direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C4⋊C8, C42.199D6, D6.5M4(2), C4⋊3(S3×C8), (C4×S3)⋊1C8, C12⋊1(C2×C8), D6.8(C2×C8), C4.54(S3×Q8), D6.9(C4⋊C4), Dic3⋊3(C2×C8), C12⋊C8⋊11C2, (C4×S3).53D4, C4.204(S3×D4), (C2×C8).214D6, C6.9(C22×C8), (C4×S3).10Q8, Dic3⋊C8⋊27C2, C12.363(C2×D4), (S3×C42).1C2, C12.112(C2×Q8), (C4×Dic3).5C4, C2.5(S3×M4(2)), (C4×C12).58C22, C6.25(C2×M4(2)), Dic3.10(C4⋊C4), (C2×C24).251C22, (C2×C12).829C23, (C4×Dic3).275C22, C3⋊1(C2×C4⋊C8), C6.7(C2×C4⋊C4), C2.3(S3×C4⋊C4), (S3×C2×C4).5C4, C2.11(S3×C2×C8), (C3×C4⋊C8)⋊18C2, (S3×C2×C8).13C2, C22.46(S3×C2×C4), (C2×C4).144(C4×S3), (C2×C12).68(C2×C4), (C2×C3⋊C8).304C22, (S3×C2×C4).307C22, (C2×C6).84(C22×C4), (C22×S3).72(C2×C4), (C2×C4).771(C22×S3), (C2×Dic3).87(C2×C4), SmallGroup(192,391)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C4⋊C8
G = < a,b,c,d | a3=b2=c4=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 280 in 138 conjugacy classes, 73 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C12, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×C12, C22×S3, C4⋊C8, C4⋊C8, C2×C42, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, C2×C4⋊C8, C12⋊C8, Dic3⋊C8, C3×C4⋊C8, S3×C42, S3×C2×C8, S3×C4⋊C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, D4, Q8, C23, D6, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C4⋊C8, C2×C4⋊C4, C22×C8, C2×M4(2), S3×C8, S3×C2×C4, S3×D4, S3×Q8, C2×C4⋊C8, S3×C4⋊C4, S3×C2×C8, S3×M4(2), S3×C4⋊C8
►On 96 points
(1 73 43)(2 74 44)(3 75 45)(4 76 46)(5 77 47)(6 78 48)(7 79 41)(8 80 42)(9 39 72)(10 40 65)(11 33 66)(12 34 67)(13 35 68)(14 36 69)(15 37 70)(16 38 71)(17 60 85)(18 61 86)(19 62 87)(20 63 88)(21 64 81)(22 57 82)(23 58 83)(24 59 84)(25 52 92)(26 53 93)(27 54 94)(28 55 95)(29 56 96)(30 49 89)(31 50 90)(32 51 91)
(1 5)(2 6)(3 7)(4 8)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 33)(16 34)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)(49 93)(50 94)(51 95)(52 96)(53 89)(54 90)(55 91)(56 92)(57 86)(58 87)(59 88)(60 81)(61 82)(62 83)(63 84)(64 85)(65 69)(66 70)(67 71)(68 72)
(1 31 23 68)(2 69 24 32)(3 25 17 70)(4 71 18 26)(5 27 19 72)(6 65 20 28)(7 29 21 66)(8 67 22 30)(9 77 54 62)(10 63 55 78)(11 79 56 64)(12 57 49 80)(13 73 50 58)(14 59 51 74)(15 75 52 60)(16 61 53 76)(33 41 96 81)(34 82 89 42)(35 43 90 83)(36 84 91 44)(37 45 92 85)(38 86 93 46)(39 47 94 87)(40 88 95 48)
(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)
G:=sub<Sym(96)| (1,73,43)(2,74,44)(3,75,45)(4,76,46)(5,77,47)(6,78,48)(7,79,41)(8,80,42)(9,39,72)(10,40,65)(11,33,66)(12,34,67)(13,35,68)(14,36,69)(15,37,70)(16,38,71)(17,60,85)(18,61,86)(19,62,87)(20,63,88)(21,64,81)(22,57,82)(23,58,83)(24,59,84)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,49,89)(31,50,90)(32,51,91), (1,5)(2,6)(3,7)(4,8)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,69)(66,70)(67,71)(68,72), (1,31,23,68)(2,69,24,32)(3,25,17,70)(4,71,18,26)(5,27,19,72)(6,65,20,28)(7,29,21,66)(8,67,22,30)(9,77,54,62)(10,63,55,78)(11,79,56,64)(12,57,49,80)(13,73,50,58)(14,59,51,74)(15,75,52,60)(16,61,53,76)(33,41,96,81)(34,82,89,42)(35,43,90,83)(36,84,91,44)(37,45,92,85)(38,86,93,46)(39,47,94,87)(40,88,95,48), (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)>;
G:=Group( (1,73,43)(2,74,44)(3,75,45)(4,76,46)(5,77,47)(6,78,48)(7,79,41)(8,80,42)(9,39,72)(10,40,65)(11,33,66)(12,34,67)(13,35,68)(14,36,69)(15,37,70)(16,38,71)(17,60,85)(18,61,86)(19,62,87)(20,63,88)(21,64,81)(22,57,82)(23,58,83)(24,59,84)(25,52,92)(26,53,93)(27,54,94)(28,55,95)(29,56,96)(30,49,89)(31,50,90)(32,51,91), (1,5)(2,6)(3,7)(4,8)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,33)(16,34)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,86)(58,87)(59,88)(60,81)(61,82)(62,83)(63,84)(64,85)(65,69)(66,70)(67,71)(68,72), (1,31,23,68)(2,69,24,32)(3,25,17,70)(4,71,18,26)(5,27,19,72)(6,65,20,28)(7,29,21,66)(8,67,22,30)(9,77,54,62)(10,63,55,78)(11,79,56,64)(12,57,49,80)(13,73,50,58)(14,59,51,74)(15,75,52,60)(16,61,53,76)(33,41,96,81)(34,82,89,42)(35,43,90,83)(36,84,91,44)(37,45,92,85)(38,86,93,46)(39,47,94,87)(40,88,95,48), (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) );
G=PermutationGroup([[(1,73,43),(2,74,44),(3,75,45),(4,76,46),(5,77,47),(6,78,48),(7,79,41),(8,80,42),(9,39,72),(10,40,65),(11,33,66),(12,34,67),(13,35,68),(14,36,69),(15,37,70),(16,38,71),(17,60,85),(18,61,86),(19,62,87),(20,63,88),(21,64,81),(22,57,82),(23,58,83),(24,59,84),(25,52,92),(26,53,93),(27,54,94),(28,55,95),(29,56,96),(30,49,89),(31,50,90),(32,51,91)], [(1,5),(2,6),(3,7),(4,8),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,33),(16,34),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74),(49,93),(50,94),(51,95),(52,96),(53,89),(54,90),(55,91),(56,92),(57,86),(58,87),(59,88),(60,81),(61,82),(62,83),(63,84),(64,85),(65,69),(66,70),(67,71),(68,72)], [(1,31,23,68),(2,69,24,32),(3,25,17,70),(4,71,18,26),(5,27,19,72),(6,65,20,28),(7,29,21,66),(8,67,22,30),(9,77,54,62),(10,63,55,78),(11,79,56,64),(12,57,49,80),(13,73,50,58),(14,59,51,74),(15,75,52,60),(16,61,53,76),(33,41,96,81),(34,82,89,42),(35,43,90,83),(36,84,91,44),(37,45,92,85),(38,86,93,46),(39,47,94,87),(40,88,95,48)], [(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)]])
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 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | Q8 | D6 | D6 | M4(2) | C4×S3 | S3×C8 | S3×D4 | S3×Q8 | S3×M4(2) |
| kernel | S3×C4⋊C8 | C12⋊C8 | Dic3⋊C8 | C3×C4⋊C8 | S3×C42 | S3×C2×C8 | C4×Dic3 | S3×C2×C4 | C4×S3 | C4⋊C8 | C4×S3 | C4×S3 | C42 | C2×C8 | D6 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 4 | 4 | 16 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of S3×C4⋊C8 ►in GL4(𝔽73) generated by
| 72 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 50 |
| 0 | 0 | 36 | 43 |
| 51 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 46 | 0 |
G:=sub<GL(4,GF(73))| [72,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[72,1,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,30,36,0,0,50,43],[51,0,0,0,0,51,0,0,0,0,0,46,0,0,1,0] >;
S3×C4⋊C8 in GAP, Magma, Sage, TeX
S_3\times C_4\rtimes C_8
% in TeX
G:=Group("S3xC4:C8"); // GroupNames label
G:=SmallGroup(192,391);
// by ID
G=gap.SmallGroup(192,391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^4=d^8=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^-1>;
// generators/relations