direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8.7D6, SD16⋊12D6, C12.8C24, C24.44C23, D12.4C23, Dic6.4C23, C6⋊3(C4○D8), C4.45(S3×D4), D6.10(C2×D4), (C4×S3).29D4, (C2×C8).265D6, C12.83(C2×D4), C3⋊C8.21C23, C4.8(S3×C23), (S3×C8)⋊18C22, D4⋊S3⋊11C22, (C6×SD16)⋊11C2, (C2×SD16)⋊16S3, (C2×D4).184D6, C8.41(C22×S3), (C2×Q8).175D6, (C3×D4).6C23, D4.6(C22×S3), C3⋊Q16⋊7C22, C24⋊C2⋊18C22, (C3×Q8).2C23, D4⋊2S3⋊7C22, (C4×S3).26C23, Dic3.69(C2×D4), Q8⋊3S3⋊6C22, (C22×S3).62D4, C22.141(S3×D4), C6.109(C22×D4), Q8.12(C22×S3), (C2×C12).525C23, (C2×C24).166C22, (C2×Dic3).216D4, (C3×SD16)⋊13C22, (C6×D4).166C22, (C6×Q8).148C22, (C2×D12).178C22, (C2×Dic6).197C22, C3⋊3(C2×C4○D8), (S3×C2×C8)⋊10C2, C2.82(C2×S3×D4), (C2×D4⋊S3)⋊28C2, (C2×C24⋊C2)⋊32C2, (C2×C3⋊Q16)⋊26C2, (C2×C6).398(C2×D4), (C2×D4⋊2S3)⋊25C2, (C2×Q8⋊3S3)⋊15C2, (C2×C3⋊C8).284C22, (S3×C2×C4).258C22, (C2×C4).614(C22×S3), SmallGroup(192,1320)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8.7D6
G = < a,b,c,d,e | a2=b4=d6=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >
Subgroups: 696 in 266 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C8, C2×C8, D8, SD16, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×S3, C22×C6, C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8, C2×C4○D4, S3×C8, C24⋊C2, C2×C3⋊C8, D4⋊S3, C3⋊Q16, C2×C24, C3×SD16, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, D4⋊2S3, D4⋊2S3, Q8⋊3S3, Q8⋊3S3, C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C2×C4○D8, S3×C2×C8, C2×C24⋊C2, Q8.7D6, C2×D4⋊S3, C2×C3⋊Q16, C6×SD16, C2×D4⋊2S3, C2×Q8⋊3S3, C2×Q8.7D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C4○D8, C22×D4, S3×D4, S3×C23, C2×C4○D8, Q8.7D6, C2×S3×D4, C2×Q8.7D6
►On 96 points
(1 46)(2 47)(3 48)(4 43)(5 44)(6 45)(7 89)(8 90)(9 85)(10 86)(11 87)(12 88)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(25 39)(26 40)(27 41)(28 42)(29 37)(30 38)(31 68)(32 69)(33 70)(34 71)(35 72)(36 67)(49 59)(50 60)(51 55)(52 56)(53 57)(54 58)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(73 82)(74 83)(75 84)(76 79)(77 80)(78 81)
(1 26 31 89)(2 90 32 27)(3 28 33 85)(4 86 34 29)(5 30 35 87)(6 88 36 25)(7 46 40 68)(8 69 41 47)(9 48 42 70)(10 71 37 43)(11 44 38 72)(12 67 39 45)(13 53 96 84)(14 79 91 54)(15 49 92 80)(16 81 93 50)(17 51 94 82)(18 83 95 52)(19 59 62 77)(20 78 63 60)(21 55 64 73)(22 74 65 56)(23 57 66 75)(24 76 61 58)
(1 74 31 56)(2 66 32 23)(3 76 33 58)(4 62 34 19)(5 78 35 60)(6 64 36 21)(7 95 40 18)(8 53 41 84)(9 91 42 14)(10 49 37 80)(11 93 38 16)(12 51 39 82)(13 47 96 69)(15 43 92 71)(17 45 94 67)(20 87 63 30)(22 89 65 26)(24 85 61 28)(25 73 88 55)(27 75 90 57)(29 77 86 59)(44 81 72 50)(46 83 68 52)(48 79 70 54)
(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 55 31 73)(2 78 32 60)(3 59 33 77)(4 76 34 58)(5 57 35 75)(6 74 36 56)(7 94 40 17)(8 16 41 93)(9 92 42 15)(10 14 37 91)(11 96 38 13)(12 18 39 95)(19 85 62 28)(20 27 63 90)(21 89 64 26)(22 25 65 88)(23 87 66 30)(24 29 61 86)(43 79 71 54)(44 53 72 84)(45 83 67 52)(46 51 68 82)(47 81 69 50)(48 49 70 80)
G:=sub<Sym(96)| (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,89)(8,90)(9,85)(10,86)(11,87)(12,88)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(49,59)(50,60)(51,55)(52,56)(53,57)(54,58)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81), (1,26,31,89)(2,90,32,27)(3,28,33,85)(4,86,34,29)(5,30,35,87)(6,88,36,25)(7,46,40,68)(8,69,41,47)(9,48,42,70)(10,71,37,43)(11,44,38,72)(12,67,39,45)(13,53,96,84)(14,79,91,54)(15,49,92,80)(16,81,93,50)(17,51,94,82)(18,83,95,52)(19,59,62,77)(20,78,63,60)(21,55,64,73)(22,74,65,56)(23,57,66,75)(24,76,61,58), (1,74,31,56)(2,66,32,23)(3,76,33,58)(4,62,34,19)(5,78,35,60)(6,64,36,21)(7,95,40,18)(8,53,41,84)(9,91,42,14)(10,49,37,80)(11,93,38,16)(12,51,39,82)(13,47,96,69)(15,43,92,71)(17,45,94,67)(20,87,63,30)(22,89,65,26)(24,85,61,28)(25,73,88,55)(27,75,90,57)(29,77,86,59)(44,81,72,50)(46,83,68,52)(48,79,70,54), (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), (1,55,31,73)(2,78,32,60)(3,59,33,77)(4,76,34,58)(5,57,35,75)(6,74,36,56)(7,94,40,17)(8,16,41,93)(9,92,42,15)(10,14,37,91)(11,96,38,13)(12,18,39,95)(19,85,62,28)(20,27,63,90)(21,89,64,26)(22,25,65,88)(23,87,66,30)(24,29,61,86)(43,79,71,54)(44,53,72,84)(45,83,67,52)(46,51,68,82)(47,81,69,50)(48,49,70,80)>;
G:=Group( (1,46)(2,47)(3,48)(4,43)(5,44)(6,45)(7,89)(8,90)(9,85)(10,86)(11,87)(12,88)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,39)(26,40)(27,41)(28,42)(29,37)(30,38)(31,68)(32,69)(33,70)(34,71)(35,72)(36,67)(49,59)(50,60)(51,55)(52,56)(53,57)(54,58)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81), (1,26,31,89)(2,90,32,27)(3,28,33,85)(4,86,34,29)(5,30,35,87)(6,88,36,25)(7,46,40,68)(8,69,41,47)(9,48,42,70)(10,71,37,43)(11,44,38,72)(12,67,39,45)(13,53,96,84)(14,79,91,54)(15,49,92,80)(16,81,93,50)(17,51,94,82)(18,83,95,52)(19,59,62,77)(20,78,63,60)(21,55,64,73)(22,74,65,56)(23,57,66,75)(24,76,61,58), (1,74,31,56)(2,66,32,23)(3,76,33,58)(4,62,34,19)(5,78,35,60)(6,64,36,21)(7,95,40,18)(8,53,41,84)(9,91,42,14)(10,49,37,80)(11,93,38,16)(12,51,39,82)(13,47,96,69)(15,43,92,71)(17,45,94,67)(20,87,63,30)(22,89,65,26)(24,85,61,28)(25,73,88,55)(27,75,90,57)(29,77,86,59)(44,81,72,50)(46,83,68,52)(48,79,70,54), (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), (1,55,31,73)(2,78,32,60)(3,59,33,77)(4,76,34,58)(5,57,35,75)(6,74,36,56)(7,94,40,17)(8,16,41,93)(9,92,42,15)(10,14,37,91)(11,96,38,13)(12,18,39,95)(19,85,62,28)(20,27,63,90)(21,89,64,26)(22,25,65,88)(23,87,66,30)(24,29,61,86)(43,79,71,54)(44,53,72,84)(45,83,67,52)(46,51,68,82)(47,81,69,50)(48,49,70,80) );
G=PermutationGroup([[(1,46),(2,47),(3,48),(4,43),(5,44),(6,45),(7,89),(8,90),(9,85),(10,86),(11,87),(12,88),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(25,39),(26,40),(27,41),(28,42),(29,37),(30,38),(31,68),(32,69),(33,70),(34,71),(35,72),(36,67),(49,59),(50,60),(51,55),(52,56),(53,57),(54,58),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(73,82),(74,83),(75,84),(76,79),(77,80),(78,81)], [(1,26,31,89),(2,90,32,27),(3,28,33,85),(4,86,34,29),(5,30,35,87),(6,88,36,25),(7,46,40,68),(8,69,41,47),(9,48,42,70),(10,71,37,43),(11,44,38,72),(12,67,39,45),(13,53,96,84),(14,79,91,54),(15,49,92,80),(16,81,93,50),(17,51,94,82),(18,83,95,52),(19,59,62,77),(20,78,63,60),(21,55,64,73),(22,74,65,56),(23,57,66,75),(24,76,61,58)], [(1,74,31,56),(2,66,32,23),(3,76,33,58),(4,62,34,19),(5,78,35,60),(6,64,36,21),(7,95,40,18),(8,53,41,84),(9,91,42,14),(10,49,37,80),(11,93,38,16),(12,51,39,82),(13,47,96,69),(15,43,92,71),(17,45,94,67),(20,87,63,30),(22,89,65,26),(24,85,61,28),(25,73,88,55),(27,75,90,57),(29,77,86,59),(44,81,72,50),(46,83,68,52),(48,79,70,54)], [(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)], [(1,55,31,73),(2,78,32,60),(3,59,33,77),(4,76,34,58),(5,57,35,75),(6,74,36,56),(7,94,40,17),(8,16,41,93),(9,92,42,15),(10,14,37,91),(11,96,38,13),(12,18,39,95),(19,85,62,28),(20,27,63,90),(21,89,64,26),(22,25,65,88),(23,87,66,30),(24,29,61,86),(43,79,71,54),(44,53,72,84),(45,83,67,52),(46,51,68,82),(47,81,69,50),(48,49,70,80)]])
42 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 | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | D6 | C4○D8 | S3×D4 | S3×D4 | Q8.7D6 |
| kernel | C2×Q8.7D6 | S3×C2×C8 | C2×C24⋊C2 | Q8.7D6 | C2×D4⋊S3 | C2×C3⋊Q16 | C6×SD16 | C2×D4⋊2S3 | C2×Q8⋊3S3 | C2×SD16 | C4×S3 | C2×Dic3 | C22×S3 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 1 | 1 | 4 |
Matrix representation of C2×Q8.7D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 1 | 0 | 0 |
| 71 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 54 | 27 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 16 | 0 | 0 |
| 32 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 72 | 0 |
| 12 | 6 | 0 | 0 |
| 61 | 61 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[1,71,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[46,54,0,0,0,27,0,0,0,0,72,0,0,0,0,72],[0,32,0,0,16,0,0,0,0,0,72,72,0,0,1,0],[12,61,0,0,6,61,0,0,0,0,1,1,0,0,0,72] >;
C2×Q8.7D6 in GAP, Magma, Sage, TeX
C_2\times Q_8._7D_6
% in TeX
G:=Group("C2xQ8.7D6"); // GroupNames label
G:=SmallGroup(192,1320);
// by ID
G=gap.SmallGroup(192,1320);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,1123,185,136,438,235,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^6=1,c^2=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b^-1*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d^-1>;
// generators/relations