direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C8⋊6D4, C24⋊28D4, C12⋊6M4(2), C8⋊6(C3×D4), C4⋊C8⋊16C6, (C4×C8)⋊15C6, (C4×C24)⋊31C2, C4⋊C4.8C12, (C4×D4).3C6, C4.81(C6×D4), C22⋊C8⋊14C6, (C6×D4).21C4, (C2×D4).9C12, C6.113(C4×D4), C2.11(D4×C12), C4⋊1(C3×M4(2)), (D4×C12).18C2, C6.50(C8○D4), C12.486(C2×D4), C22⋊C4.5C12, C42.69(C2×C6), (C2×M4(2))⋊15C6, (C6×M4(2))⋊33C2, C23.12(C2×C12), C6.54(C2×M4(2)), C2.10(C6×M4(2)), C12.355(C4○D4), (C4×C12).354C22, (C2×C24).328C22, (C2×C12).992C23, C22.48(C22×C12), (C22×C12).418C22, (C3×C4⋊C8)⋊35C2, C2.8(C3×C8○D4), (C3×C4⋊C4).20C4, (C2×C8).53(C2×C6), C4.53(C3×C4○D4), (C3×C22⋊C8)⋊31C2, (C2×C4).29(C2×C12), (C2×C12).213(C2×C4), (C3×C22⋊C4).12C4, (C22×C6).24(C2×C4), (C22×C4).41(C2×C6), (C2×C4).160(C22×C6), (C2×C6).242(C22×C4), SmallGroup(192,869)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊6D4
G = < a,b,c,d | a3=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 178 in 122 conjugacy classes, 74 normal (38 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, C23, C12, C12, C12, 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, C2×C12, C3×D4, C22×C6, C4×C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C2×C24, C3×M4(2), C22×C12, C6×D4, C8⋊6D4, C4×C24, C3×C22⋊C8, C3×C4⋊C8, D4×C12, C6×M4(2), C3×C8⋊6D4
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⋊6D4, D4×C12, C6×M4(2), C3×C8○D4, C3×C8⋊6D4
►On 96 points
(1 39 16)(2 40 9)(3 33 10)(4 34 11)(5 35 12)(6 36 13)(7 37 14)(8 38 15)(17 56 41)(18 49 42)(19 50 43)(20 51 44)(21 52 45)(22 53 46)(23 54 47)(24 55 48)(25 93 68)(26 94 69)(27 95 70)(28 96 71)(29 89 72)(30 90 65)(31 91 66)(32 92 67)(57 77 82)(58 78 83)(59 79 84)(60 80 85)(61 73 86)(62 74 87)(63 75 88)(64 76 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 65 19 59)(2 66 20 60)(3 67 21 61)(4 68 22 62)(5 69 23 63)(6 70 24 64)(7 71 17 57)(8 72 18 58)(9 91 44 85)(10 92 45 86)(11 93 46 87)(12 94 47 88)(13 95 48 81)(14 96 41 82)(15 89 42 83)(16 90 43 84)(25 53 74 34)(26 54 75 35)(27 55 76 36)(28 56 77 37)(29 49 78 38)(30 50 79 39)(31 51 80 40)(32 52 73 33)
(1 59)(2 64)(3 61)(4 58)(5 63)(6 60)(7 57)(8 62)(9 81)(10 86)(11 83)(12 88)(13 85)(14 82)(15 87)(16 84)(17 71)(18 68)(19 65)(20 70)(21 67)(22 72)(23 69)(24 66)(25 49)(26 54)(27 51)(28 56)(29 53)(30 50)(31 55)(32 52)(33 73)(34 78)(35 75)(36 80)(37 77)(38 74)(39 79)(40 76)(41 96)(42 93)(43 90)(44 95)(45 92)(46 89)(47 94)(48 91)
G:=sub<Sym(96)| (1,39,16)(2,40,9)(3,33,10)(4,34,11)(5,35,12)(6,36,13)(7,37,14)(8,38,15)(17,56,41)(18,49,42)(19,50,43)(20,51,44)(21,52,45)(22,53,46)(23,54,47)(24,55,48)(25,93,68)(26,94,69)(27,95,70)(28,96,71)(29,89,72)(30,90,65)(31,91,66)(32,92,67)(57,77,82)(58,78,83)(59,79,84)(60,80,85)(61,73,86)(62,74,87)(63,75,88)(64,76,81), (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), (1,65,19,59)(2,66,20,60)(3,67,21,61)(4,68,22,62)(5,69,23,63)(6,70,24,64)(7,71,17,57)(8,72,18,58)(9,91,44,85)(10,92,45,86)(11,93,46,87)(12,94,47,88)(13,95,48,81)(14,96,41,82)(15,89,42,83)(16,90,43,84)(25,53,74,34)(26,54,75,35)(27,55,76,36)(28,56,77,37)(29,49,78,38)(30,50,79,39)(31,51,80,40)(32,52,73,33), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,81)(10,86)(11,83)(12,88)(13,85)(14,82)(15,87)(16,84)(17,71)(18,68)(19,65)(20,70)(21,67)(22,72)(23,69)(24,66)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,73)(34,78)(35,75)(36,80)(37,77)(38,74)(39,79)(40,76)(41,96)(42,93)(43,90)(44,95)(45,92)(46,89)(47,94)(48,91)>;
G:=Group( (1,39,16)(2,40,9)(3,33,10)(4,34,11)(5,35,12)(6,36,13)(7,37,14)(8,38,15)(17,56,41)(18,49,42)(19,50,43)(20,51,44)(21,52,45)(22,53,46)(23,54,47)(24,55,48)(25,93,68)(26,94,69)(27,95,70)(28,96,71)(29,89,72)(30,90,65)(31,91,66)(32,92,67)(57,77,82)(58,78,83)(59,79,84)(60,80,85)(61,73,86)(62,74,87)(63,75,88)(64,76,81), (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), (1,65,19,59)(2,66,20,60)(3,67,21,61)(4,68,22,62)(5,69,23,63)(6,70,24,64)(7,71,17,57)(8,72,18,58)(9,91,44,85)(10,92,45,86)(11,93,46,87)(12,94,47,88)(13,95,48,81)(14,96,41,82)(15,89,42,83)(16,90,43,84)(25,53,74,34)(26,54,75,35)(27,55,76,36)(28,56,77,37)(29,49,78,38)(30,50,79,39)(31,51,80,40)(32,52,73,33), (1,59)(2,64)(3,61)(4,58)(5,63)(6,60)(7,57)(8,62)(9,81)(10,86)(11,83)(12,88)(13,85)(14,82)(15,87)(16,84)(17,71)(18,68)(19,65)(20,70)(21,67)(22,72)(23,69)(24,66)(25,49)(26,54)(27,51)(28,56)(29,53)(30,50)(31,55)(32,52)(33,73)(34,78)(35,75)(36,80)(37,77)(38,74)(39,79)(40,76)(41,96)(42,93)(43,90)(44,95)(45,92)(46,89)(47,94)(48,91) );
G=PermutationGroup([[(1,39,16),(2,40,9),(3,33,10),(4,34,11),(5,35,12),(6,36,13),(7,37,14),(8,38,15),(17,56,41),(18,49,42),(19,50,43),(20,51,44),(21,52,45),(22,53,46),(23,54,47),(24,55,48),(25,93,68),(26,94,69),(27,95,70),(28,96,71),(29,89,72),(30,90,65),(31,91,66),(32,92,67),(57,77,82),(58,78,83),(59,79,84),(60,80,85),(61,73,86),(62,74,87),(63,75,88),(64,76,81)], [(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)], [(1,65,19,59),(2,66,20,60),(3,67,21,61),(4,68,22,62),(5,69,23,63),(6,70,24,64),(7,71,17,57),(8,72,18,58),(9,91,44,85),(10,92,45,86),(11,93,46,87),(12,94,47,88),(13,95,48,81),(14,96,41,82),(15,89,42,83),(16,90,43,84),(25,53,74,34),(26,54,75,35),(27,55,76,36),(28,56,77,37),(29,49,78,38),(30,50,79,39),(31,51,80,40),(32,52,73,33)], [(1,59),(2,64),(3,61),(4,58),(5,63),(6,60),(7,57),(8,62),(9,81),(10,86),(11,83),(12,88),(13,85),(14,82),(15,87),(16,84),(17,71),(18,68),(19,65),(20,70),(21,67),(22,72),(23,69),(24,66),(25,49),(26,54),(27,51),(28,56),(29,53),(30,50),(31,55),(32,52),(33,73),(34,78),(35,75),(36,80),(37,77),(38,74),(39,79),(40,76),(41,96),(42,93),(43,90),(44,95),(45,92),(46,89),(47,94),(48,91)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | 12R | 12S | 12T | 24A | ··· | 24P | 24Q | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C12 | D4 | M4(2) | C4○D4 | C3×D4 | C8○D4 | C3×M4(2) | C3×C4○D4 | C3×C8○D4 |
| kernel | C3×C8⋊6D4 | C4×C24 | C3×C22⋊C8 | C3×C4⋊C8 | D4×C12 | C6×M4(2) | C8⋊6D4 | C3×C22⋊C4 | C3×C4⋊C4 | C6×D4 | C4×C8 | C22⋊C8 | C4⋊C8 | C4×D4 | C2×M4(2) | C22⋊C4 | C4⋊C4 | C2×D4 | C24 | C12 | C12 | C8 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 2 | 2 | 4 | 8 | 4 | 4 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C3×C8⋊6D4 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 0 | 36 | 0 | 0 |
| 54 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,8,0,0,0,0,8],[0,54,0,0,36,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[1,0,0,0,0,72,0,0,0,0,0,1,0,0,1,0] >;
C3×C8⋊6D4 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes_6D_4
% in TeX
G:=Group("C3xC8:6D4"); // GroupNames label
G:=SmallGroup(192,869);
// by ID
G=gap.SmallGroup(192,869);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,2102,1059,268,124]);
// 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,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations