metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊C8⋊40C2, C4.89(C2×D12), (C2×C8).189D6, C4.12(D6⋊C4), (C2×D12).14C4, C6.34(C8○D4), C12.444(C2×D4), (C2×C4).152D12, (C2×C12).171D4, (C2×M4(2))⋊9S3, C23.34(C4×S3), (C6×M4(2))⋊17C2, C22.2(D6⋊C4), (C2×Dic6).14C4, (C22×C4).366D6, C12.26(C22⋊C4), (C2×C12).869C23, C2.19(D12.C4), (C2×C24).319C22, (C22×C12).186C22, (C22×C3⋊C8)⋊6C2, (C2×C4).84(C4×S3), C2.28(C2×D6⋊C4), (C2×C3⋊D4).12C4, C4.135(C2×C3⋊D4), C6.56(C2×C22⋊C4), C22.148(S3×C2×C4), (C2×C12).106(C2×C4), C3⋊3((C22×C8)⋊C2), (C2×C4○D12).11C2, (C2×C3⋊C8).325C22, (S3×C2×C4).186C22, (C22×C6).69(C2×C4), (C2×C4).141(C3⋊D4), (C2×C6).20(C22⋊C4), (C22×S3).26(C2×C4), (C2×C6).139(C22×C4), (C2×C4).811(C22×S3), (C2×Dic3).35(C2×C4), SmallGroup(192,688)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊C8⋊40C2
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a3b, dbd=bc4, dcd=c5 >
Subgroups: 408 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, C22×C8, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C2×C3⋊C8, C2×C24, C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, (C22×C8)⋊C2, D6⋊C8, C22×C3⋊C8, C6×M4(2), C2×C4○D12, D6⋊C8⋊40C2
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C8○D4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, (C22×C8)⋊C2, D12.C4, C2×D6⋊C4, D6⋊C8⋊40C2
►On 96 points
(1 88 51 31 76 14)(2 81 52 32 77 15)(3 82 53 25 78 16)(4 83 54 26 79 9)(5 84 55 27 80 10)(6 85 56 28 73 11)(7 86 49 29 74 12)(8 87 50 30 75 13)(17 89 34 59 44 71)(18 90 35 60 45 72)(19 91 36 61 46 65)(20 92 37 62 47 66)(21 93 38 63 48 67)(22 94 39 64 41 68)(23 95 40 57 42 69)(24 96 33 58 43 70)
(1 66)(2 38)(3 68)(4 40)(5 70)(6 34)(7 72)(8 36)(9 57)(10 24)(11 59)(12 18)(13 61)(14 20)(15 63)(16 22)(17 56)(19 50)(21 52)(23 54)(25 39)(26 69)(27 33)(28 71)(29 35)(30 65)(31 37)(32 67)(41 82)(42 79)(43 84)(44 73)(45 86)(46 75)(47 88)(48 77)(49 60)(51 62)(53 64)(55 58)(74 90)(76 92)(78 94)(80 96)(81 93)(83 95)(85 89)(87 91)
(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 18)(2 23)(3 20)(4 17)(5 22)(6 19)(7 24)(8 21)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(25 62)(26 59)(27 64)(28 61)(29 58)(30 63)(31 60)(32 57)(33 49)(34 54)(35 51)(36 56)(37 53)(38 50)(39 55)(40 52)(41 80)(42 77)(43 74)(44 79)(45 76)(46 73)(47 78)(48 75)(81 95)(82 92)(83 89)(84 94)(85 91)(86 96)(87 93)(88 90)
G:=sub<Sym(96)| (1,88,51,31,76,14)(2,81,52,32,77,15)(3,82,53,25,78,16)(4,83,54,26,79,9)(5,84,55,27,80,10)(6,85,56,28,73,11)(7,86,49,29,74,12)(8,87,50,30,75,13)(17,89,34,59,44,71)(18,90,35,60,45,72)(19,91,36,61,46,65)(20,92,37,62,47,66)(21,93,38,63,48,67)(22,94,39,64,41,68)(23,95,40,57,42,69)(24,96,33,58,43,70), (1,66)(2,38)(3,68)(4,40)(5,70)(6,34)(7,72)(8,36)(9,57)(10,24)(11,59)(12,18)(13,61)(14,20)(15,63)(16,22)(17,56)(19,50)(21,52)(23,54)(25,39)(26,69)(27,33)(28,71)(29,35)(30,65)(31,37)(32,67)(41,82)(42,79)(43,84)(44,73)(45,86)(46,75)(47,88)(48,77)(49,60)(51,62)(53,64)(55,58)(74,90)(76,92)(78,94)(80,96)(81,93)(83,95)(85,89)(87,91), (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), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(33,49)(34,54)(35,51)(36,56)(37,53)(38,50)(39,55)(40,52)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75)(81,95)(82,92)(83,89)(84,94)(85,91)(86,96)(87,93)(88,90)>;
G:=Group( (1,88,51,31,76,14)(2,81,52,32,77,15)(3,82,53,25,78,16)(4,83,54,26,79,9)(5,84,55,27,80,10)(6,85,56,28,73,11)(7,86,49,29,74,12)(8,87,50,30,75,13)(17,89,34,59,44,71)(18,90,35,60,45,72)(19,91,36,61,46,65)(20,92,37,62,47,66)(21,93,38,63,48,67)(22,94,39,64,41,68)(23,95,40,57,42,69)(24,96,33,58,43,70), (1,66)(2,38)(3,68)(4,40)(5,70)(6,34)(7,72)(8,36)(9,57)(10,24)(11,59)(12,18)(13,61)(14,20)(15,63)(16,22)(17,56)(19,50)(21,52)(23,54)(25,39)(26,69)(27,33)(28,71)(29,35)(30,65)(31,37)(32,67)(41,82)(42,79)(43,84)(44,73)(45,86)(46,75)(47,88)(48,77)(49,60)(51,62)(53,64)(55,58)(74,90)(76,92)(78,94)(80,96)(81,93)(83,95)(85,89)(87,91), (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), (1,18)(2,23)(3,20)(4,17)(5,22)(6,19)(7,24)(8,21)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(25,62)(26,59)(27,64)(28,61)(29,58)(30,63)(31,60)(32,57)(33,49)(34,54)(35,51)(36,56)(37,53)(38,50)(39,55)(40,52)(41,80)(42,77)(43,74)(44,79)(45,76)(46,73)(47,78)(48,75)(81,95)(82,92)(83,89)(84,94)(85,91)(86,96)(87,93)(88,90) );
G=PermutationGroup([[(1,88,51,31,76,14),(2,81,52,32,77,15),(3,82,53,25,78,16),(4,83,54,26,79,9),(5,84,55,27,80,10),(6,85,56,28,73,11),(7,86,49,29,74,12),(8,87,50,30,75,13),(17,89,34,59,44,71),(18,90,35,60,45,72),(19,91,36,61,46,65),(20,92,37,62,47,66),(21,93,38,63,48,67),(22,94,39,64,41,68),(23,95,40,57,42,69),(24,96,33,58,43,70)], [(1,66),(2,38),(3,68),(4,40),(5,70),(6,34),(7,72),(8,36),(9,57),(10,24),(11,59),(12,18),(13,61),(14,20),(15,63),(16,22),(17,56),(19,50),(21,52),(23,54),(25,39),(26,69),(27,33),(28,71),(29,35),(30,65),(31,37),(32,67),(41,82),(42,79),(43,84),(44,73),(45,86),(46,75),(47,88),(48,77),(49,60),(51,62),(53,64),(55,58),(74,90),(76,92),(78,94),(80,96),(81,93),(83,95),(85,89),(87,91)], [(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)], [(1,18),(2,23),(3,20),(4,17),(5,22),(6,19),(7,24),(8,21),(9,71),(10,68),(11,65),(12,70),(13,67),(14,72),(15,69),(16,66),(25,62),(26,59),(27,64),(28,61),(29,58),(30,63),(31,60),(32,57),(33,49),(34,54),(35,51),(36,56),(37,53),(38,50),(39,55),(40,52),(41,80),(42,77),(43,74),(44,79),(45,76),(46,73),(47,78),(48,75),(81,95),(82,92),(83,89),(84,94),(85,91),(86,96),(87,93),(88,90)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 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 | 2 | 3 | 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 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | C8○D4 | D12.C4 |
| kernel | D6⋊C8⋊40C2 | D6⋊C8 | C22×C3⋊C8 | C6×M4(2) | C2×C4○D12 | C2×Dic6 | C2×D12 | C2×C3⋊D4 | C2×M4(2) | C2×C12 | C2×C8 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 1 | 4 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 4 |
Matrix representation of D6⋊C8⋊40C2 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 48 | 22 | 0 | 0 |
| 0 | 0 | 38 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 72 | 0 | 0 |
| 0 | 0 | 58 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 22 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 26 | 0 | 0 |
| 0 | 0 | 69 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,22,0,0,0,0,10,0,0,0,0,0,0,0,48,38,0,0,0,0,22,25,0,0,0,0,0,0,72,72,0,0,0,0,0,1],[10,0,0,0,0,0,0,63,0,0,0,0,0,0,31,58,0,0,0,0,72,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,10,0,0,0,0,22,0,0,0,0,0,0,0,18,69,0,0,0,0,26,55,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D6⋊C8⋊40C2 in GAP, Magma, Sage, TeX
D_6\rtimes C_8\rtimes_{40}C_2 % in TeX
G:=Group("D6:C8:40C2"); // GroupNames label
G:=SmallGroup(192,688);
// by ID
G=gap.SmallGroup(192,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,422,387,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,d*b*d=b*c^4,d*c*d=c^5>;
// generators/relations