direct product, metabelian, supersoluble, monomial
Aliases: C3×Q16⋊S3, C24.43D6, C8.3(S3×C6), (S3×Q8)⋊6C6, C24⋊C2⋊4C6, C8⋊S3⋊4C6, (C3×Q16)⋊4C6, Q16⋊2(C3×S3), (C3×Q16)⋊6S3, D6.9(C3×D4), C3⋊Q16⋊4C6, C6.35(C6×D4), C24.10(C2×C6), Q8⋊2S3⋊3C6, (S3×C6).45D4, D12.4(C2×C6), C6.195(S3×D4), (C3×Q8).51D6, Q8.14(S3×C6), C12.9(C22×C6), Q8⋊3S3.2C6, Dic6.5(C2×C6), (C32×Q16)⋊8C2, (C3×C24).35C22, (C3×C12).80C23, Dic3.11(C3×D4), (C3×Dic3).48D4, (S3×C12).29C22, C12.160(C22×S3), (C3×D12).28C22, C32⋊20(C8.C22), (C3×Dic6).28C22, (Q8×C32).14C22, C4.9(S3×C2×C6), (C3×S3×Q8)⋊6C2, C3⋊C8.2(C2×C6), C2.23(C3×S3×D4), (C3×C24⋊C2)⋊8C2, (C3×C8⋊S3)⋊8C2, (C4×S3).4(C2×C6), C3⋊3(C3×C8.C22), (C3×Q8).9(C2×C6), (C3×C3⋊Q16)⋊12C2, (C3×C6).223(C2×D4), (C3×C3⋊C8).21C22, (C3×Q8⋊2S3)⋊10C2, (C3×Q8⋊3S3).2C2, SmallGroup(288,689)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q16⋊S3
G = < a,b,c,d,e | a3=b8=d3=e2=1, c2=b4, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, ebe=b5, cd=dc, ece=b4c, ede=d-1 >
Subgroups: 314 in 129 conjugacy classes, 54 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, M4(2), SD16, Q16, Q16, C2×Q8, C4○D4, C3×S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, D12, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C8.C22, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C8⋊S3, C24⋊C2, Q8⋊2S3, C3⋊Q16, C3×M4(2), C3×SD16, C3×Q16, C3×Q16, S3×Q8, Q8⋊3S3, C6×Q8, C3×C4○D4, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C3×D12, C3×D12, Q8×C32, Q16⋊S3, C3×C8.C22, C3×C8⋊S3, C3×C24⋊C2, C3×Q8⋊2S3, C3×C3⋊Q16, C32×Q16, C3×S3×Q8, C3×Q8⋊3S3, C3×Q16⋊S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C8.C22, S3×C6, S3×D4, C6×D4, S3×C2×C6, Q16⋊S3, C3×C8.C22, C3×S3×D4, C3×Q16⋊S3
►On 96 points
(1 75 49)(2 76 50)(3 77 51)(4 78 52)(5 79 53)(6 80 54)(7 73 55)(8 74 56)(9 90 57)(10 91 58)(11 92 59)(12 93 60)(13 94 61)(14 95 62)(15 96 63)(16 89 64)(17 65 28)(18 66 29)(19 67 30)(20 68 31)(21 69 32)(22 70 25)(23 71 26)(24 72 27)(33 85 42)(34 86 43)(35 87 44)(36 88 45)(37 81 46)(38 82 47)(39 83 48)(40 84 41)
(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 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 85 13 81)(10 84 14 88)(11 83 15 87)(12 82 16 86)(25 54 29 50)(26 53 30 49)(27 52 31 56)(28 51 32 55)(33 61 37 57)(34 60 38 64)(35 59 39 63)(36 58 40 62)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(65 77 69 73)(66 76 70 80)(67 75 71 79)(68 74 72 78)
(1 49 75)(2 50 76)(3 51 77)(4 52 78)(5 53 79)(6 54 80)(7 55 73)(8 56 74)(9 90 57)(10 91 58)(11 92 59)(12 93 60)(13 94 61)(14 95 62)(15 96 63)(16 89 64)(17 28 65)(18 29 66)(19 30 67)(20 31 68)(21 32 69)(22 25 70)(23 26 71)(24 27 72)(33 85 42)(34 86 43)(35 87 44)(36 88 45)(37 81 46)(38 82 47)(39 83 48)(40 84 41)
(1 33)(2 38)(3 35)(4 40)(5 37)(6 34)(7 39)(8 36)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(17 59)(18 64)(19 61)(20 58)(21 63)(22 60)(23 57)(24 62)(25 93)(26 90)(27 95)(28 92)(29 89)(30 94)(31 91)(32 96)(41 52)(42 49)(43 54)(44 51)(45 56)(46 53)(47 50)(48 55)(73 83)(74 88)(75 85)(76 82)(77 87)(78 84)(79 81)(80 86)
G:=sub<Sym(96)| (1,75,49)(2,76,50)(3,77,51)(4,78,52)(5,79,53)(6,80,54)(7,73,55)(8,74,56)(9,90,57)(10,91,58)(11,92,59)(12,93,60)(13,94,61)(14,95,62)(15,96,63)(16,89,64)(17,65,28)(18,66,29)(19,67,30)(20,68,31)(21,69,32)(22,70,25)(23,71,26)(24,72,27)(33,85,42)(34,86,43)(35,87,44)(36,88,45)(37,81,46)(38,82,47)(39,83,48)(40,84,41), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,85,13,81)(10,84,14,88)(11,83,15,87)(12,82,16,86)(25,54,29,50)(26,53,30,49)(27,52,31,56)(28,51,32,55)(33,61,37,57)(34,60,38,64)(35,59,39,63)(36,58,40,62)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78), (1,49,75)(2,50,76)(3,51,77)(4,52,78)(5,53,79)(6,54,80)(7,55,73)(8,56,74)(9,90,57)(10,91,58)(11,92,59)(12,93,60)(13,94,61)(14,95,62)(15,96,63)(16,89,64)(17,28,65)(18,29,66)(19,30,67)(20,31,68)(21,32,69)(22,25,70)(23,26,71)(24,27,72)(33,85,42)(34,86,43)(35,87,44)(36,88,45)(37,81,46)(38,82,47)(39,83,48)(40,84,41), (1,33)(2,38)(3,35)(4,40)(5,37)(6,34)(7,39)(8,36)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(25,93)(26,90)(27,95)(28,92)(29,89)(30,94)(31,91)(32,96)(41,52)(42,49)(43,54)(44,51)(45,56)(46,53)(47,50)(48,55)(73,83)(74,88)(75,85)(76,82)(77,87)(78,84)(79,81)(80,86)>;
G:=Group( (1,75,49)(2,76,50)(3,77,51)(4,78,52)(5,79,53)(6,80,54)(7,73,55)(8,74,56)(9,90,57)(10,91,58)(11,92,59)(12,93,60)(13,94,61)(14,95,62)(15,96,63)(16,89,64)(17,65,28)(18,66,29)(19,67,30)(20,68,31)(21,69,32)(22,70,25)(23,71,26)(24,72,27)(33,85,42)(34,86,43)(35,87,44)(36,88,45)(37,81,46)(38,82,47)(39,83,48)(40,84,41), (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), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,85,13,81)(10,84,14,88)(11,83,15,87)(12,82,16,86)(25,54,29,50)(26,53,30,49)(27,52,31,56)(28,51,32,55)(33,61,37,57)(34,60,38,64)(35,59,39,63)(36,58,40,62)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(65,77,69,73)(66,76,70,80)(67,75,71,79)(68,74,72,78), (1,49,75)(2,50,76)(3,51,77)(4,52,78)(5,53,79)(6,54,80)(7,55,73)(8,56,74)(9,90,57)(10,91,58)(11,92,59)(12,93,60)(13,94,61)(14,95,62)(15,96,63)(16,89,64)(17,28,65)(18,29,66)(19,30,67)(20,31,68)(21,32,69)(22,25,70)(23,26,71)(24,27,72)(33,85,42)(34,86,43)(35,87,44)(36,88,45)(37,81,46)(38,82,47)(39,83,48)(40,84,41), (1,33)(2,38)(3,35)(4,40)(5,37)(6,34)(7,39)(8,36)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(17,59)(18,64)(19,61)(20,58)(21,63)(22,60)(23,57)(24,62)(25,93)(26,90)(27,95)(28,92)(29,89)(30,94)(31,91)(32,96)(41,52)(42,49)(43,54)(44,51)(45,56)(46,53)(47,50)(48,55)(73,83)(74,88)(75,85)(76,82)(77,87)(78,84)(79,81)(80,86) );
G=PermutationGroup([[(1,75,49),(2,76,50),(3,77,51),(4,78,52),(5,79,53),(6,80,54),(7,73,55),(8,74,56),(9,90,57),(10,91,58),(11,92,59),(12,93,60),(13,94,61),(14,95,62),(15,96,63),(16,89,64),(17,65,28),(18,66,29),(19,67,30),(20,68,31),(21,69,32),(22,70,25),(23,71,26),(24,72,27),(33,85,42),(34,86,43),(35,87,44),(36,88,45),(37,81,46),(38,82,47),(39,83,48),(40,84,41)], [(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)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,85,13,81),(10,84,14,88),(11,83,15,87),(12,82,16,86),(25,54,29,50),(26,53,30,49),(27,52,31,56),(28,51,32,55),(33,61,37,57),(34,60,38,64),(35,59,39,63),(36,58,40,62),(41,95,45,91),(42,94,46,90),(43,93,47,89),(44,92,48,96),(65,77,69,73),(66,76,70,80),(67,75,71,79),(68,74,72,78)], [(1,49,75),(2,50,76),(3,51,77),(4,52,78),(5,53,79),(6,54,80),(7,55,73),(8,56,74),(9,90,57),(10,91,58),(11,92,59),(12,93,60),(13,94,61),(14,95,62),(15,96,63),(16,89,64),(17,28,65),(18,29,66),(19,30,67),(20,31,68),(21,32,69),(22,25,70),(23,26,71),(24,27,72),(33,85,42),(34,86,43),(35,87,44),(36,88,45),(37,81,46),(38,82,47),(39,83,48),(40,84,41)], [(1,33),(2,38),(3,35),(4,40),(5,37),(6,34),(7,39),(8,36),(9,71),(10,68),(11,65),(12,70),(13,67),(14,72),(15,69),(16,66),(17,59),(18,64),(19,61),(20,58),(21,63),(22,60),(23,57),(24,62),(25,93),(26,90),(27,95),(28,92),(29,89),(30,94),(31,91),(32,96),(41,52),(42,49),(43,54),(44,51),(45,56),(46,53),(47,50),(48,55),(73,83),(74,88),(75,85),(76,82),(77,87),(78,84),(79,81),(80,86)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 12A | 12B | 12C | ··· | 12I | 12J | 12K | 12L | ··· | 12Q | 12R | 12S | 24A | ··· | 24H | 24I | 24J |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 4 | 12 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 4 | ··· | 4 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | C8.C22 | S3×D4 | Q16⋊S3 | C3×C8.C22 | C3×S3×D4 | C3×Q16⋊S3 |
| kernel | C3×Q16⋊S3 | C3×C8⋊S3 | C3×C24⋊C2 | C3×Q8⋊2S3 | C3×C3⋊Q16 | C32×Q16 | C3×S3×Q8 | C3×Q8⋊3S3 | Q16⋊S3 | C8⋊S3 | C24⋊C2 | Q8⋊2S3 | C3⋊Q16 | C3×Q16 | S3×Q8 | Q8⋊3S3 | C3×Q16 | C3×Dic3 | S3×C6 | C24 | C3×Q8 | Q16 | Dic3 | D6 | C8 | Q8 | C32 | C6 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C3×Q16⋊S3 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 1 | 3 | 3 | 3 |
| 1 | 6 | 5 | 3 |
| 5 | 4 | 1 | 1 |
| 4 | 0 | 4 | 6 |
| 4 | 4 | 4 | 4 |
| 6 | 3 | 3 | 5 |
| 5 | 3 | 6 | 4 |
| 4 | 4 | 1 | 1 |
| 2 | 0 | 0 | 0 |
| 6 | 5 | 1 | 1 |
| 0 | 5 | 2 | 5 |
| 1 | 6 | 6 | 3 |
| 5 | 4 | 2 | 0 |
| 4 | 2 | 6 | 1 |
| 1 | 0 | 4 | 5 |
| 1 | 2 | 5 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,1,5,4,3,6,4,0,3,5,1,4,3,3,1,6],[4,6,5,4,4,3,3,4,4,3,6,1,4,5,4,1],[2,6,0,1,0,5,5,6,0,1,2,6,0,1,5,3],[5,4,1,1,4,2,0,2,2,6,4,5,0,1,5,3] >;
C3×Q16⋊S3 in GAP, Magma, Sage, TeX
C_3\times Q_{16}\rtimes S_3 % in TeX
G:=Group("C3xQ16:S3"); // GroupNames label
G:=SmallGroup(288,689);
// by ID
G=gap.SmallGroup(288,689);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,1094,303,268,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^8=d^3=e^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,e*b*e=b^5,c*d=d*c,e*c*e=b^4*c,e*d*e=d^-1>;
// generators/relations