73)(61,74)(62,75)(63,76)(64,77)(65,78), (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), (1,13)(2,12)(3,11)(4,10)(5,9)(6,8)(14,18)(15,17)(19,26)(20,25)(21,24)(22,23)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34)(40,48)(41,47)(42,46)(43,45)(49,52)(50,51)(53,59)(54,58)(55,57)(60,65)(61,64)(62,63)(66,72)(67,71)(68,70)(73,78)(74,77)(75,76) );
G=PermutationGroup([[(1,51),(2,52),(3,40),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,53),(27,76),(28,77),(29,78),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75)], [(1,23,27),(2,24,28),(3,25,29),(4,26,30),(5,14,31),(6,15,32),(7,16,33),(8,17,34),(9,18,35),(10,19,36),(11,20,37),(12,21,38),(13,22,39),(40,65,78),(41,53,66),(42,54,67),(43,55,68),(44,56,69),(45,57,70),(46,58,71),(47,59,72),(48,60,73),(49,61,74),(50,62,75),(51,63,76),(52,64,77)], [(14,31),(15,32),(16,33),(17,34),(18,35),(19,36),(20,37),(21,38),(22,39),(23,27),(24,28),(25,29),(26,30),(53,66),(54,67),(55,68),(56,69),(57,70),(58,71),(59,72),(60,73),(61,74),(62,75),(63,76),(64,77),(65,78)], [(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)], [(1,13),(2,12),(3,11),(4,10),(5,9),(6,8),(14,18),(15,17),(19,26),(20,25),(21,24),(22,23),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34),(40,48),(41,47),(42,46),(43,45),(49,52),(50,51),(53,59),(54,58),(55,57),(60,65),(61,64),(62,63),(66,72),(67,71),(68,70),(73,78),(74,77),(75,76)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 6A | 6B | 6C | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 39A | ··· | 39F | 78A | ··· | 78F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 6 | 6 | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 3 | 3 | 13 | 13 | 39 | 39 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D13 | D26 | D26 | S3×D13 | C2×S3×D13 |
| kernel | C2×S3×D13 | S3×D13 | C6×D13 | S3×C26 | D78 | D26 | D13 | C26 | D6 | S3 | C6 | C2 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 1 | 6 | 12 | 6 | 6 | 6 |
Matrix representation of C2×S3×D13 ►in GL5(𝔽79)
| 78 | 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 78 | 78 |
| 0 | 0 | 0 | 1 | 0 |
| 78 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 78 | 78 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 18 | 1 | 0 | 0 |
| 0 | 78 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 78 | 0 | 0 | 0 | 0 |
| 0 | 50 | 72 | 0 | 0 |
| 0 | 41 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(5,GF(79))| [78,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,78,1,0,0,0,78,0],[78,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,78,0,0,0,0,78],[1,0,0,0,0,0,18,78,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[78,0,0,0,0,0,50,41,0,0,0,72,29,0,0,0,0,0,1,0,0,0,0,0,1] >;
C2×S3×D13 in GAP, Magma, Sage, TeX
C_2\times S_3\times D_{13} % in TeX
G:=Group("C2xS3xD13"); // GroupNames label
G:=SmallGroup(312,54);
// by ID
G=gap.SmallGroup(312,54);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-13,168,7204]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^13=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations