direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C8⋊C4, D6.7C42, C42.181D6, D6.3M4(2), Dic3.7C42, (S3×C8)⋊6C4, C8⋊15(C4×S3), C24⋊19(C2×C4), C24⋊C4⋊24C2, (C2×C8).269D6, C2.9(S3×C42), C6.8(C2×C42), C2.1(S3×M4(2)), (S3×C42).13C2, (C4×Dic3).14C4, C6.15(C2×M4(2)), (C4×C12).226C22, (C2×C12).810C23, C42.S3⋊17C2, (C2×C24).268C22, C12.125(C22×C4), (C4×Dic3).266C22, C3⋊1(C2×C8⋊C4), C3⋊C8⋊25(C2×C4), C4.99(S3×C2×C4), (S3×C2×C8).16C2, (S3×C2×C4).14C4, (C3×C8⋊C4)⋊11C2, C22.39(S3×C2×C4), (C4×S3).37(C2×C4), (C2×C4).126(C4×S3), (C2×C12).145(C2×C4), (C2×C3⋊C8).294C22, (S3×C2×C4).305C22, (C2×C6).65(C22×C4), (C22×S3).71(C2×C4), (C2×C4).752(C22×S3), (C2×Dic3).107(C2×C4), SmallGroup(192,263)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C8⋊C4
G = < a,b,c,d | a3=b2=c8=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 280 in 146 conjugacy classes, 83 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, C8⋊C4, C2×C42, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, S3×C2×C4, C2×C8⋊C4, C42.S3, C24⋊C4, C3×C8⋊C4, S3×C42, S3×C2×C8, S3×C8⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, M4(2), C22×C4, C4×S3, C22×S3, C8⋊C4, C2×C42, C2×M4(2), S3×C2×C4, C2×C8⋊C4, S3×C42, S3×M4(2), S3×C8⋊C4
►On 96 points
(1 76 58)(2 77 59)(3 78 60)(4 79 61)(5 80 62)(6 73 63)(7 74 64)(8 75 57)(9 72 27)(10 65 28)(11 66 29)(12 67 30)(13 68 31)(14 69 32)(15 70 25)(16 71 26)(17 86 38)(18 87 39)(19 88 40)(20 81 33)(21 82 34)(22 83 35)(23 84 36)(24 85 37)(41 94 55)(42 95 56)(43 96 49)(44 89 50)(45 90 51)(46 91 52)(47 92 53)(48 93 54)
(1 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 33)(8 34)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(25 47)(26 48)(27 41)(28 42)(29 43)(30 44)(31 45)(32 46)(65 95)(66 96)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)(73 88)(74 81)(75 82)(76 83)(77 84)(78 85)(79 86)(80 87)
(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)
(1 9 39 51)(2 14 40 56)(3 11 33 53)(4 16 34 50)(5 13 35 55)(6 10 36 52)(7 15 37 49)(8 12 38 54)(17 48 75 67)(18 45 76 72)(19 42 77 69)(20 47 78 66)(21 44 79 71)(22 41 80 68)(23 46 73 65)(24 43 74 70)(25 85 96 64)(26 82 89 61)(27 87 90 58)(28 84 91 63)(29 81 92 60)(30 86 93 57)(31 83 94 62)(32 88 95 59)
G:=sub<Sym(96)| (1,76,58)(2,77,59)(3,78,60)(4,79,61)(5,80,62)(6,73,63)(7,74,64)(8,75,57)(9,72,27)(10,65,28)(11,66,29)(12,67,30)(13,68,31)(14,69,32)(15,70,25)(16,71,26)(17,86,38)(18,87,39)(19,88,40)(20,81,33)(21,82,34)(22,83,35)(23,84,36)(24,85,37)(41,94,55)(42,95,56)(43,96,49)(44,89,50)(45,90,51)(46,91,52)(47,92,53)(48,93,54), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (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), (1,9,39,51)(2,14,40,56)(3,11,33,53)(4,16,34,50)(5,13,35,55)(6,10,36,52)(7,15,37,49)(8,12,38,54)(17,48,75,67)(18,45,76,72)(19,42,77,69)(20,47,78,66)(21,44,79,71)(22,41,80,68)(23,46,73,65)(24,43,74,70)(25,85,96,64)(26,82,89,61)(27,87,90,58)(28,84,91,63)(29,81,92,60)(30,86,93,57)(31,83,94,62)(32,88,95,59)>;
G:=Group( (1,76,58)(2,77,59)(3,78,60)(4,79,61)(5,80,62)(6,73,63)(7,74,64)(8,75,57)(9,72,27)(10,65,28)(11,66,29)(12,67,30)(13,68,31)(14,69,32)(15,70,25)(16,71,26)(17,86,38)(18,87,39)(19,88,40)(20,81,33)(21,82,34)(22,83,35)(23,84,36)(24,85,37)(41,94,55)(42,95,56)(43,96,49)(44,89,50)(45,90,51)(46,91,52)(47,92,53)(48,93,54), (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,33)(8,34)(9,55)(10,56)(11,49)(12,50)(13,51)(14,52)(15,53)(16,54)(17,61)(18,62)(19,63)(20,64)(21,57)(22,58)(23,59)(24,60)(25,47)(26,48)(27,41)(28,42)(29,43)(30,44)(31,45)(32,46)(65,95)(66,96)(67,89)(68,90)(69,91)(70,92)(71,93)(72,94)(73,88)(74,81)(75,82)(76,83)(77,84)(78,85)(79,86)(80,87), (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), (1,9,39,51)(2,14,40,56)(3,11,33,53)(4,16,34,50)(5,13,35,55)(6,10,36,52)(7,15,37,49)(8,12,38,54)(17,48,75,67)(18,45,76,72)(19,42,77,69)(20,47,78,66)(21,44,79,71)(22,41,80,68)(23,46,73,65)(24,43,74,70)(25,85,96,64)(26,82,89,61)(27,87,90,58)(28,84,91,63)(29,81,92,60)(30,86,93,57)(31,83,94,62)(32,88,95,59) );
G=PermutationGroup([[(1,76,58),(2,77,59),(3,78,60),(4,79,61),(5,80,62),(6,73,63),(7,74,64),(8,75,57),(9,72,27),(10,65,28),(11,66,29),(12,67,30),(13,68,31),(14,69,32),(15,70,25),(16,71,26),(17,86,38),(18,87,39),(19,88,40),(20,81,33),(21,82,34),(22,83,35),(23,84,36),(24,85,37),(41,94,55),(42,95,56),(43,96,49),(44,89,50),(45,90,51),(46,91,52),(47,92,53),(48,93,54)], [(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,33),(8,34),(9,55),(10,56),(11,49),(12,50),(13,51),(14,52),(15,53),(16,54),(17,61),(18,62),(19,63),(20,64),(21,57),(22,58),(23,59),(24,60),(25,47),(26,48),(27,41),(28,42),(29,43),(30,44),(31,45),(32,46),(65,95),(66,96),(67,89),(68,90),(69,91),(70,92),(71,93),(72,94),(73,88),(74,81),(75,82),(76,83),(77,84),(78,85),(79,86),(80,87)], [(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)], [(1,9,39,51),(2,14,40,56),(3,11,33,53),(4,16,34,50),(5,13,35,55),(6,10,36,52),(7,15,37,49),(8,12,38,54),(17,48,75,67),(18,45,76,72),(19,42,77,69),(20,47,78,66),(21,44,79,71),(22,41,80,68),(23,46,73,65),(24,43,74,70),(25,85,96,64),(26,82,89,61),(27,87,90,58),(28,84,91,63),(29,81,92,60),(30,86,93,57),(31,83,94,62),(32,88,95,59)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | S3×M4(2) |
| kernel | S3×C8⋊C4 | C42.S3 | C24⋊C4 | C3×C8⋊C4 | S3×C42 | S3×C2×C8 | S3×C8 | C4×Dic3 | S3×C2×C4 | C8⋊C4 | C42 | C2×C8 | D6 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 1 | 1 | 2 | 8 | 8 | 4 | 4 |
Matrix representation of S3×C8⋊C4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 51 | 28 | 0 | 0 |
| 28 | 22 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 5 | 44 | 0 | 0 |
| 16 | 68 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,72,0],[51,28,0,0,28,22,0,0,0,0,72,0,0,0,0,72],[5,16,0,0,44,68,0,0,0,0,27,0,0,0,0,27] >;
S3×C8⋊C4 in GAP, Magma, Sage, TeX
S_3\times C_8\rtimes C_4
% in TeX
G:=Group("S3xC8:C4"); // GroupNames label
G:=SmallGroup(192,263);
// by ID
G=gap.SmallGroup(192,263);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^8=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations