direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: S3×C2.D8, D6.11D8, D6.4Q16, (S3×C8)⋊1C4, C8⋊13(C4×S3), C24⋊5(C2×C4), C2.4(S3×D8), C6.28(C2×D8), (C4×S3).5Q8, C4.29(S3×Q8), C2.4(S3×Q16), C4⋊C4.169D6, C24⋊1C4⋊20C2, (C2×C8).229D6, C6.23(C2×Q16), C12.20(C2×Q8), D6.11(C4⋊C4), C6.Q16⋊19C2, C22.89(S3×D4), Dic3.6(C4⋊C4), C12.48(C22×C4), (C2×C24).81C22, (C2×C12).295C23, (C2×Dic3).102D4, (C22×S3).107D4, C4⋊Dic3.121C22, C3⋊1(C2×C2.D8), C3⋊C8⋊22(C2×C4), (S3×C2×C8).2C2, C4.79(S3×C2×C4), C6.13(C2×C4⋊C4), (S3×C4⋊C4).7C2, C2.14(S3×C4⋊C4), (C3×C2.D8)⋊3C2, (C4×S3).27(C2×C4), (C2×C6).300(C2×D4), (C3×C4⋊C4).88C22, (C2×C3⋊C8).233C22, (S3×C2×C4).234C22, (C2×C4).398(C22×S3), SmallGroup(192,438)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C2.D8
G = < a,b,c,d,e | a3=b2=c2=d8=1, e2=c, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 336 in 130 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C2.D8, C2.D8, C2×C4⋊C4, C22×C8, S3×C8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×C2.D8, C6.Q16, C24⋊1C4, C3×C2.D8, S3×C4⋊C4, S3×C2×C8, S3×C2.D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, C23, D6, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C4×S3, C22×S3, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, S3×C2×C4, S3×D4, S3×Q8, C2×C2.D8, S3×C4⋊C4, S3×D8, S3×Q16, S3×C2.D8
►On 96 points
Generators in S96
(1 34 79)(2 35 80)(3 36 73)(4 37 74)(5 38 75)(6 39 76)(7 40 77)(8 33 78)(9 24 84)(10 17 85)(11 18 86)(12 19 87)(13 20 88)(14 21 81)(15 22 82)(16 23 83)(25 89 47)(26 90 48)(27 91 41)(28 92 42)(29 93 43)(30 94 44)(31 95 45)(32 96 46)(49 70 61)(50 71 62)(51 72 63)(52 65 64)(53 66 57)(54 67 58)(55 68 59)(56 69 60)
(1 5)(2 6)(3 7)(4 8)(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)(25 93)(26 94)(27 95)(28 96)(29 89)(30 90)(31 91)(32 92)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 70)(58 71)(59 72)(60 65)(61 66)(62 67)(63 68)(64 69)(81 85)(82 86)(83 87)(84 88)
(1 45)(2 46)(3 47)(4 48)(5 41)(6 42)(7 43)(8 44)(9 70)(10 71)(11 72)(12 65)(13 66)(14 67)(15 68)(16 69)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(49 84)(50 85)(51 86)(52 87)(53 88)(54 81)(55 82)(56 83)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 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 56 45 83)(2 55 46 82)(3 54 47 81)(4 53 48 88)(5 52 41 87)(6 51 42 86)(7 50 43 85)(8 49 44 84)(9 33 70 30)(10 40 71 29)(11 39 72 28)(12 38 65 27)(13 37 66 26)(14 36 67 25)(15 35 68 32)(16 34 69 31)(17 77 62 93)(18 76 63 92)(19 75 64 91)(20 74 57 90)(21 73 58 89)(22 80 59 96)(23 79 60 95)(24 78 61 94)
G:=sub<Sym(96)| (1,34,79)(2,35,80)(3,36,73)(4,37,74)(5,38,75)(6,39,76)(7,40,77)(8,33,78)(9,24,84)(10,17,85)(11,18,86)(12,19,87)(13,20,88)(14,21,81)(15,22,82)(16,23,83)(25,89,47)(26,90,48)(27,91,41)(28,92,42)(29,93,43)(30,94,44)(31,95,45)(32,96,46)(49,70,61)(50,71,62)(51,72,63)(52,65,64)(53,66,57)(54,67,58)(55,68,59)(56,69,60), (1,5)(2,6)(3,7)(4,8)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,70)(58,71)(59,72)(60,65)(61,66)(62,67)(63,68)(64,69)(81,85)(82,86)(83,87)(84,88), (1,45)(2,46)(3,47)(4,48)(5,41)(6,42)(7,43)(8,44)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,84)(50,85)(51,86)(52,87)(53,88)(54,81)(55,82)(56,83)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (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), (1,56,45,83)(2,55,46,82)(3,54,47,81)(4,53,48,88)(5,52,41,87)(6,51,42,86)(7,50,43,85)(8,49,44,84)(9,33,70,30)(10,40,71,29)(11,39,72,28)(12,38,65,27)(13,37,66,26)(14,36,67,25)(15,35,68,32)(16,34,69,31)(17,77,62,93)(18,76,63,92)(19,75,64,91)(20,74,57,90)(21,73,58,89)(22,80,59,96)(23,79,60,95)(24,78,61,94)>;
G:=Group( (1,34,79)(2,35,80)(3,36,73)(4,37,74)(5,38,75)(6,39,76)(7,40,77)(8,33,78)(9,24,84)(10,17,85)(11,18,86)(12,19,87)(13,20,88)(14,21,81)(15,22,82)(16,23,83)(25,89,47)(26,90,48)(27,91,41)(28,92,42)(29,93,43)(30,94,44)(31,95,45)(32,96,46)(49,70,61)(50,71,62)(51,72,63)(52,65,64)(53,66,57)(54,67,58)(55,68,59)(56,69,60), (1,5)(2,6)(3,7)(4,8)(9,20)(10,21)(11,22)(12,23)(13,24)(14,17)(15,18)(16,19)(25,93)(26,94)(27,95)(28,96)(29,89)(30,90)(31,91)(32,92)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,70)(58,71)(59,72)(60,65)(61,66)(62,67)(63,68)(64,69)(81,85)(82,86)(83,87)(84,88), (1,45)(2,46)(3,47)(4,48)(5,41)(6,42)(7,43)(8,44)(9,70)(10,71)(11,72)(12,65)(13,66)(14,67)(15,68)(16,69)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(49,84)(50,85)(51,86)(52,87)(53,88)(54,81)(55,82)(56,83)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96), (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), (1,56,45,83)(2,55,46,82)(3,54,47,81)(4,53,48,88)(5,52,41,87)(6,51,42,86)(7,50,43,85)(8,49,44,84)(9,33,70,30)(10,40,71,29)(11,39,72,28)(12,38,65,27)(13,37,66,26)(14,36,67,25)(15,35,68,32)(16,34,69,31)(17,77,62,93)(18,76,63,92)(19,75,64,91)(20,74,57,90)(21,73,58,89)(22,80,59,96)(23,79,60,95)(24,78,61,94) );
G=PermutationGroup([[(1,34,79),(2,35,80),(3,36,73),(4,37,74),(5,38,75),(6,39,76),(7,40,77),(8,33,78),(9,24,84),(10,17,85),(11,18,86),(12,19,87),(13,20,88),(14,21,81),(15,22,82),(16,23,83),(25,89,47),(26,90,48),(27,91,41),(28,92,42),(29,93,43),(30,94,44),(31,95,45),(32,96,46),(49,70,61),(50,71,62),(51,72,63),(52,65,64),(53,66,57),(54,67,58),(55,68,59),(56,69,60)], [(1,5),(2,6),(3,7),(4,8),(9,20),(10,21),(11,22),(12,23),(13,24),(14,17),(15,18),(16,19),(25,93),(26,94),(27,95),(28,96),(29,89),(30,90),(31,91),(32,92),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,70),(58,71),(59,72),(60,65),(61,66),(62,67),(63,68),(64,69),(81,85),(82,86),(83,87),(84,88)], [(1,45),(2,46),(3,47),(4,48),(5,41),(6,42),(7,43),(8,44),(9,70),(10,71),(11,72),(12,65),(13,66),(14,67),(15,68),(16,69),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(49,84),(50,85),(51,86),(52,87),(53,88),(54,81),(55,82),(56,83),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96)], [(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)], [(1,56,45,83),(2,55,46,82),(3,54,47,81),(4,53,48,88),(5,52,41,87),(6,51,42,86),(7,50,43,85),(8,49,44,84),(9,33,70,30),(10,40,71,29),(11,39,72,28),(12,38,65,27),(13,37,66,26),(14,36,67,25),(15,35,68,32),(16,34,69,31),(17,77,62,93),(18,76,63,92),(19,75,64,91),(20,74,57,90),(21,73,58,89),(22,80,59,96),(23,79,60,95),(24,78,61,94)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 2 | 2 | 3 | 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 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | + | - | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | Q8 | D4 | D4 | D6 | D6 | D8 | Q16 | C4×S3 | S3×Q8 | S3×D4 | S3×D8 | S3×Q16 |
| kernel | S3×C2.D8 | C6.Q16 | C24⋊1C4 | C3×C2.D8 | S3×C4⋊C4 | S3×C2×C8 | S3×C8 | C2.D8 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | D6 | D6 | C8 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of S3×C2.D8 ►in GL5(𝔽73)
| 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 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 38 | 0 | 0 |
| 0 | 48 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 | 0 |
| 0 | 1 | 67 | 0 | 0 |
| 0 | 49 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(5,GF(73))| [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],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,72,0,0,0,0,72],[72,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,32,48,0,0,0,38,0,0,0,0,0,0,1,0,0,0,0,0,1],[46,0,0,0,0,0,1,49,0,0,0,67,72,0,0,0,0,0,72,0,0,0,0,0,72] >;
S3×C2.D8 in GAP, Magma, Sage, TeX
S_3\times C_2.D_8
% in TeX
G:=Group("S3xC2.D8"); // GroupNames label
G:=SmallGroup(192,438);
// by ID
G=gap.SmallGroup(192,438);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,219,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=1,e^2=c,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations