metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2.D8⋊5S3, D6⋊C8⋊16C2, C4⋊C4.50D6, (C2×C8).29D6, C12⋊D4.8C2, C6.76(C4○D8), C2.D24⋊15C2, C6.D8⋊22C2, C4.82(C4○D12), C12.40(C4○D4), C2.23(D8⋊S3), C6.42(C8⋊C22), C12.Q8⋊21C2, (C22×S3).28D4, C22.231(S3×D4), (C2×C24).171C22, (C2×C12).301C23, C4.30(Q8⋊3S3), (C2×Dic3).168D4, (C2×D12).83C22, C3⋊5(C23.19D4), C2.14(D24⋊C2), C2.17(D6.D4), C4⋊Dic3.126C22, C6.47(C22.D4), C4⋊C4⋊7S3⋊7C2, (C3×C2.D8)⋊13C2, (C2×C6).306(C2×D4), (C2×C3⋊C8).71C22, (S3×C2×C4).40C22, (C3×C4⋊C4).94C22, (C2×C4).404(C22×S3), SmallGroup(192,444)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2.D8⋊S3
G = < a,b,c,d,e | a2=b8=d3=e2=1, c2=a, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=ab5, cd=dc, ede=d-1 >
Subgroups: 368 in 106 conjugacy classes, 37 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C3⋊C8, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22⋊C8, D4⋊C4, C4.Q8, C2.D8, C42⋊C2, C4⋊D4, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C2×D12, C23.19D4, C12.Q8, C6.D8, D6⋊C8, C2.D24, C3×C2.D8, C4⋊C4⋊7S3, C12⋊D4, C2.D8⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, C4○D8, C8⋊C22, C4○D12, S3×D4, Q8⋊3S3, C23.19D4, D6.D4, D8⋊S3, D24⋊C2, C2.D8⋊S3
►On 96 points
Generators in S96
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 53)(18 54)(19 55)(20 56)(21 49)(22 50)(23 51)(24 52)(25 90)(26 91)(27 92)(28 93)(29 94)(30 95)(31 96)(32 89)(33 87)(34 88)(35 81)(36 82)(37 83)(38 84)(39 85)(40 86)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 57)
(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 12 77 68)(2 11 78 67)(3 10 79 66)(4 9 80 65)(5 16 73 72)(6 15 74 71)(7 14 75 70)(8 13 76 69)(17 38 53 84)(18 37 54 83)(19 36 55 82)(20 35 56 81)(21 34 49 88)(22 33 50 87)(23 40 51 86)(24 39 52 85)(25 60 90 43)(26 59 91 42)(27 58 92 41)(28 57 93 48)(29 64 94 47)(30 63 95 46)(31 62 96 45)(32 61 89 44)
(1 49 41)(2 50 42)(3 51 43)(4 52 44)(5 53 45)(6 54 46)(7 55 47)(8 56 48)(9 85 32)(10 86 25)(11 87 26)(12 88 27)(13 81 28)(14 82 29)(15 83 30)(16 84 31)(17 62 73)(18 63 74)(19 64 75)(20 57 76)(21 58 77)(22 59 78)(23 60 79)(24 61 80)(33 91 67)(34 92 68)(35 93 69)(36 94 70)(37 95 71)(38 96 72)(39 89 65)(40 90 66)
(1 70)(2 11)(3 72)(4 13)(5 66)(6 15)(7 68)(8 9)(10 73)(12 75)(14 77)(16 79)(17 25)(18 95)(19 27)(20 89)(21 29)(22 91)(23 31)(24 93)(26 50)(28 52)(30 54)(32 56)(33 59)(34 47)(35 61)(36 41)(37 63)(38 43)(39 57)(40 45)(42 87)(44 81)(46 83)(48 85)(49 94)(51 96)(53 90)(55 92)(58 82)(60 84)(62 86)(64 88)(65 76)(67 78)(69 80)(71 74)
G:=sub<Sym(96)| (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,89)(33,87)(34,88)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,57), (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), (1,12,77,68)(2,11,78,67)(3,10,79,66)(4,9,80,65)(5,16,73,72)(6,15,74,71)(7,14,75,70)(8,13,76,69)(17,38,53,84)(18,37,54,83)(19,36,55,82)(20,35,56,81)(21,34,49,88)(22,33,50,87)(23,40,51,86)(24,39,52,85)(25,60,90,43)(26,59,91,42)(27,58,92,41)(28,57,93,48)(29,64,94,47)(30,63,95,46)(31,62,96,45)(32,61,89,44), (1,49,41)(2,50,42)(3,51,43)(4,52,44)(5,53,45)(6,54,46)(7,55,47)(8,56,48)(9,85,32)(10,86,25)(11,87,26)(12,88,27)(13,81,28)(14,82,29)(15,83,30)(16,84,31)(17,62,73)(18,63,74)(19,64,75)(20,57,76)(21,58,77)(22,59,78)(23,60,79)(24,61,80)(33,91,67)(34,92,68)(35,93,69)(36,94,70)(37,95,71)(38,96,72)(39,89,65)(40,90,66), (1,70)(2,11)(3,72)(4,13)(5,66)(6,15)(7,68)(8,9)(10,73)(12,75)(14,77)(16,79)(17,25)(18,95)(19,27)(20,89)(21,29)(22,91)(23,31)(24,93)(26,50)(28,52)(30,54)(32,56)(33,59)(34,47)(35,61)(36,41)(37,63)(38,43)(39,57)(40,45)(42,87)(44,81)(46,83)(48,85)(49,94)(51,96)(53,90)(55,92)(58,82)(60,84)(62,86)(64,88)(65,76)(67,78)(69,80)(71,74)>;
G:=Group( (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,53)(18,54)(19,55)(20,56)(21,49)(22,50)(23,51)(24,52)(25,90)(26,91)(27,92)(28,93)(29,94)(30,95)(31,96)(32,89)(33,87)(34,88)(35,81)(36,82)(37,83)(38,84)(39,85)(40,86)(41,58)(42,59)(43,60)(44,61)(45,62)(46,63)(47,64)(48,57), (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), (1,12,77,68)(2,11,78,67)(3,10,79,66)(4,9,80,65)(5,16,73,72)(6,15,74,71)(7,14,75,70)(8,13,76,69)(17,38,53,84)(18,37,54,83)(19,36,55,82)(20,35,56,81)(21,34,49,88)(22,33,50,87)(23,40,51,86)(24,39,52,85)(25,60,90,43)(26,59,91,42)(27,58,92,41)(28,57,93,48)(29,64,94,47)(30,63,95,46)(31,62,96,45)(32,61,89,44), (1,49,41)(2,50,42)(3,51,43)(4,52,44)(5,53,45)(6,54,46)(7,55,47)(8,56,48)(9,85,32)(10,86,25)(11,87,26)(12,88,27)(13,81,28)(14,82,29)(15,83,30)(16,84,31)(17,62,73)(18,63,74)(19,64,75)(20,57,76)(21,58,77)(22,59,78)(23,60,79)(24,61,80)(33,91,67)(34,92,68)(35,93,69)(36,94,70)(37,95,71)(38,96,72)(39,89,65)(40,90,66), (1,70)(2,11)(3,72)(4,13)(5,66)(6,15)(7,68)(8,9)(10,73)(12,75)(14,77)(16,79)(17,25)(18,95)(19,27)(20,89)(21,29)(22,91)(23,31)(24,93)(26,50)(28,52)(30,54)(32,56)(33,59)(34,47)(35,61)(36,41)(37,63)(38,43)(39,57)(40,45)(42,87)(44,81)(46,83)(48,85)(49,94)(51,96)(53,90)(55,92)(58,82)(60,84)(62,86)(64,88)(65,76)(67,78)(69,80)(71,74) );
G=PermutationGroup([[(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,53),(18,54),(19,55),(20,56),(21,49),(22,50),(23,51),(24,52),(25,90),(26,91),(27,92),(28,93),(29,94),(30,95),(31,96),(32,89),(33,87),(34,88),(35,81),(36,82),(37,83),(38,84),(39,85),(40,86),(41,58),(42,59),(43,60),(44,61),(45,62),(46,63),(47,64),(48,57)], [(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)], [(1,12,77,68),(2,11,78,67),(3,10,79,66),(4,9,80,65),(5,16,73,72),(6,15,74,71),(7,14,75,70),(8,13,76,69),(17,38,53,84),(18,37,54,83),(19,36,55,82),(20,35,56,81),(21,34,49,88),(22,33,50,87),(23,40,51,86),(24,39,52,85),(25,60,90,43),(26,59,91,42),(27,58,92,41),(28,57,93,48),(29,64,94,47),(30,63,95,46),(31,62,96,45),(32,61,89,44)], [(1,49,41),(2,50,42),(3,51,43),(4,52,44),(5,53,45),(6,54,46),(7,55,47),(8,56,48),(9,85,32),(10,86,25),(11,87,26),(12,88,27),(13,81,28),(14,82,29),(15,83,30),(16,84,31),(17,62,73),(18,63,74),(19,64,75),(20,57,76),(21,58,77),(22,59,78),(23,60,79),(24,61,80),(33,91,67),(34,92,68),(35,93,69),(36,94,70),(37,95,71),(38,96,72),(39,89,65),(40,90,66)], [(1,70),(2,11),(3,72),(4,13),(5,66),(6,15),(7,68),(8,9),(10,73),(12,75),(14,77),(16,79),(17,25),(18,95),(19,27),(20,89),(21,29),(22,91),(23,31),(24,93),(26,50),(28,52),(30,54),(32,56),(33,59),(34,47),(35,61),(36,41),(37,63),(38,43),(39,57),(40,45),(42,87),(44,81),(46,83),(48,85),(49,94),(51,96),(53,90),(55,92),(58,82),(60,84),(62,86),(64,88),(65,76),(67,78),(69,80),(71,74)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 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 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C4○D8 | C4○D12 | C8⋊C22 | Q8⋊3S3 | S3×D4 | D8⋊S3 | D24⋊C2 |
| kernel | C2.D8⋊S3 | C12.Q8 | C6.D8 | D6⋊C8 | C2.D24 | C3×C2.D8 | C4⋊C4⋊7S3 | C12⋊D4 | C2.D8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | C6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C2.D8⋊S3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 57 | 16 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 43 | 13 |
| 0 | 0 | 60 | 30 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 |
| 0 | 0 | 1 | 0 |
| 0 | 46 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 66 | 7 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[57,57,0,0,16,57,0,0,0,0,43,60,0,0,13,30],[46,0,0,0,0,27,0,0,0,0,66,14,0,0,59,7],[1,0,0,0,0,1,0,0,0,0,72,1,0,0,72,0],[0,27,0,0,46,0,0,0,0,0,66,66,0,0,59,7] >;
C2.D8⋊S3 in GAP, Magma, Sage, TeX
C_2.D_8\rtimes S_3
% in TeX
G:=Group("C2.D8:S3"); // GroupNames label
G:=SmallGroup(192,444);
// by ID
G=gap.SmallGroup(192,444);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,64,926,219,268,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=d^3=e^2=1,c^2=a,a*b=b*a,e*c*e=a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=a*b^5,c*d=d*c,e*d*e=d^-1>;
// generators/relations