direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C8○D12, C12.67C24, C24.79C23, (C2×C8)⋊37D6, C6⋊1(C8○D4), (C22×C8)⋊17S3, C4○D12.7C4, C3⋊C8.31C23, (S3×C8)⋊19C22, (C22×C24)⋊22C2, (C2×C24)⋊51C22, (C2×D12).19C4, D12.30(C2×C4), C23.44(C4×S3), C8.65(C22×S3), C6.30(C23×C4), C4.66(S3×C23), C8⋊S3⋊21C22, (C4×S3).34C23, Dic6.32(C2×C4), (C2×Dic6).19C4, D6.12(C22×C4), (C22×C4).457D6, C12.121(C22×C4), (C2×C12).880C23, C4○D12.58C22, C4.Dic3⋊39C22, Dic3.12(C22×C4), (C22×C12).544C22, C3⋊1(C2×C8○D4), (S3×C2×C8)⋊25C2, C4.121(S3×C2×C4), C3⋊D4.5(C2×C4), (C2×C8⋊S3)⋊29C2, C22.11(S3×C2×C4), C2.31(S3×C22×C4), (C4×S3).23(C2×C4), (C2×C4).119(C4×S3), (C2×C3⋊D4).17C4, (C2×C12).236(C2×C4), (C2×C4○D12).27C2, (C2×C3⋊C8).327C22, (S3×C2×C4).302C22, (C2×C4.Dic3)⋊33C2, (C22×S3).45(C2×C4), (C22×C6).103(C2×C4), (C2×C4).824(C22×S3), (C2×C6).156(C22×C4), (C2×Dic3).71(C2×C4), SmallGroup(192,1297)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 504 in 266 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×4], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊C8 [×4], C24 [×4], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×4], C22×S3 [×2], C22×C6, C22×C8, C22×C8 [×2], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, S3×C8 [×8], C8⋊S3 [×8], C2×C3⋊C8 [×2], C4.Dic3 [×4], C2×C24 [×2], C2×C24 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C2×C8○D4, S3×C2×C8 [×2], C2×C8⋊S3 [×2], C8○D12 [×8], C2×C4.Dic3, C22×C24, C2×C4○D12, C2×C8○D12
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C8○D4 [×2], C23×C4, S3×C2×C4 [×6], S3×C23, C2×C8○D4, C8○D12 [×2], S3×C22×C4, C2×C8○D12
Generators and relations
G = < a,b,c,d | a2=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >
►On 96 points
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 81)(14 82)(15 83)(16 84)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 55)(38 56)(39 57)(40 58)(41 59)(42 60)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)
(1 84 47 61 7 78 41 67)(2 73 48 62 8 79 42 68)(3 74 37 63 9 80 43 69)(4 75 38 64 10 81 44 70)(5 76 39 65 11 82 45 71)(6 77 40 66 12 83 46 72)(13 50 27 88 19 56 33 94)(14 51 28 89 20 57 34 95)(15 52 29 90 21 58 35 96)(16 53 30 91 22 59 36 85)(17 54 31 92 23 60 25 86)(18 55 32 93 24 49 26 87)
(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 90)(2 89)(3 88)(4 87)(5 86)(6 85)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 84)(22 83)(23 82)(24 81)(25 71)(26 70)(27 69)(28 68)(29 67)(30 66)(31 65)(32 64)(33 63)(34 62)(35 61)(36 72)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)
G:=sub<Sym(96)| (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,84,47,61,7,78,41,67)(2,73,48,62,8,79,42,68)(3,74,37,63,9,80,43,69)(4,75,38,64,10,81,44,70)(5,76,39,65,11,82,45,71)(6,77,40,66,12,83,46,72)(13,50,27,88,19,56,33,94)(14,51,28,89,20,57,34,95)(15,52,29,90,21,58,35,96)(16,53,30,91,22,59,36,85)(17,54,31,92,23,60,25,86)(18,55,32,93,24,49,26,87), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,72)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)>;
G:=Group( (1,85)(2,86)(3,87)(4,88)(5,89)(6,90)(7,91)(8,92)(9,93)(10,94)(11,95)(12,96)(13,81)(14,82)(15,83)(16,84)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,68)(26,69)(27,70)(28,71)(29,72)(30,61)(31,62)(32,63)(33,64)(34,65)(35,66)(36,67)(37,55)(38,56)(39,57)(40,58)(41,59)(42,60)(43,49)(44,50)(45,51)(46,52)(47,53)(48,54), (1,84,47,61,7,78,41,67)(2,73,48,62,8,79,42,68)(3,74,37,63,9,80,43,69)(4,75,38,64,10,81,44,70)(5,76,39,65,11,82,45,71)(6,77,40,66,12,83,46,72)(13,50,27,88,19,56,33,94)(14,51,28,89,20,57,34,95)(15,52,29,90,21,58,35,96)(16,53,30,91,22,59,36,85)(17,54,31,92,23,60,25,86)(18,55,32,93,24,49,26,87), (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), (1,90)(2,89)(3,88)(4,87)(5,86)(6,85)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,84)(22,83)(23,82)(24,81)(25,71)(26,70)(27,69)(28,68)(29,67)(30,66)(31,65)(32,64)(33,63)(34,62)(35,61)(36,72)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57) );
G=PermutationGroup([(1,85),(2,86),(3,87),(4,88),(5,89),(6,90),(7,91),(8,92),(9,93),(10,94),(11,95),(12,96),(13,81),(14,82),(15,83),(16,84),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,68),(26,69),(27,70),(28,71),(29,72),(30,61),(31,62),(32,63),(33,64),(34,65),(35,66),(36,67),(37,55),(38,56),(39,57),(40,58),(41,59),(42,60),(43,49),(44,50),(45,51),(46,52),(47,53),(48,54)], [(1,84,47,61,7,78,41,67),(2,73,48,62,8,79,42,68),(3,74,37,63,9,80,43,69),(4,75,38,64,10,81,44,70),(5,76,39,65,11,82,45,71),(6,77,40,66,12,83,46,72),(13,50,27,88,19,56,33,94),(14,51,28,89,20,57,34,95),(15,52,29,90,21,58,35,96),(16,53,30,91,22,59,36,85),(17,54,31,92,23,60,25,86),(18,55,32,93,24,49,26,87)], [(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)], [(1,90),(2,89),(3,88),(4,87),(5,86),(6,85),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,84),(22,83),(23,82),(24,81),(25,71),(26,70),(27,69),(28,68),(29,67),(30,66),(31,65),(32,64),(33,63),(34,62),(35,61),(36,72),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57)])
Matrix representation ►G ⊆ 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 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 |
| 0 | 0 | 0 | 0 | 51 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 62 | 8 |
| 0 | 0 | 0 | 3 | 11 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 11 | 65 |
| 0 | 0 | 0 | 15 | 62 |
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],[27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,51,0,0,0,0,0,51],[72,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,62,3,0,0,0,8,11],[72,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,11,15,0,0,0,65,62] >;
72 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 | ··· | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 8M | ··· | 8T | 12A | ··· | 12H | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D6 | D6 | C4×S3 | C4×S3 | C8○D4 | C8○D12 |
| kernel | C2×C8○D12 | S3×C2×C8 | C2×C8⋊S3 | C8○D12 | C2×C4.Dic3 | C22×C24 | C2×C4○D12 | C2×Dic6 | C2×D12 | C4○D12 | C2×C3⋊D4 | C22×C8 | C2×C8 | C22×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 2 | 2 | 8 | 1 | 1 | 1 | 2 | 2 | 8 | 4 | 1 | 6 | 1 | 6 | 2 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2\times C_8\circ D_{12} % in TeX
G:=Group("C2xC8oD12"); // GroupNames label
G:=SmallGroup(192,1297);
// by ID
G=gap.SmallGroup(192,1297);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,80,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^6=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^5>;
// generators/relations