metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8⋊3(C4×S3), C6.37(C4×D4), C4⋊C4.155D6, Q8⋊2S3⋊2C4, (Q8×Dic3)⋊2C2, D12.4(C2×C4), C24⋊C4⋊20C2, (C2×C8).179D6, Q8⋊C4⋊20S3, C6.Q16⋊13C2, C2.4(Q8⋊3D6), (C2×Q8).135D6, C22.80(S3×D4), Dic3⋊5D4.2C2, C3⋊3(SD16⋊C4), C6.65(C8⋊C22), C12.16(C22×C4), C2.D24.12C2, C2.4(Q16⋊S3), (C6×Q8).40C22, C12.165(C4○D4), C4.58(D4⋊2S3), (C2×C12).257C23, (C2×C24).247C22, (C2×Dic3).158D4, (C2×D12).67C22, C6.65(C8.C22), C4⋊Dic3.101C22, (C4×Dic3).24C22, C2.21(Dic3⋊4D4), C3⋊C8⋊3(C2×C4), C4.16(S3×C2×C4), (C3×Q8)⋊4(C2×C4), (C2×C6).270(C2×D4), (C2×C3⋊C8).47C22, (C3×Q8⋊C4)⋊30C2, (C3×C4⋊C4).58C22, (C2×Q8⋊2S3).2C2, (C2×C4).364(C22×S3), SmallGroup(192,376)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊3(C4×S3)
G = < a,b,c,d,e | a4=c4=d3=e2=1, b2=a2, bab-1=cac-1=eae=a-1, ad=da, cbc-1=ab, bd=db, ebe=a-1b, cd=dc, ce=ec, ede=d-1 >
Subgroups: 344 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, Q8⋊2S3, C3×C4⋊C4, C2×C24, S3×C2×C4, C2×D12, C6×Q8, SD16⋊C4, C6.Q16, C24⋊C4, C2.D24, C3×Q8⋊C4, Dic3⋊5D4, C2×Q8⋊2S3, Q8×Dic3, Q8⋊3(C4×S3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, D4⋊2S3, SD16⋊C4, Dic3⋊4D4, Q8⋊3D6, Q16⋊S3, Q8⋊3(C4×S3)
►On 96 points
Generators in S96
(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 72 3 70)(2 71 4 69)(5 44 7 42)(6 43 8 41)(9 40 11 38)(10 39 12 37)(13 75 15 73)(14 74 16 76)(17 77 19 79)(18 80 20 78)(21 57 23 59)(22 60 24 58)(25 63 27 61)(26 62 28 64)(29 65 31 67)(30 68 32 66)(33 93 35 95)(34 96 36 94)(45 81 47 83)(46 84 48 82)(49 90 51 92)(50 89 52 91)(53 86 55 88)(54 85 56 87)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 76 85 67)(6 75 86 66)(7 74 87 65)(8 73 88 68)(9 78 89 63)(10 77 90 62)(11 80 91 61)(12 79 92 64)(13 56 32 44)(14 55 29 43)(15 54 30 42)(16 53 31 41)(17 52 26 40)(18 51 27 39)(19 50 28 38)(20 49 25 37)(57 94 69 82)(58 93 70 81)(59 96 71 84)(60 95 72 83)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(2 4)(5 9)(6 12)(7 11)(8 10)(13 20)(14 19)(15 18)(16 17)(21 23)(25 32)(26 31)(27 30)(28 29)(33 35)(37 44)(38 43)(39 42)(40 41)(45 47)(49 56)(50 55)(51 54)(52 53)(57 60)(58 59)(61 65)(62 68)(63 67)(64 66)(69 72)(70 71)(73 77)(74 80)(75 79)(76 78)(81 84)(82 83)(85 89)(86 92)(87 91)(88 90)(93 96)(94 95)
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), (1,72,3,70)(2,71,4,69)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,75,15,73)(14,74,16,76)(17,77,19,79)(18,80,20,78)(21,57,23,59)(22,60,24,58)(25,63,27,61)(26,62,28,64)(29,65,31,67)(30,68,32,66)(33,93,35,95)(34,96,36,94)(45,81,47,83)(46,84,48,82)(49,90,51,92)(50,89,52,91)(53,86,55,88)(54,85,56,87), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,23)(25,32)(26,31)(27,30)(28,29)(33,35)(37,44)(38,43)(39,42)(40,41)(45,47)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,65)(62,68)(63,67)(64,66)(69,72)(70,71)(73,77)(74,80)(75,79)(76,78)(81,84)(82,83)(85,89)(86,92)(87,91)(88,90)(93,96)(94,95)>;
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), (1,72,3,70)(2,71,4,69)(5,44,7,42)(6,43,8,41)(9,40,11,38)(10,39,12,37)(13,75,15,73)(14,74,16,76)(17,77,19,79)(18,80,20,78)(21,57,23,59)(22,60,24,58)(25,63,27,61)(26,62,28,64)(29,65,31,67)(30,68,32,66)(33,93,35,95)(34,96,36,94)(45,81,47,83)(46,84,48,82)(49,90,51,92)(50,89,52,91)(53,86,55,88)(54,85,56,87), (1,46,22,34)(2,45,23,33)(3,48,24,36)(4,47,21,35)(5,76,85,67)(6,75,86,66)(7,74,87,65)(8,73,88,68)(9,78,89,63)(10,77,90,62)(11,80,91,61)(12,79,92,64)(13,56,32,44)(14,55,29,43)(15,54,30,42)(16,53,31,41)(17,52,26,40)(18,51,27,39)(19,50,28,38)(20,49,25,37)(57,94,69,82)(58,93,70,81)(59,96,71,84)(60,95,72,83), (1,17,16)(2,18,13)(3,19,14)(4,20,15)(5,95,10)(6,96,11)(7,93,12)(8,94,9)(21,25,30)(22,26,31)(23,27,32)(24,28,29)(33,39,44)(34,40,41)(35,37,42)(36,38,43)(45,51,56)(46,52,53)(47,49,54)(48,50,55)(57,63,68)(58,64,65)(59,61,66)(60,62,67)(69,78,73)(70,79,74)(71,80,75)(72,77,76)(81,92,87)(82,89,88)(83,90,85)(84,91,86), (2,4)(5,9)(6,12)(7,11)(8,10)(13,20)(14,19)(15,18)(16,17)(21,23)(25,32)(26,31)(27,30)(28,29)(33,35)(37,44)(38,43)(39,42)(40,41)(45,47)(49,56)(50,55)(51,54)(52,53)(57,60)(58,59)(61,65)(62,68)(63,67)(64,66)(69,72)(70,71)(73,77)(74,80)(75,79)(76,78)(81,84)(82,83)(85,89)(86,92)(87,91)(88,90)(93,96)(94,95) );
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)], [(1,72,3,70),(2,71,4,69),(5,44,7,42),(6,43,8,41),(9,40,11,38),(10,39,12,37),(13,75,15,73),(14,74,16,76),(17,77,19,79),(18,80,20,78),(21,57,23,59),(22,60,24,58),(25,63,27,61),(26,62,28,64),(29,65,31,67),(30,68,32,66),(33,93,35,95),(34,96,36,94),(45,81,47,83),(46,84,48,82),(49,90,51,92),(50,89,52,91),(53,86,55,88),(54,85,56,87)], [(1,46,22,34),(2,45,23,33),(3,48,24,36),(4,47,21,35),(5,76,85,67),(6,75,86,66),(7,74,87,65),(8,73,88,68),(9,78,89,63),(10,77,90,62),(11,80,91,61),(12,79,92,64),(13,56,32,44),(14,55,29,43),(15,54,30,42),(16,53,31,41),(17,52,26,40),(18,51,27,39),(19,50,28,38),(20,49,25,37),(57,94,69,82),(58,93,70,81),(59,96,71,84),(60,95,72,83)], [(1,17,16),(2,18,13),(3,19,14),(4,20,15),(5,95,10),(6,96,11),(7,93,12),(8,94,9),(21,25,30),(22,26,31),(23,27,32),(24,28,29),(33,39,44),(34,40,41),(35,37,42),(36,38,43),(45,51,56),(46,52,53),(47,49,54),(48,50,55),(57,63,68),(58,64,65),(59,61,66),(60,62,67),(69,78,73),(70,79,74),(71,80,75),(72,77,76),(81,92,87),(82,89,88),(83,90,85),(84,91,86)], [(2,4),(5,9),(6,12),(7,11),(8,10),(13,20),(14,19),(15,18),(16,17),(21,23),(25,32),(26,31),(27,30),(28,29),(33,35),(37,44),(38,43),(39,42),(40,41),(45,47),(49,56),(50,55),(51,54),(52,53),(57,60),(58,59),(61,65),(62,68),(63,67),(64,66),(69,72),(70,71),(73,77),(74,80),(75,79),(76,78),(81,84),(82,83),(85,89),(86,92),(87,91),(88,90),(93,96),(94,95)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 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 | 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 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | C8⋊C22 | C8.C22 | D4⋊2S3 | S3×D4 | Q8⋊3D6 | Q16⋊S3 |
| kernel | Q8⋊3(C4×S3) | C6.Q16 | C24⋊C4 | C2.D24 | C3×Q8⋊C4 | Dic3⋊5D4 | C2×Q8⋊2S3 | Q8×Dic3 | Q8⋊2S3 | Q8⋊C4 | C2×Dic3 | C4⋊C4 | C2×C8 | C2×Q8 | C12 | Q8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Q8⋊3(C4×S3) ►in GL8(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 72 |
| 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 |
| 38 | 70 | 0 | 0 | 0 | 0 | 0 | 0 |
| 43 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 20 | 16 | 57 |
| 0 | 0 | 0 | 0 | 2 | 0 | 57 | 0 |
| 0 | 0 | 0 | 0 | 2 | 71 | 2 | 71 |
| 0 | 0 | 0 | 0 | 16 | 18 | 71 | 55 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 46 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 71 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 |
G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,1,0,0,0,0,0,71,1,1,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0],[38,43,0,0,0,0,0,0,70,35,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,16,2,2,16,0,0,0,0,20,0,71,18,0,0,0,0,16,57,2,71,0,0,0,0,57,0,71,55],[27,27,0,0,0,0,0,0,0,46,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,71,1,1,72,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,72,0,1,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72] >;
Q8⋊3(C4×S3) in GAP, Magma, Sage, TeX
Q_8\rtimes_3(C_4\times S_3)
% in TeX
G:=Group("Q8:3(C4xS3)"); // GroupNames label
G:=SmallGroup(192,376);
// by ID
G=gap.SmallGroup(192,376);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,219,58,1684,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=c^4=d^3=e^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=e*a*e=a^-1,a*d=d*a,c*b*c^-1=a*b,b*d=d*b,e*b*e=a^-1*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations