direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8⋊9D4, C24⋊35D4, C8⋊9(C3×D4), C4⋊C8⋊15C6, C8⋊C4⋊9C6, C4⋊C4.7C12, (C4×D4).2C6, C4.80(C6×D4), C22⋊C8⋊13C6, (C22×C8)⋊15C6, (C6×D4).20C4, (C2×D4).8C12, C2.10(D4×C12), C6.112(C4×D4), (C2×C6)⋊4M4(2), C42.7(C2×C6), (C22×C24)⋊25C2, (D4×C12).17C2, C6.49(C8○D4), C12.485(C2×D4), C22⋊C4.4C12, C2.9(C6×M4(2)), (C6×M4(2))⋊32C2, (C2×M4(2))⋊14C6, C23.24(C2×C12), C6.53(C2×M4(2)), C22⋊2(C3×M4(2)), C12.354(C4○D4), (C2×C24).446C22, (C2×C12).991C23, (C4×C12).248C22, C22.47(C22×C12), (C22×C12).417C22, (C3×C4⋊C8)⋊34C2, C2.7(C3×C8○D4), (C3×C4⋊C4).19C4, (C2×C8).52(C2×C6), (C3×C8⋊C4)⋊23C2, C4.52(C3×C4○D4), (C3×C22⋊C8)⋊30C2, (C2×C4).28(C2×C12), (C2×C12).201(C2×C4), (C3×C22⋊C4).11C4, (C22×C6).23(C2×C4), (C22×C4).103(C2×C6), (C2×C6).241(C22×C4), (C2×C4).159(C22×C6), SmallGroup(192,868)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊9D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b5, dcd=c-1 >
Subgroups: 178 in 124 conjugacy classes, 74 normal (66 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C8⋊C4, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C22×C12, C6×D4, C8⋊9D4, C3×C8⋊C4, C3×C22⋊C8, C3×C4⋊C8, D4×C12, C22×C24, C6×M4(2), C3×C8⋊9D4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, M4(2), C22×C4, C2×D4, C4○D4, C2×C12, C3×D4, C22×C6, C4×D4, C2×M4(2), C8○D4, C3×M4(2), C22×C12, C6×D4, C3×C4○D4, C8⋊9D4, D4×C12, C6×M4(2), C3×C8○D4, C3×C8⋊9D4
►On 96 points
(1 16 33)(2 9 34)(3 10 35)(4 11 36)(5 12 37)(6 13 38)(7 14 39)(8 15 40)(17 49 41)(18 50 42)(19 51 43)(20 52 44)(21 53 45)(22 54 46)(23 55 47)(24 56 48)(25 93 69)(26 94 70)(27 95 71)(28 96 72)(29 89 65)(30 90 66)(31 91 67)(32 92 68)(57 78 82)(58 79 83)(59 80 84)(60 73 85)(61 74 86)(62 75 87)(63 76 88)(64 77 81)
(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 81 23 71)(2 86 24 68)(3 83 17 65)(4 88 18 70)(5 85 19 67)(6 82 20 72)(7 87 21 69)(8 84 22 66)(9 61 56 32)(10 58 49 29)(11 63 50 26)(12 60 51 31)(13 57 52 28)(14 62 53 25)(15 59 54 30)(16 64 55 27)(33 77 47 95)(34 74 48 92)(35 79 41 89)(36 76 42 94)(37 73 43 91)(38 78 44 96)(39 75 45 93)(40 80 46 90)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(65 83)(66 88)(67 85)(68 82)(69 87)(70 84)(71 81)(72 86)(73 91)(74 96)(75 93)(76 90)(77 95)(78 92)(79 89)(80 94)
G:=sub<Sym(96)| (1,16,33)(2,9,34)(3,10,35)(4,11,36)(5,12,37)(6,13,38)(7,14,39)(8,15,40)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,93,69)(26,94,70)(27,95,71)(28,96,72)(29,89,65)(30,90,66)(31,91,67)(32,92,68)(57,78,82)(58,79,83)(59,80,84)(60,73,85)(61,74,86)(62,75,87)(63,76,88)(64,77,81), (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), (1,81,23,71)(2,86,24,68)(3,83,17,65)(4,88,18,70)(5,85,19,67)(6,82,20,72)(7,87,21,69)(8,84,22,66)(9,61,56,32)(10,58,49,29)(11,63,50,26)(12,60,51,31)(13,57,52,28)(14,62,53,25)(15,59,54,30)(16,64,55,27)(33,77,47,95)(34,74,48,92)(35,79,41,89)(36,76,42,94)(37,73,43,91)(38,78,44,96)(39,75,45,93)(40,80,46,90), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(65,83)(66,88)(67,85)(68,82)(69,87)(70,84)(71,81)(72,86)(73,91)(74,96)(75,93)(76,90)(77,95)(78,92)(79,89)(80,94)>;
G:=Group( (1,16,33)(2,9,34)(3,10,35)(4,11,36)(5,12,37)(6,13,38)(7,14,39)(8,15,40)(17,49,41)(18,50,42)(19,51,43)(20,52,44)(21,53,45)(22,54,46)(23,55,47)(24,56,48)(25,93,69)(26,94,70)(27,95,71)(28,96,72)(29,89,65)(30,90,66)(31,91,67)(32,92,68)(57,78,82)(58,79,83)(59,80,84)(60,73,85)(61,74,86)(62,75,87)(63,76,88)(64,77,81), (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), (1,81,23,71)(2,86,24,68)(3,83,17,65)(4,88,18,70)(5,85,19,67)(6,82,20,72)(7,87,21,69)(8,84,22,66)(9,61,56,32)(10,58,49,29)(11,63,50,26)(12,60,51,31)(13,57,52,28)(14,62,53,25)(15,59,54,30)(16,64,55,27)(33,77,47,95)(34,74,48,92)(35,79,41,89)(36,76,42,94)(37,73,43,91)(38,78,44,96)(39,75,45,93)(40,80,46,90), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(65,83)(66,88)(67,85)(68,82)(69,87)(70,84)(71,81)(72,86)(73,91)(74,96)(75,93)(76,90)(77,95)(78,92)(79,89)(80,94) );
G=PermutationGroup([[(1,16,33),(2,9,34),(3,10,35),(4,11,36),(5,12,37),(6,13,38),(7,14,39),(8,15,40),(17,49,41),(18,50,42),(19,51,43),(20,52,44),(21,53,45),(22,54,46),(23,55,47),(24,56,48),(25,93,69),(26,94,70),(27,95,71),(28,96,72),(29,89,65),(30,90,66),(31,91,67),(32,92,68),(57,78,82),(58,79,83),(59,80,84),(60,73,85),(61,74,86),(62,75,87),(63,76,88),(64,77,81)], [(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)], [(1,81,23,71),(2,86,24,68),(3,83,17,65),(4,88,18,70),(5,85,19,67),(6,82,20,72),(7,87,21,69),(8,84,22,66),(9,61,56,32),(10,58,49,29),(11,63,50,26),(12,60,51,31),(13,57,52,28),(14,62,53,25),(15,59,54,30),(16,64,55,27),(33,77,47,95),(34,74,48,92),(35,79,41,89),(36,76,42,94),(37,73,43,91),(38,78,44,96),(39,75,45,93),(40,80,46,90)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,62),(26,59),(27,64),(28,61),(29,58),(30,63),(31,60),(32,57),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(65,83),(66,88),(67,85),(68,82),(69,87),(70,84),(71,81),(72,86),(73,91),(74,96),(75,93),(76,90),(77,95),(78,92),(79,89),(80,94)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12R | 24A | ··· | 24P | 24Q | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | C4○D4 | M4(2) | C3×D4 | C8○D4 | C3×C4○D4 | C3×M4(2) | C3×C8○D4 |
| kernel | C3×C8⋊9D4 | C3×C8⋊C4 | C3×C22⋊C8 | C3×C4⋊C8 | D4×C12 | C22×C24 | C6×M4(2) | C8⋊9D4 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C4×D4 | C22×C8 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C24 | C12 | C2×C6 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C8⋊9D4 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 46 | 71 | 0 | 0 |
| 59 | 27 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 46 | 72 | 0 | 0 |
| 0 | 0 | 72 | 71 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 46 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 72 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[46,59,0,0,71,27,0,0,0,0,1,0,0,0,0,1],[1,46,0,0,0,72,0,0,0,0,72,1,0,0,71,1],[1,46,0,0,0,72,0,0,0,0,1,72,0,0,0,72] >;
C3×C8⋊9D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_9D_4
% in TeX
G:=Group("C3xC8:9D4"); // GroupNames label
G:=SmallGroup(192,868);
// by ID
G=gap.SmallGroup(192,868);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,268,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^5,d*c*d=c^-1>;
// generators/relations