direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D8⋊3S3, D8⋊12D6, C12.3C24, C24.32C23, Dic6.1C23, Dic12⋊13C22, (C6×D8)⋊8C2, (C2×D8)⋊13S3, C6⋊2(C4○D8), D6.9(C2×D4), C4.41(S3×D4), (C4×S3).27D4, C12.78(C2×D4), (C2×C8).245D6, C3⋊C8.20C23, C4.3(S3×C23), (S3×C8)⋊14C22, (C2×D4).180D6, (C3×D8)⋊10C22, C8.38(C22×S3), D4.S3⋊8C22, (C3×D4).1C23, D4.1(C22×S3), (C2×Dic12)⋊19C2, D4⋊2S3⋊6C22, (C4×S3).24C23, (C2×C24).97C22, Dic3.68(C2×D4), (C22×S3).61D4, C6.104(C22×D4), C22.137(S3×D4), (C2×C12).520C23, (C2×Dic3).215D4, (C6×D4).162C22, (C2×Dic6).195C22, (S3×C2×C8)⋊5C2, C3⋊2(C2×C4○D8), C2.77(C2×S3×D4), (C2×D4.S3)⋊26C2, (C2×C6).393(C2×D4), (C2×D4⋊2S3)⋊24C2, (C2×C3⋊C8).283C22, (S3×C2×C4).256C22, (C2×C4).610(C22×S3), SmallGroup(192,1315)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 664 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×2], C6, C6 [×2], C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×4], D4 [×10], Q8 [×6], C23 [×3], Dic3 [×2], Dic3 [×4], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×8], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×8], Q16 [×4], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×12], C3⋊C8 [×2], C24 [×2], Dic6 [×4], Dic6 [×2], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×S3, C22×C6 [×2], C22×C8, C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], S3×C8 [×4], Dic12 [×4], C2×C3⋊C8, D4.S3 [×8], C2×C24, C3×D8 [×4], C2×Dic6 [×2], S3×C2×C4, D4⋊2S3 [×8], D4⋊2S3 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], C2×C4○D8, S3×C2×C8, C2×Dic12, D8⋊3S3 [×8], C2×D4.S3 [×2], C6×D8, C2×D4⋊2S3 [×2], C2×D8⋊3S3
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C4○D8 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C4○D8, D8⋊3S3 [×2], C2×S3×D4, C2×D8⋊3S3
Generators and relations
G = < a,b,c,d,e | a2=b8=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b4c, ede=d-1 >
►On 96 points
(1 37)(2 38)(3 39)(4 40)(5 33)(6 34)(7 35)(8 36)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 84)
(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 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 77)(10 76)(11 75)(12 74)(13 73)(14 80)(15 79)(16 78)(17 36)(18 35)(19 34)(20 33)(21 40)(22 39)(23 38)(24 37)(25 43)(26 42)(27 41)(28 48)(29 47)(30 46)(31 45)(32 44)(49 91)(50 90)(51 89)(52 96)(53 95)(54 94)(55 93)(56 92)(65 83)(66 82)(67 81)(68 88)(69 87)(70 86)(71 85)(72 84)
(1 54 73)(2 55 74)(3 56 75)(4 49 76)(5 50 77)(6 51 78)(7 52 79)(8 53 80)(9 60 90)(10 61 91)(11 62 92)(12 63 93)(13 64 94)(14 57 95)(15 58 96)(16 59 89)(17 31 72)(18 32 65)(19 25 66)(20 26 67)(21 27 68)(22 28 69)(23 29 70)(24 30 71)(33 42 81)(34 43 82)(35 44 83)(36 45 84)(37 46 85)(38 47 86)(39 48 87)(40 41 88)
(9 94)(10 95)(11 96)(12 89)(13 90)(14 91)(15 92)(16 93)(17 21)(18 22)(19 23)(20 24)(25 70)(26 71)(27 72)(28 65)(29 66)(30 67)(31 68)(32 69)(41 88)(42 81)(43 82)(44 83)(45 84)(46 85)(47 86)(48 87)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)(57 61)(58 62)(59 63)(60 64)
G:=sub<Sym(96)| (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,77)(10,76)(11,75)(12,74)(13,73)(14,80)(15,79)(16,78)(17,36)(18,35)(19,34)(20,33)(21,40)(22,39)(23,38)(24,37)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(49,91)(50,90)(51,89)(52,96)(53,95)(54,94)(55,93)(56,92)(65,83)(66,82)(67,81)(68,88)(69,87)(70,86)(71,85)(72,84), (1,54,73)(2,55,74)(3,56,75)(4,49,76)(5,50,77)(6,51,78)(7,52,79)(8,53,80)(9,60,90)(10,61,91)(11,62,92)(12,63,93)(13,64,94)(14,57,95)(15,58,96)(16,59,89)(17,31,72)(18,32,65)(19,25,66)(20,26,67)(21,27,68)(22,28,69)(23,29,70)(24,30,71)(33,42,81)(34,43,82)(35,44,83)(36,45,84)(37,46,85)(38,47,86)(39,48,87)(40,41,88), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,21)(18,22)(19,23)(20,24)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,61)(58,62)(59,63)(60,64)>;
G:=Group( (1,37)(2,38)(3,39)(4,40)(5,33)(6,34)(7,35)(8,36)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,89)(26,90)(27,91)(28,92)(29,93)(30,94)(31,95)(32,96)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,85)(74,86)(75,87)(76,88)(77,81)(78,82)(79,83)(80,84), (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), (1,64)(2,63)(3,62)(4,61)(5,60)(6,59)(7,58)(8,57)(9,77)(10,76)(11,75)(12,74)(13,73)(14,80)(15,79)(16,78)(17,36)(18,35)(19,34)(20,33)(21,40)(22,39)(23,38)(24,37)(25,43)(26,42)(27,41)(28,48)(29,47)(30,46)(31,45)(32,44)(49,91)(50,90)(51,89)(52,96)(53,95)(54,94)(55,93)(56,92)(65,83)(66,82)(67,81)(68,88)(69,87)(70,86)(71,85)(72,84), (1,54,73)(2,55,74)(3,56,75)(4,49,76)(5,50,77)(6,51,78)(7,52,79)(8,53,80)(9,60,90)(10,61,91)(11,62,92)(12,63,93)(13,64,94)(14,57,95)(15,58,96)(16,59,89)(17,31,72)(18,32,65)(19,25,66)(20,26,67)(21,27,68)(22,28,69)(23,29,70)(24,30,71)(33,42,81)(34,43,82)(35,44,83)(36,45,84)(37,46,85)(38,47,86)(39,48,87)(40,41,88), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,21)(18,22)(19,23)(20,24)(25,70)(26,71)(27,72)(28,65)(29,66)(30,67)(31,68)(32,69)(41,88)(42,81)(43,82)(44,83)(45,84)(46,85)(47,86)(48,87)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75)(57,61)(58,62)(59,63)(60,64) );
G=PermutationGroup([(1,37),(2,38),(3,39),(4,40),(5,33),(6,34),(7,35),(8,36),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,65),(16,66),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,89),(26,90),(27,91),(28,92),(29,93),(30,94),(31,95),(32,96),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(73,85),(74,86),(75,87),(76,88),(77,81),(78,82),(79,83),(80,84)], [(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)], [(1,64),(2,63),(3,62),(4,61),(5,60),(6,59),(7,58),(8,57),(9,77),(10,76),(11,75),(12,74),(13,73),(14,80),(15,79),(16,78),(17,36),(18,35),(19,34),(20,33),(21,40),(22,39),(23,38),(24,37),(25,43),(26,42),(27,41),(28,48),(29,47),(30,46),(31,45),(32,44),(49,91),(50,90),(51,89),(52,96),(53,95),(54,94),(55,93),(56,92),(65,83),(66,82),(67,81),(68,88),(69,87),(70,86),(71,85),(72,84)], [(1,54,73),(2,55,74),(3,56,75),(4,49,76),(5,50,77),(6,51,78),(7,52,79),(8,53,80),(9,60,90),(10,61,91),(11,62,92),(12,63,93),(13,64,94),(14,57,95),(15,58,96),(16,59,89),(17,31,72),(18,32,65),(19,25,66),(20,26,67),(21,27,68),(22,28,69),(23,29,70),(24,30,71),(33,42,81),(34,43,82),(35,44,83),(36,45,84),(37,46,85),(38,47,86),(39,48,87),(40,41,88)], [(9,94),(10,95),(11,96),(12,89),(13,90),(14,91),(15,92),(16,93),(17,21),(18,22),(19,23),(20,24),(25,70),(26,71),(27,72),(28,65),(29,66),(30,67),(31,68),(32,69),(41,88),(42,81),(43,82),(44,83),(45,84),(46,85),(47,86),(48,87),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75),(57,61),(58,62),(59,63),(60,64)])
Matrix representation ►G ⊆ GL5(𝔽73)
| 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 71 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 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 | 1 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,10,0,0,0,0,6,22,0,0,0,0,0,72,0,0,0,0,0,72],[72,0,0,0,0,0,72,71,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72],[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],[72,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,1,0] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 6 | 6 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 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 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D8 | S3×D4 | S3×D4 | D8⋊3S3 |
| kernel | C2×D8⋊3S3 | S3×C2×C8 | C2×Dic12 | D8⋊3S3 | C2×D4.S3 | C6×D8 | C2×D4⋊2S3 | C2×D8 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | D8 | C2×D4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 1 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_2\times D_8\rtimes_3S_3
% in TeX
G:=Group("C2xD8:3S3"); // GroupNames label
G:=SmallGroup(192,1315);
// by ID
G=gap.SmallGroup(192,1315);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,1123,185,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^8=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations