metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (S3×C8)⋊4C4, C8.30(C4×S3), C4.Q8⋊14S3, (C4×S3).4Q8, C4.24(S3×Q8), C24.34(C2×C4), D6.3(C4⋊C4), C4⋊C4.161D6, C8⋊Dic3⋊25C2, (C2×C8).259D6, C12.13(C2×Q8), C6.54(C4○D8), C6.Q16⋊14C2, C22.84(S3×D4), C12.43(C22×C4), (C22×S3).52D4, Dic3.11(C4⋊C4), (C2×C24).160C22, (C2×C12).276C23, (C2×Dic3).206D4, C2.6(Q8.7D6), C3⋊1(C23.25D4), C4⋊Dic3.108C22, (S3×C2×C8).8C2, C4.77(S3×C2×C4), C6.11(C2×C4⋊C4), C2.12(S3×C4⋊C4), C3⋊C8.16(C2×C4), (C3×C4.Q8)⋊8C2, C4⋊C4⋊7S3.4C2, (C4×S3).26(C2×C4), (C2×C6).281(C2×D4), (C3×C4⋊C4).69C22, (C2×C3⋊C8).227C22, (S3×C2×C4).232C22, (C2×C4).379(C22×S3), SmallGroup(192,419)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (S3×C8)⋊C4
G = < a,b,c,d | a8=b3=c2=d4=1, ab=ba, ac=ca, dad-1=a3, cbc=b-1, bd=db, dcd-1=a4c >
Subgroups: 272 in 114 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4.Q8, C4.Q8, C2.D8, C42⋊C2, C22×C8, S3×C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C23.25D4, C6.Q16, C8⋊Dic3, C3×C4.Q8, C4⋊C4⋊7S3, S3×C2×C8, (S3×C8)⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2×C4⋊C4, C4○D8, S3×C2×C4, S3×D4, S3×Q8, C23.25D4, S3×C4⋊C4, Q8.7D6, (S3×C8)⋊C4
►On 96 points
(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 71 82)(2 72 83)(3 65 84)(4 66 85)(5 67 86)(6 68 87)(7 69 88)(8 70 81)(9 40 77)(10 33 78)(11 34 79)(12 35 80)(13 36 73)(14 37 74)(15 38 75)(16 39 76)(17 49 58)(18 50 59)(19 51 60)(20 52 61)(21 53 62)(22 54 63)(23 55 64)(24 56 57)(25 47 89)(26 48 90)(27 41 91)(28 42 92)(29 43 93)(30 44 94)(31 45 95)(32 46 96)
(1 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 73)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(33 53)(34 54)(35 55)(36 56)(37 49)(38 50)(39 51)(40 52)(81 90)(82 91)(83 92)(84 93)(85 94)(86 95)(87 96)(88 89)
(1 52 45 40)(2 55 46 35)(3 50 47 38)(4 53 48 33)(5 56 41 36)(6 51 42 39)(7 54 43 34)(8 49 44 37)(9 82 20 31)(10 85 21 26)(11 88 22 29)(12 83 23 32)(13 86 24 27)(14 81 17 30)(15 84 18 25)(16 87 19 28)(57 91 73 67)(58 94 74 70)(59 89 75 65)(60 92 76 68)(61 95 77 71)(62 90 78 66)(63 93 79 69)(64 96 80 72)
G:=sub<Sym(96)| (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), (1,71,82)(2,72,83)(3,65,84)(4,66,85)(5,67,86)(6,68,87)(7,69,88)(8,70,81)(9,40,77)(10,33,78)(11,34,79)(12,35,80)(13,36,73)(14,37,74)(15,38,75)(16,39,76)(17,49,58)(18,50,59)(19,51,60)(20,52,61)(21,53,62)(22,54,63)(23,55,64)(24,56,57)(25,47,89)(26,48,90)(27,41,91)(28,42,92)(29,43,93)(30,44,94)(31,45,95)(32,46,96), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (1,52,45,40)(2,55,46,35)(3,50,47,38)(4,53,48,33)(5,56,41,36)(6,51,42,39)(7,54,43,34)(8,49,44,37)(9,82,20,31)(10,85,21,26)(11,88,22,29)(12,83,23,32)(13,86,24,27)(14,81,17,30)(15,84,18,25)(16,87,19,28)(57,91,73,67)(58,94,74,70)(59,89,75,65)(60,92,76,68)(61,95,77,71)(62,90,78,66)(63,93,79,69)(64,96,80,72)>;
G:=Group( (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), (1,71,82)(2,72,83)(3,65,84)(4,66,85)(5,67,86)(6,68,87)(7,69,88)(8,70,81)(9,40,77)(10,33,78)(11,34,79)(12,35,80)(13,36,73)(14,37,74)(15,38,75)(16,39,76)(17,49,58)(18,50,59)(19,51,60)(20,52,61)(21,53,62)(22,54,63)(23,55,64)(24,56,57)(25,47,89)(26,48,90)(27,41,91)(28,42,92)(29,43,93)(30,44,94)(31,45,95)(32,46,96), (1,41)(2,42)(3,43)(4,44)(5,45)(6,46)(7,47)(8,48)(9,61)(10,62)(11,63)(12,64)(13,57)(14,58)(15,59)(16,60)(17,74)(18,75)(19,76)(20,77)(21,78)(22,79)(23,80)(24,73)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,53)(34,54)(35,55)(36,56)(37,49)(38,50)(39,51)(40,52)(81,90)(82,91)(83,92)(84,93)(85,94)(86,95)(87,96)(88,89), (1,52,45,40)(2,55,46,35)(3,50,47,38)(4,53,48,33)(5,56,41,36)(6,51,42,39)(7,54,43,34)(8,49,44,37)(9,82,20,31)(10,85,21,26)(11,88,22,29)(12,83,23,32)(13,86,24,27)(14,81,17,30)(15,84,18,25)(16,87,19,28)(57,91,73,67)(58,94,74,70)(59,89,75,65)(60,92,76,68)(61,95,77,71)(62,90,78,66)(63,93,79,69)(64,96,80,72) );
G=PermutationGroup([[(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)], [(1,71,82),(2,72,83),(3,65,84),(4,66,85),(5,67,86),(6,68,87),(7,69,88),(8,70,81),(9,40,77),(10,33,78),(11,34,79),(12,35,80),(13,36,73),(14,37,74),(15,38,75),(16,39,76),(17,49,58),(18,50,59),(19,51,60),(20,52,61),(21,53,62),(22,54,63),(23,55,64),(24,56,57),(25,47,89),(26,48,90),(27,41,91),(28,42,92),(29,43,93),(30,44,94),(31,45,95),(32,46,96)], [(1,41),(2,42),(3,43),(4,44),(5,45),(6,46),(7,47),(8,48),(9,61),(10,62),(11,63),(12,64),(13,57),(14,58),(15,59),(16,60),(17,74),(18,75),(19,76),(20,77),(21,78),(22,79),(23,80),(24,73),(25,69),(26,70),(27,71),(28,72),(29,65),(30,66),(31,67),(32,68),(33,53),(34,54),(35,55),(36,56),(37,49),(38,50),(39,51),(40,52),(81,90),(82,91),(83,92),(84,93),(85,94),(86,95),(87,96),(88,89)], [(1,52,45,40),(2,55,46,35),(3,50,47,38),(4,53,48,33),(5,56,41,36),(6,51,42,39),(7,54,43,34),(8,49,44,37),(9,82,20,31),(10,85,21,26),(11,88,22,29),(12,83,23,32),(13,86,24,27),(14,81,17,30),(15,84,18,25),(16,87,19,28),(57,91,73,67),(58,94,74,70),(59,89,75,65),(60,92,76,68),(61,95,77,71),(62,90,78,66),(63,93,79,69),(64,96,80,72)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 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 | S3 | Q8 | D4 | D4 | D6 | D6 | C4×S3 | C4○D8 | S3×Q8 | S3×D4 | Q8.7D6 |
| kernel | (S3×C8)⋊C4 | C6.Q16 | C8⋊Dic3 | C3×C4.Q8 | C4⋊C4⋊7S3 | S3×C2×C8 | S3×C8 | C4.Q8 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 4 |
Matrix representation of (S3×C8)⋊C4 ►in GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 10 | 12 | 0 | 0 |
| 0 | 0 | 51 | 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 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 54 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 55 | 10 | 0 | 0 |
| 0 | 48 | 18 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,10,0,0,0,0,12,51,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,72,0,0,0,0,54,1,0,0,0,0,0,0,1,0,0,0,1,0],[27,0,0,0,0,0,55,48,0,0,0,10,18,0,0,0,0,0,72,0,0,0,0,0,72] >;
(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(192,419);
// by ID
G=gap.SmallGroup(192,419);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,555,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^3=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^4*c>;
// generators/relations