metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3⋊C8⋊26D4, D6⋊C8⋊26C2, C3⋊3(C8⋊9D4), C6.29(C4×D4), C22⋊C8⋊15S3, D6⋊C4.10C4, C24⋊C4⋊14C2, (C2×C8).164D6, C4.199(S3×D4), (C2×C6)⋊2M4(2), Dic3⋊C8⋊26C2, C6.25(C8○D4), C12.358(C2×D4), C23.29(C4×S3), C6.6(C2×M4(2)), C22⋊2(C8⋊S3), Dic3⋊C4.10C4, C6.D4.7C4, (C22×C4).321D6, C12.300(C4○D4), (C2×C12).825C23, (C2×C24).217C22, C2.11(D12.C4), C4.126(D4⋊2S3), C2.13(Dic3⋊4D4), (C22×C12).339C22, (C4×Dic3).183C22, (C22×C3⋊C8)⋊17C2, (C2×C4).64(C4×S3), (C2×C3⋊D4).7C4, (C2×C8⋊S3)⋊14C2, (C3×C22⋊C8)⋊24C2, C2.10(C2×C8⋊S3), (C4×C3⋊D4).14C2, C22.107(S3×C2×C4), (C2×C12).155(C2×C4), (C2×C3⋊C8).302C22, (S3×C2×C4).180C22, (C22×C6).43(C2×C4), (C2×C6).80(C22×C4), (C22×S3).14(C2×C4), (C2×C4).767(C22×S3), (C2×Dic3).18(C2×C4), SmallGroup(192,289)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3⋊C8⋊26D4
G = < a,b,c,d | a3=b8=c4=d2=1, bab-1=cac-1=a-1, ad=da, cbc-1=b5, bd=db, dcd=c-1 >
Subgroups: 288 in 124 conjugacy classes, 53 normal (47 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, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C22×C8, C2×M4(2), C8⋊S3, C2×C3⋊C8, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8⋊9D4, Dic3⋊C8, C24⋊C4, D6⋊C8, C3×C22⋊C8, C2×C8⋊S3, C22×C3⋊C8, C4×C3⋊D4, C3⋊C8⋊26D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×M4(2), C8○D4, C8⋊S3, S3×C2×C4, S3×D4, D4⋊2S3, C8⋊9D4, Dic3⋊4D4, C2×C8⋊S3, D12.C4, C3⋊C8⋊26D4
►On 96 points
(1 52 70)(2 71 53)(3 54 72)(4 65 55)(5 56 66)(6 67 49)(7 50 68)(8 69 51)(9 44 89)(10 90 45)(11 46 91)(12 92 47)(13 48 93)(14 94 41)(15 42 95)(16 96 43)(17 85 80)(18 73 86)(19 87 74)(20 75 88)(21 81 76)(22 77 82)(23 83 78)(24 79 84)(25 39 57)(26 58 40)(27 33 59)(28 60 34)(29 35 61)(30 62 36)(31 37 63)(32 64 38)
(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 88 30)(2 95 81 27)(3 92 82 32)(4 89 83 29)(5 94 84 26)(6 91 85 31)(7 96 86 28)(8 93 87 25)(9 23 35 55)(10 20 36 52)(11 17 37 49)(12 22 38 54)(13 19 39 51)(14 24 40 56)(15 21 33 53)(16 18 34 50)(41 79 58 66)(42 76 59 71)(43 73 60 68)(44 78 61 65)(45 75 62 70)(46 80 63 67)(47 77 64 72)(48 74 57 69)
(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 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 81)(32 82)(33 80)(34 73)(35 74)(36 75)(37 76)(38 77)(39 78)(40 79)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)
G:=sub<Sym(96)| (1,52,70)(2,71,53)(3,54,72)(4,65,55)(5,56,66)(6,67,49)(7,50,68)(8,69,51)(9,44,89)(10,90,45)(11,46,91)(12,92,47)(13,48,93)(14,94,41)(15,42,95)(16,96,43)(17,85,80)(18,73,86)(19,87,74)(20,75,88)(21,81,76)(22,77,82)(23,83,78)(24,79,84)(25,39,57)(26,58,40)(27,33,59)(28,60,34)(29,35,61)(30,62,36)(31,37,63)(32,64,38), (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), (1,90,88,30)(2,95,81,27)(3,92,82,32)(4,89,83,29)(5,94,84,26)(6,91,85,31)(7,96,86,28)(8,93,87,25)(9,23,35,55)(10,20,36,52)(11,17,37,49)(12,22,38,54)(13,19,39,51)(14,24,40,56)(15,21,33,53)(16,18,34,50)(41,79,58,66)(42,76,59,71)(43,73,60,68)(44,78,61,65)(45,75,62,70)(46,80,63,67)(47,77,64,72)(48,74,57,69), (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)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)>;
G:=Group( (1,52,70)(2,71,53)(3,54,72)(4,65,55)(5,56,66)(6,67,49)(7,50,68)(8,69,51)(9,44,89)(10,90,45)(11,46,91)(12,92,47)(13,48,93)(14,94,41)(15,42,95)(16,96,43)(17,85,80)(18,73,86)(19,87,74)(20,75,88)(21,81,76)(22,77,82)(23,83,78)(24,79,84)(25,39,57)(26,58,40)(27,33,59)(28,60,34)(29,35,61)(30,62,36)(31,37,63)(32,64,38), (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), (1,90,88,30)(2,95,81,27)(3,92,82,32)(4,89,83,29)(5,94,84,26)(6,91,85,31)(7,96,86,28)(8,93,87,25)(9,23,35,55)(10,20,36,52)(11,17,37,49)(12,22,38,54)(13,19,39,51)(14,24,40,56)(15,21,33,53)(16,18,34,50)(41,79,58,66)(42,76,59,71)(43,73,60,68)(44,78,61,65)(45,75,62,70)(46,80,63,67)(47,77,64,72)(48,74,57,69), (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)(17,59)(18,60)(19,61)(20,62)(21,63)(22,64)(23,57)(24,58)(25,83)(26,84)(27,85)(28,86)(29,87)(30,88)(31,81)(32,82)(33,80)(34,73)(35,74)(36,75)(37,76)(38,77)(39,78)(40,79)(41,56)(42,49)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55) );
G=PermutationGroup([[(1,52,70),(2,71,53),(3,54,72),(4,65,55),(5,56,66),(6,67,49),(7,50,68),(8,69,51),(9,44,89),(10,90,45),(11,46,91),(12,92,47),(13,48,93),(14,94,41),(15,42,95),(16,96,43),(17,85,80),(18,73,86),(19,87,74),(20,75,88),(21,81,76),(22,77,82),(23,83,78),(24,79,84),(25,39,57),(26,58,40),(27,33,59),(28,60,34),(29,35,61),(30,62,36),(31,37,63),(32,64,38)], [(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)], [(1,90,88,30),(2,95,81,27),(3,92,82,32),(4,89,83,29),(5,94,84,26),(6,91,85,31),(7,96,86,28),(8,93,87,25),(9,23,35,55),(10,20,36,52),(11,17,37,49),(12,22,38,54),(13,19,39,51),(14,24,40,56),(15,21,33,53),(16,18,34,50),(41,79,58,66),(42,76,59,71),(43,73,60,68),(44,78,61,65),(45,75,62,70),(46,80,63,67),(47,77,64,72),(48,74,57,69)], [(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),(17,59),(18,60),(19,61),(20,62),(21,63),(22,64),(23,57),(24,58),(25,83),(26,84),(27,85),(28,86),(29,87),(30,88),(31,81),(32,82),(33,80),(34,73),(35,74),(36,75),(37,76),(38,77),(39,78),(40,79),(41,56),(42,49),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4○D4 | M4(2) | C4×S3 | C4×S3 | C8○D4 | C8⋊S3 | S3×D4 | D4⋊2S3 | D12.C4 |
| kernel | C3⋊C8⋊26D4 | Dic3⋊C8 | C24⋊C4 | D6⋊C8 | C3×C22⋊C8 | C2×C8⋊S3 | C22×C3⋊C8 | C4×C3⋊D4 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22⋊C8 | C3⋊C8 | C2×C8 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C6 | C22 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of C3⋊C8⋊26D4 ►in GL4(𝔽73) generated by
| 0 | 72 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 16 | 65 | 0 | 0 |
| 8 | 57 | 0 | 0 |
| 0 | 0 | 69 | 67 |
| 0 | 0 | 59 | 4 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 58 | 14 |
| 0 | 0 | 36 | 15 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 15 | 59 |
| 0 | 0 | 16 | 58 |
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[16,8,0,0,65,57,0,0,0,0,69,59,0,0,67,4],[0,72,0,0,72,0,0,0,0,0,58,36,0,0,14,15],[72,0,0,0,0,72,0,0,0,0,15,16,0,0,59,58] >;
C3⋊C8⋊26D4 in GAP, Magma, Sage, TeX
C_3\rtimes C_8\rtimes_{26}D_4 % in TeX
G:=Group("C3:C8:26D4"); // GroupNames label
G:=SmallGroup(192,289);
// by ID
G=gap.SmallGroup(192,289);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^4=d^2=1,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=b^5,b*d=d*b,d*c*d=c^-1>;
// generators/relations