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), (C4×S3).29D4, D6.10(C2×D4), (C2×C8).265D6, C12.83(C2×D4), C3⋊C8.21C23, C4.8(S3×C23), (S3×C8)⋊18C22, D4⋊S3⋊11C22, (C2×SD16)⋊16S3, (C6×SD16)⋊11C2, (C2×D4).184D6, C8.41(C22×S3), (C2×Q8).175D6, C3⋊Q16⋊7C22, D4.6(C22×S3), (C3×D4).6C23, C24⋊C2⋊18C22, (C3×Q8).2C23, D4⋊2S3⋊7C22, (C4×S3).26C23, Dic3.69(C2×D4), Q8⋊3S3⋊6C22, (C22×S3).62D4, C6.109(C22×D4), C22.141(S3×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
Subgroups: 696 in 266 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], S3 [×4], C6, C6 [×2], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×15], D4 [×2], D4 [×12], Q8 [×2], Q8 [×4], C23 [×3], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C2×C8, C2×C8 [×5], D8 [×4], SD16 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C2×D4, C2×D4 [×3], C2×Q8, C2×Q8, C4○D4 [×12], C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6, C4×S3 [×4], C4×S3 [×4], D12 [×2], D12 [×5], C2×Dic3, C2×Dic3 [×5], C3⋊D4 [×4], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×S3, C22×S3, C22×C6, C22×C8, C2×D8, C2×SD16, C2×SD16, C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], S3×C8 [×4], C24⋊C2 [×4], C2×C3⋊C8, D4⋊S3 [×4], C3⋊Q16 [×4], C2×C24, C3×SD16 [×4], C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, D4⋊2S3 [×4], D4⋊2S3 [×2], Q8⋊3S3 [×4], Q8⋊3S3 [×2], C22×Dic3, C2×C3⋊D4, C6×D4, C6×Q8, C2×C4○D8, S3×C2×C8, C2×C24⋊C2, Q8.7D6 [×8], C2×D4⋊S3, C2×C3⋊Q16, C6×SD16, C2×D4⋊2S3, C2×Q8⋊3S3, C2×Q8.7D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C4○D8 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C4○D8, Q8.7D6 [×2], C2×S3×D4, C2×Q8.7D6
Generators and relations
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 >
►On 96 points
(1 40)(2 41)(3 42)(4 37)(5 38)(6 39)(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 46)(26 47)(27 48)(28 43)(29 44)(30 45)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(49 70)(50 71)(51 72)(52 67)(53 68)(54 69)(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 53 89)(2 90 54 27)(3 28 49 85)(4 86 50 29)(5 30 51 87)(6 88 52 25)(7 40 47 68)(8 69 48 41)(9 42 43 70)(10 71 44 37)(11 38 45 72)(12 67 46 39)(13 36 96 84)(14 79 91 31)(15 32 92 80)(16 81 93 33)(17 34 94 82)(18 83 95 35)(19 56 62 77)(20 78 63 57)(21 58 64 73)(22 74 65 59)(23 60 66 75)(24 76 61 55)
(1 74 53 59)(2 66 54 23)(3 76 49 55)(4 62 50 19)(5 78 51 57)(6 64 52 21)(7 95 47 18)(8 36 48 84)(9 91 43 14)(10 32 44 80)(11 93 45 16)(12 34 46 82)(13 41 96 69)(15 37 92 71)(17 39 94 67)(20 87 63 30)(22 89 65 26)(24 85 61 28)(25 73 88 58)(27 75 90 60)(29 77 86 56)(31 42 79 70)(33 38 81 72)(35 40 83 68)
(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 58 53 73)(2 78 54 57)(3 56 49 77)(4 76 50 55)(5 60 51 75)(6 74 52 59)(7 94 47 17)(8 16 48 93)(9 92 43 15)(10 14 44 91)(11 96 45 13)(12 18 46 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)(31 37 79 71)(32 70 80 42)(33 41 81 69)(34 68 82 40)(35 39 83 67)(36 72 84 38)
G:=sub<Sym(96)| (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,89)(8,90)(9,85)(10,86)(11,87)(12,88)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81), (1,26,53,89)(2,90,54,27)(3,28,49,85)(4,86,50,29)(5,30,51,87)(6,88,52,25)(7,40,47,68)(8,69,48,41)(9,42,43,70)(10,71,44,37)(11,38,45,72)(12,67,46,39)(13,36,96,84)(14,79,91,31)(15,32,92,80)(16,81,93,33)(17,34,94,82)(18,83,95,35)(19,56,62,77)(20,78,63,57)(21,58,64,73)(22,74,65,59)(23,60,66,75)(24,76,61,55), (1,74,53,59)(2,66,54,23)(3,76,49,55)(4,62,50,19)(5,78,51,57)(6,64,52,21)(7,95,47,18)(8,36,48,84)(9,91,43,14)(10,32,44,80)(11,93,45,16)(12,34,46,82)(13,41,96,69)(15,37,92,71)(17,39,94,67)(20,87,63,30)(22,89,65,26)(24,85,61,28)(25,73,88,58)(27,75,90,60)(29,77,86,56)(31,42,79,70)(33,38,81,72)(35,40,83,68), (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), (1,58,53,73)(2,78,54,57)(3,56,49,77)(4,76,50,55)(5,60,51,75)(6,74,52,59)(7,94,47,17)(8,16,48,93)(9,92,43,15)(10,14,44,91)(11,96,45,13)(12,18,46,95)(19,85,62,28)(20,27,63,90)(21,89,64,26)(22,25,65,88)(23,87,66,30)(24,29,61,86)(31,37,79,71)(32,70,80,42)(33,41,81,69)(34,68,82,40)(35,39,83,67)(36,72,84,38)>;
G:=Group( (1,40)(2,41)(3,42)(4,37)(5,38)(6,39)(7,89)(8,90)(9,85)(10,86)(11,87)(12,88)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(25,46)(26,47)(27,48)(28,43)(29,44)(30,45)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(49,70)(50,71)(51,72)(52,67)(53,68)(54,69)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(73,82)(74,83)(75,84)(76,79)(77,80)(78,81), (1,26,53,89)(2,90,54,27)(3,28,49,85)(4,86,50,29)(5,30,51,87)(6,88,52,25)(7,40,47,68)(8,69,48,41)(9,42,43,70)(10,71,44,37)(11,38,45,72)(12,67,46,39)(13,36,96,84)(14,79,91,31)(15,32,92,80)(16,81,93,33)(17,34,94,82)(18,83,95,35)(19,56,62,77)(20,78,63,57)(21,58,64,73)(22,74,65,59)(23,60,66,75)(24,76,61,55), (1,74,53,59)(2,66,54,23)(3,76,49,55)(4,62,50,19)(5,78,51,57)(6,64,52,21)(7,95,47,18)(8,36,48,84)(9,91,43,14)(10,32,44,80)(11,93,45,16)(12,34,46,82)(13,41,96,69)(15,37,92,71)(17,39,94,67)(20,87,63,30)(22,89,65,26)(24,85,61,28)(25,73,88,58)(27,75,90,60)(29,77,86,56)(31,42,79,70)(33,38,81,72)(35,40,83,68), (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), (1,58,53,73)(2,78,54,57)(3,56,49,77)(4,76,50,55)(5,60,51,75)(6,74,52,59)(7,94,47,17)(8,16,48,93)(9,92,43,15)(10,14,44,91)(11,96,45,13)(12,18,46,95)(19,85,62,28)(20,27,63,90)(21,89,64,26)(22,25,65,88)(23,87,66,30)(24,29,61,86)(31,37,79,71)(32,70,80,42)(33,41,81,69)(34,68,82,40)(35,39,83,67)(36,72,84,38) );
G=PermutationGroup([(1,40),(2,41),(3,42),(4,37),(5,38),(6,39),(7,89),(8,90),(9,85),(10,86),(11,87),(12,88),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(25,46),(26,47),(27,48),(28,43),(29,44),(30,45),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(49,70),(50,71),(51,72),(52,67),(53,68),(54,69),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(73,82),(74,83),(75,84),(76,79),(77,80),(78,81)], [(1,26,53,89),(2,90,54,27),(3,28,49,85),(4,86,50,29),(5,30,51,87),(6,88,52,25),(7,40,47,68),(8,69,48,41),(9,42,43,70),(10,71,44,37),(11,38,45,72),(12,67,46,39),(13,36,96,84),(14,79,91,31),(15,32,92,80),(16,81,93,33),(17,34,94,82),(18,83,95,35),(19,56,62,77),(20,78,63,57),(21,58,64,73),(22,74,65,59),(23,60,66,75),(24,76,61,55)], [(1,74,53,59),(2,66,54,23),(3,76,49,55),(4,62,50,19),(5,78,51,57),(6,64,52,21),(7,95,47,18),(8,36,48,84),(9,91,43,14),(10,32,44,80),(11,93,45,16),(12,34,46,82),(13,41,96,69),(15,37,92,71),(17,39,94,67),(20,87,63,30),(22,89,65,26),(24,85,61,28),(25,73,88,58),(27,75,90,60),(29,77,86,56),(31,42,79,70),(33,38,81,72),(35,40,83,68)], [(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)], [(1,58,53,73),(2,78,54,57),(3,56,49,77),(4,76,50,55),(5,60,51,75),(6,74,52,59),(7,94,47,17),(8,16,48,93),(9,92,43,15),(10,14,44,91),(11,96,45,13),(12,18,46,95),(19,85,62,28),(20,27,63,90),(21,89,64,26),(22,25,65,88),(23,87,66,30),(24,29,61,86),(31,37,79,71),(32,70,80,42),(33,41,81,69),(34,68,82,40),(35,39,83,67),(36,72,84,38)])
Matrix representation ►G ⊆ 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] >;
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 |
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