direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: S3×C4×C8, D6.6C42, C42.281D6, Dic3.6C42, C12⋊5(C2×C8), (C4×C24)⋊22C2, C24⋊25(C2×C4), D6.7(C2×C8), Dic3⋊5(C2×C8), (C2×C8).338D6, C6.5(C2×C42), C2.1(S3×C42), C6.2(C22×C8), (C8×Dic3)⋊33C2, (S3×C42).16C2, (C4×Dic3).23C4, (C2×C12).802C23, (C2×C24).423C22, C12.122(C22×C4), (C4×C12).337C22, (C4×Dic3).298C22, C3⋊1(C2×C4×C8), C2.1(S3×C2×C8), (C4×C3⋊C8)⋊27C2, C3⋊C8⋊28(C2×C4), C4.96(S3×C2×C4), (S3×C2×C4).23C4, (S3×C2×C8).19C2, C22.34(S3×C2×C4), (C4×S3).36(C2×C4), (C2×C4).172(C4×S3), (C2×C12).245(C2×C4), (C2×C3⋊C8).328C22, (S3×C2×C4).304C22, (C2×C6).57(C22×C4), (C22×S3).70(C2×C4), (C2×C4).744(C22×S3), (C2×Dic3).106(C2×C4), SmallGroup(192,243)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C4×C8 |
Generators and relations for S3×C4×C8
G = < a,b,c,d | a4=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 280 in 162 conjugacy classes, 103 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, C12, D6, C2×C6, C42, C42, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C4×C8, C2×C42, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, S3×C2×C4, C2×C4×C8, C4×C3⋊C8, C8×Dic3, C4×C24, S3×C42, S3×C2×C8, S3×C4×C8
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C42, C2×C8, C22×C4, C4×S3, C22×S3, C4×C8, C2×C42, C22×C8, S3×C8, S3×C2×C4, C2×C4×C8, S3×C42, S3×C2×C8, S3×C4×C8
►On 96 points
(1 15 62 94)(2 16 63 95)(3 9 64 96)(4 10 57 89)(5 11 58 90)(6 12 59 91)(7 13 60 92)(8 14 61 93)(17 38 50 69)(18 39 51 70)(19 40 52 71)(20 33 53 72)(21 34 54 65)(22 35 55 66)(23 36 56 67)(24 37 49 68)(25 78 41 87)(26 79 42 88)(27 80 43 81)(28 73 44 82)(29 74 45 83)(30 75 46 84)(31 76 47 85)(32 77 48 86)
(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 55 83)(2 56 84)(3 49 85)(4 50 86)(5 51 87)(6 52 88)(7 53 81)(8 54 82)(9 68 31)(10 69 32)(11 70 25)(12 71 26)(13 72 27)(14 65 28)(15 66 29)(16 67 30)(17 77 57)(18 78 58)(19 79 59)(20 80 60)(21 73 61)(22 74 62)(23 75 63)(24 76 64)(33 43 92)(34 44 93)(35 45 94)(36 46 95)(37 47 96)(38 48 89)(39 41 90)(40 42 91)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 92)(10 93)(11 94)(12 95)(13 96)(14 89)(15 90)(16 91)(17 82)(18 83)(19 84)(20 85)(21 86)(22 87)(23 88)(24 81)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 80)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)
G:=sub<Sym(96)| (1,15,62,94)(2,16,63,95)(3,9,64,96)(4,10,57,89)(5,11,58,90)(6,12,59,91)(7,13,60,92)(8,14,61,93)(17,38,50,69)(18,39,51,70)(19,40,52,71)(20,33,53,72)(21,34,54,65)(22,35,55,66)(23,36,56,67)(24,37,49,68)(25,78,41,87)(26,79,42,88)(27,80,43,81)(28,73,44,82)(29,74,45,83)(30,75,46,84)(31,76,47,85)(32,77,48,86), (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), (1,55,83)(2,56,84)(3,49,85)(4,50,86)(5,51,87)(6,52,88)(7,53,81)(8,54,82)(9,68,31)(10,69,32)(11,70,25)(12,71,26)(13,72,27)(14,65,28)(15,66,29)(16,67,30)(17,77,57)(18,78,58)(19,79,59)(20,80,60)(21,73,61)(22,74,62)(23,75,63)(24,76,64)(33,43,92)(34,44,93)(35,45,94)(36,46,95)(37,47,96)(38,48,89)(39,41,90)(40,42,91), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,92)(10,93)(11,94)(12,95)(13,96)(14,89)(15,90)(16,91)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79)>;
G:=Group( (1,15,62,94)(2,16,63,95)(3,9,64,96)(4,10,57,89)(5,11,58,90)(6,12,59,91)(7,13,60,92)(8,14,61,93)(17,38,50,69)(18,39,51,70)(19,40,52,71)(20,33,53,72)(21,34,54,65)(22,35,55,66)(23,36,56,67)(24,37,49,68)(25,78,41,87)(26,79,42,88)(27,80,43,81)(28,73,44,82)(29,74,45,83)(30,75,46,84)(31,76,47,85)(32,77,48,86), (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), (1,55,83)(2,56,84)(3,49,85)(4,50,86)(5,51,87)(6,52,88)(7,53,81)(8,54,82)(9,68,31)(10,69,32)(11,70,25)(12,71,26)(13,72,27)(14,65,28)(15,66,29)(16,67,30)(17,77,57)(18,78,58)(19,79,59)(20,80,60)(21,73,61)(22,74,62)(23,75,63)(24,76,64)(33,43,92)(34,44,93)(35,45,94)(36,46,95)(37,47,96)(38,48,89)(39,41,90)(40,42,91), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,92)(10,93)(11,94)(12,95)(13,96)(14,89)(15,90)(16,91)(17,82)(18,83)(19,84)(20,85)(21,86)(22,87)(23,88)(24,81)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(31,33)(32,34)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,80)(50,73)(51,74)(52,75)(53,76)(54,77)(55,78)(56,79) );
G=PermutationGroup([[(1,15,62,94),(2,16,63,95),(3,9,64,96),(4,10,57,89),(5,11,58,90),(6,12,59,91),(7,13,60,92),(8,14,61,93),(17,38,50,69),(18,39,51,70),(19,40,52,71),(20,33,53,72),(21,34,54,65),(22,35,55,66),(23,36,56,67),(24,37,49,68),(25,78,41,87),(26,79,42,88),(27,80,43,81),(28,73,44,82),(29,74,45,83),(30,75,46,84),(31,76,47,85),(32,77,48,86)], [(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)], [(1,55,83),(2,56,84),(3,49,85),(4,50,86),(5,51,87),(6,52,88),(7,53,81),(8,54,82),(9,68,31),(10,69,32),(11,70,25),(12,71,26),(13,72,27),(14,65,28),(15,66,29),(16,67,30),(17,77,57),(18,78,58),(19,79,59),(20,80,60),(21,73,61),(22,74,62),(23,75,63),(24,76,64),(33,43,92),(34,44,93),(35,45,94),(36,46,95),(37,47,96),(38,48,89),(39,41,90),(40,42,91)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,92),(10,93),(11,94),(12,95),(13,96),(14,89),(15,90),(16,91),(17,82),(18,83),(19,84),(20,85),(21,86),(22,87),(23,88),(24,81),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(31,33),(32,34),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(49,80),(50,73),(51,74),(52,75),(53,76),(54,77),(55,78),(56,79)]])
96 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | ··· | 4L | 4M | ··· | 4X | 6A | 6B | 6C | 8A | ··· | 8P | 8Q | ··· | 8AF | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | 2 | 2 | 1 | ··· | 1 | 3 | ··· | 3 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C8 | S3 | D6 | D6 | C4×S3 | C4×S3 | S3×C8 |
| kernel | S3×C4×C8 | C4×C3⋊C8 | C8×Dic3 | C4×C24 | S3×C42 | S3×C2×C8 | S3×C8 | C4×Dic3 | S3×C2×C4 | C4×S3 | C4×C8 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 32 | 1 | 1 | 2 | 8 | 4 | 16 |
Matrix representation of S3×C4×C8 ►in GL4(𝔽73) generated by
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 63 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [46,0,0,0,0,27,0,0,0,0,1,0,0,0,0,1],[63,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[72,0,0,0,0,1,0,0,0,0,1,72,0,0,0,72] >;
S3×C4×C8 in GAP, Magma, Sage, TeX
S_3\times C_4\times C_8
% in TeX
G:=Group("S3xC4xC8"); // GroupNames label
G:=SmallGroup(192,243);
// by ID
G=gap.SmallGroup(192,243);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations