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, (C2×C8).245D6, C12.78(C2×D4), C3⋊C8.20C23, C4.3(S3×C23), (S3×C8)⋊14C22, (C2×D4).180D6, (C3×D8)⋊10C22, C8.38(C22×S3), D4.S3⋊8C22, D4.1(C22×S3), (C3×D4).1C23, (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
Generators and relations for C2×D8⋊3S3
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 >
Subgroups: 664 in 266 conjugacy classes, 103 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, D8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, C22×C8, C2×D8, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, S3×C8, Dic12, C2×C3⋊C8, D4.S3, C2×C24, C3×D8, C2×Dic6, S3×C2×C4, D4⋊2S3, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C6×D4, C2×C4○D8, S3×C2×C8, C2×Dic12, D8⋊3S3, C2×D4.S3, C6×D8, C2×D4⋊2S3, C2×D8⋊3S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C4○D8, C22×D4, S3×D4, S3×C23, C2×C4○D8, D8⋊3S3, C2×S3×D4, C2×D8⋊3S3
►On 96 points
Generators in S96
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 65)(16 66)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 56)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(41 85)(42 86)(43 87)(44 88)(45 81)(46 82)(47 83)(48 84)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)
(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 25)(10 32)(11 31)(12 30)(13 29)(14 28)(15 27)(16 26)(17 36)(18 35)(19 34)(20 33)(21 40)(22 39)(23 38)(24 37)(41 76)(42 75)(43 74)(44 73)(45 80)(46 79)(47 78)(48 77)(49 66)(50 65)(51 72)(52 71)(53 70)(54 69)(55 68)(56 67)(81 95)(82 94)(83 93)(84 92)(85 91)(86 90)(87 89)(88 96)
(1 82 29)(2 83 30)(3 84 31)(4 85 32)(5 86 25)(6 87 26)(7 88 27)(8 81 28)(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 77 69)(18 78 70)(19 79 71)(20 80 72)(21 73 65)(22 74 66)(23 75 67)(24 76 68)(33 45 51)(34 46 52)(35 47 53)(36 48 54)(37 41 55)(38 42 56)(39 43 49)(40 44 50)
(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 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(41 55)(42 56)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(57 61)(58 62)(59 63)(60 64)(65 77)(66 78)(67 79)(68 80)(69 73)(70 74)(71 75)(72 76)
G:=sub<Sym(96)| (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95), (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)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,36)(18,35)(19,34)(20,33)(21,40)(22,39)(23,38)(24,37)(41,76)(42,75)(43,74)(44,73)(45,80)(46,79)(47,78)(48,77)(49,66)(50,65)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(88,96), (1,82,29)(2,83,30)(3,84,31)(4,85,32)(5,86,25)(6,87,26)(7,88,27)(8,81,28)(9,60,90)(10,61,91)(11,62,92)(12,63,93)(13,64,94)(14,57,95)(15,58,96)(16,59,89)(17,77,69)(18,78,70)(19,79,71)(20,80,72)(21,73,65)(22,74,66)(23,75,67)(24,76,68)(33,45,51)(34,46,52)(35,47,53)(36,48,54)(37,41,55)(38,42,56)(39,43,49)(40,44,50), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,21)(18,22)(19,23)(20,24)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(57,61)(58,62)(59,63)(60,64)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76)>;
G:=Group( (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,67)(10,68)(11,69)(12,70)(13,71)(14,72)(15,65)(16,66)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,56)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(41,85)(42,86)(43,87)(44,88)(45,81)(46,82)(47,83)(48,84)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95), (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)(9,25)(10,32)(11,31)(12,30)(13,29)(14,28)(15,27)(16,26)(17,36)(18,35)(19,34)(20,33)(21,40)(22,39)(23,38)(24,37)(41,76)(42,75)(43,74)(44,73)(45,80)(46,79)(47,78)(48,77)(49,66)(50,65)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(81,95)(82,94)(83,93)(84,92)(85,91)(86,90)(87,89)(88,96), (1,82,29)(2,83,30)(3,84,31)(4,85,32)(5,86,25)(6,87,26)(7,88,27)(8,81,28)(9,60,90)(10,61,91)(11,62,92)(12,63,93)(13,64,94)(14,57,95)(15,58,96)(16,59,89)(17,77,69)(18,78,70)(19,79,71)(20,80,72)(21,73,65)(22,74,66)(23,75,67)(24,76,68)(33,45,51)(34,46,52)(35,47,53)(36,48,54)(37,41,55)(38,42,56)(39,43,49)(40,44,50), (9,94)(10,95)(11,96)(12,89)(13,90)(14,91)(15,92)(16,93)(17,21)(18,22)(19,23)(20,24)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(41,55)(42,56)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54)(57,61)(58,62)(59,63)(60,64)(65,77)(66,78)(67,79)(68,80)(69,73)(70,74)(71,75)(72,76) );
G=PermutationGroup([[(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,67),(10,68),(11,69),(12,70),(13,71),(14,72),(15,65),(16,66),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,56),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(41,85),(42,86),(43,87),(44,88),(45,81),(46,82),(47,83),(48,84),(73,96),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95)], [(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),(9,25),(10,32),(11,31),(12,30),(13,29),(14,28),(15,27),(16,26),(17,36),(18,35),(19,34),(20,33),(21,40),(22,39),(23,38),(24,37),(41,76),(42,75),(43,74),(44,73),(45,80),(46,79),(47,78),(48,77),(49,66),(50,65),(51,72),(52,71),(53,70),(54,69),(55,68),(56,67),(81,95),(82,94),(83,93),(84,92),(85,91),(86,90),(87,89),(88,96)], [(1,82,29),(2,83,30),(3,84,31),(4,85,32),(5,86,25),(6,87,26),(7,88,27),(8,81,28),(9,60,90),(10,61,91),(11,62,92),(12,63,93),(13,64,94),(14,57,95),(15,58,96),(16,59,89),(17,77,69),(18,78,70),(19,79,71),(20,80,72),(21,73,65),(22,74,66),(23,75,67),(24,76,68),(33,45,51),(34,46,52),(35,47,53),(36,48,54),(37,41,55),(38,42,56),(39,43,49),(40,44,50)], [(9,94),(10,95),(11,96),(12,89),(13,90),(14,91),(15,92),(16,93),(17,21),(18,22),(19,23),(20,24),(25,86),(26,87),(27,88),(28,81),(29,82),(30,83),(31,84),(32,85),(41,55),(42,56),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54),(57,61),(58,62),(59,63),(60,64),(65,77),(66,78),(67,79),(68,80),(69,73),(70,74),(71,75),(72,76)]])
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 |
Matrix representation of C2×D8⋊3S3 ►in 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] >;
C2×D8⋊3S3 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