metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊C8⋊22D4, (C2×C6)⋊3D8, C4⋊D4⋊3S3, C3⋊3(C8⋊7D4), C4⋊C4.58D6, C6.55(C2×D8), (C2×D4).38D6, C4.170(S3×D4), C12⋊7D4⋊23C2, C22⋊2(D4⋊S3), C6.97(C4○D8), C12.147(C2×D4), C6.D8⋊35C2, C6.Q16⋊36C2, (C2×C12).263D4, (C22×C6).84D4, D4⋊Dic3⋊15C2, C6.93(C4⋊D4), (C6×D4).54C22, (C22×C4).356D6, C12.183(C4○D4), C4.59(D4⋊2S3), (C2×C12).357C23, (C2×D12).97C22, C23.46(C3⋊D4), C2.16(Q8.13D6), C4⋊Dic3.142C22, C2.14(C23.14D6), (C22×C12).161C22, (C22×C3⋊C8)⋊3C2, (C2×D4⋊S3)⋊10C2, (C3×C4⋊D4)⋊3C2, C2.10(C2×D4⋊S3), (C2×C6).488(C2×D4), (C2×C3⋊C8).248C22, (C2×C4).105(C3⋊D4), (C3×C4⋊C4).105C22, (C2×C4).457(C22×S3), C22.163(C2×C3⋊D4), SmallGroup(192,597)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊C8⋊22D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd=c-1 >
Subgroups: 416 in 134 conjugacy classes, 45 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C2×D4, C3⋊C8, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, D4⋊C4, C2.D8, C4⋊D4, C4⋊D4, C22×C8, C2×D8, C2×C3⋊C8, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, D4⋊S3, C3×C22⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C8⋊7D4, C6.Q16, C6.D8, D4⋊Dic3, C22×C3⋊C8, C12⋊7D4, C2×D4⋊S3, C3×C4⋊D4, C3⋊C8⋊22D4
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×D8, C4○D8, D4⋊S3, S3×D4, D4⋊2S3, C2×C3⋊D4, C8⋊7D4, C2×D4⋊S3, C23.14D6, Q8.13D6, C3⋊C8⋊22D4
►On 96 points
(1 81 61)(2 62 82)(3 83 63)(4 64 84)(5 85 57)(6 58 86)(7 87 59)(8 60 88)(9 35 52)(10 53 36)(11 37 54)(12 55 38)(13 39 56)(14 49 40)(15 33 50)(16 51 34)(17 76 68)(18 69 77)(19 78 70)(20 71 79)(21 80 72)(22 65 73)(23 74 66)(24 67 75)(25 95 42)(26 43 96)(27 89 44)(28 45 90)(29 91 46)(30 47 92)(31 93 48)(32 41 94)
(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 91 70 11)(2 90 71 10)(3 89 72 9)(4 96 65 16)(5 95 66 15)(6 94 67 14)(7 93 68 13)(8 92 69 12)(17 39 87 48)(18 38 88 47)(19 37 81 46)(20 36 82 45)(21 35 83 44)(22 34 84 43)(23 33 85 42)(24 40 86 41)(25 74 50 57)(26 73 51 64)(27 80 52 63)(28 79 53 62)(29 78 54 61)(30 77 55 60)(31 76 56 59)(32 75 49 58)
(1 90)(2 91)(3 92)(4 93)(5 94)(6 95)(7 96)(8 89)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 86)(26 87)(27 88)(28 81)(29 82)(30 83)(31 84)(32 85)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
G:=sub<Sym(96)| (1,81,61)(2,62,82)(3,83,63)(4,64,84)(5,85,57)(6,58,86)(7,87,59)(8,60,88)(9,35,52)(10,53,36)(11,37,54)(12,55,38)(13,39,56)(14,49,40)(15,33,50)(16,51,34)(17,76,68)(18,69,77)(19,78,70)(20,71,79)(21,80,72)(22,65,73)(23,74,66)(24,67,75)(25,95,42)(26,43,96)(27,89,44)(28,45,90)(29,91,46)(30,47,92)(31,93,48)(32,41,94), (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), (1,91,70,11)(2,90,71,10)(3,89,72,9)(4,96,65,16)(5,95,66,15)(6,94,67,14)(7,93,68,13)(8,92,69,12)(17,39,87,48)(18,38,88,47)(19,37,81,46)(20,36,82,45)(21,35,83,44)(22,34,84,43)(23,33,85,42)(24,40,86,41)(25,74,50,57)(26,73,51,64)(27,80,52,63)(28,79,53,62)(29,78,54,61)(30,77,55,60)(31,76,56,59)(32,75,49,58), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64)>;
G:=Group( (1,81,61)(2,62,82)(3,83,63)(4,64,84)(5,85,57)(6,58,86)(7,87,59)(8,60,88)(9,35,52)(10,53,36)(11,37,54)(12,55,38)(13,39,56)(14,49,40)(15,33,50)(16,51,34)(17,76,68)(18,69,77)(19,78,70)(20,71,79)(21,80,72)(22,65,73)(23,74,66)(24,67,75)(25,95,42)(26,43,96)(27,89,44)(28,45,90)(29,91,46)(30,47,92)(31,93,48)(32,41,94), (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), (1,91,70,11)(2,90,71,10)(3,89,72,9)(4,96,65,16)(5,95,66,15)(6,94,67,14)(7,93,68,13)(8,92,69,12)(17,39,87,48)(18,38,88,47)(19,37,81,46)(20,36,82,45)(21,35,83,44)(22,34,84,43)(23,33,85,42)(24,40,86,41)(25,74,50,57)(26,73,51,64)(27,80,52,63)(28,79,53,62)(29,78,54,61)(30,77,55,60)(31,76,56,59)(32,75,49,58), (1,90)(2,91)(3,92)(4,93)(5,94)(6,95)(7,96)(8,89)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,86)(26,87)(27,88)(28,81)(29,82)(30,83)(31,84)(32,85)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64) );
G=PermutationGroup([[(1,81,61),(2,62,82),(3,83,63),(4,64,84),(5,85,57),(6,58,86),(7,87,59),(8,60,88),(9,35,52),(10,53,36),(11,37,54),(12,55,38),(13,39,56),(14,49,40),(15,33,50),(16,51,34),(17,76,68),(18,69,77),(19,78,70),(20,71,79),(21,80,72),(22,65,73),(23,74,66),(24,67,75),(25,95,42),(26,43,96),(27,89,44),(28,45,90),(29,91,46),(30,47,92),(31,93,48),(32,41,94)], [(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)], [(1,91,70,11),(2,90,71,10),(3,89,72,9),(4,96,65,16),(5,95,66,15),(6,94,67,14),(7,93,68,13),(8,92,69,12),(17,39,87,48),(18,38,88,47),(19,37,81,46),(20,36,82,45),(21,35,83,44),(22,34,84,43),(23,33,85,42),(24,40,86,41),(25,74,50,57),(26,73,51,64),(27,80,52,63),(28,79,53,62),(29,78,54,61),(30,77,55,60),(31,76,56,59),(32,75,49,58)], [(1,90),(2,91),(3,92),(4,93),(5,94),(6,95),(7,96),(8,89),(9,69),(10,70),(11,71),(12,72),(13,65),(14,66),(15,67),(16,68),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,86),(26,87),(27,88),(28,81),(29,82),(30,83),(31,84),(32,85),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 24 | 2 | 2 | 2 | 2 | 2 | 8 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 6 | ··· | 6 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | D8 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D4 | D4⋊2S3 | D4⋊S3 | Q8.13D6 |
| kernel | C3⋊C8⋊22D4 | C6.Q16 | C6.D8 | D4⋊Dic3 | C22×C3⋊C8 | C12⋊7D4 | C2×D4⋊S3 | C3×C4⋊D4 | C4⋊D4 | C3⋊C8 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C2×D4 | C12 | C2×C6 | C2×C4 | C23 | C6 | C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3⋊C8⋊22D4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 13 | 30 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 57 | 0 | 0 |
| 0 | 0 | 16 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 30 | 13 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 6 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 46 |
| 0 | 0 | 0 | 0 | 27 | 0 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[13,43,0,0,0,0,30,60,0,0,0,0,0,0,16,16,0,0,0,0,57,16,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[30,60,0,0,0,0,13,43,0,0,0,0,0,0,67,6,0,0,0,0,6,6,0,0,0,0,0,0,27,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,46,0] >;
C3⋊C8⋊22D4 in GAP, Magma, Sage, TeX
C_3\rtimes C_8\rtimes_{22}D_4 % in TeX
G:=Group("C3:C8:22D4"); // GroupNames label
G:=SmallGroup(192,597);
// by ID
G=gap.SmallGroup(192,597);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations