direct product, metabelian, supersoluble, monomial
Aliases: C3×S3×Q16, C24.57D6, Dic12⋊5C6, C8.9(S3×C6), C3⋊2(C6×Q16), (S3×C8).1C6, C24.7(C2×C6), (C3×Q16)⋊2C6, C3⋊Q16⋊3C6, C6.34(C6×D4), (S3×Q8).2C6, (S3×C24).2C2, (S3×C6).50D4, D6.14(C3×D4), C6.194(S3×D4), (C3×Q8).50D6, Q8.13(S3×C6), C32⋊11(C2×Q16), C12.8(C22×C6), Dic3.5(C3×D4), Dic6.4(C2×C6), (C32×Q16)⋊3C2, (C3×Dic12)⋊13C2, (C3×C24).27C22, (C3×C12).79C23, (C3×Dic3).32D4, (S3×C12).51C22, C12.159(C22×S3), (C3×Dic6).27C22, (Q8×C32).13C22, C4.8(S3×C2×C6), C3⋊C8.7(C2×C6), C2.22(C3×S3×D4), (C3×S3×Q8).2C2, (C3×Q8).8(C2×C6), (C4×S3).11(C2×C6), (C3×C3⋊Q16)⋊10C2, (C3×C6).222(C2×D4), (C3×C3⋊C8).35C22, SmallGroup(288,688)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×S3×Q16
G = < a,b,c,d,e | a3=b3=c2=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 282 in 129 conjugacy classes, 58 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, Q16, Q16, C2×Q8, C3×S3, C3×C6, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, C4×S3, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C8, Dic12, C3⋊Q16, C2×C24, C3×Q16, C3×Q16, S3×Q8, C6×Q8, C3×C3⋊C8, C3×C24, C3×Dic6, C3×Dic6, S3×C12, S3×C12, Q8×C32, S3×Q16, C6×Q16, S3×C24, C3×Dic12, C3×C3⋊Q16, C32×Q16, C3×S3×Q8, C3×S3×Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, Q16, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C2×Q16, S3×C6, C3×Q16, S3×D4, C6×D4, S3×C2×C6, S3×Q16, C6×Q16, C3×S3×D4, C3×S3×Q16
►On 96 points
Generators in S96
(1 29 82)(2 30 83)(3 31 84)(4 32 85)(5 25 86)(6 26 87)(7 27 88)(8 28 81)(9 94 64)(10 95 57)(11 96 58)(12 89 59)(13 90 60)(14 91 61)(15 92 62)(16 93 63)(17 72 73)(18 65 74)(19 66 75)(20 67 76)(21 68 77)(22 69 78)(23 70 79)(24 71 80)(33 51 45)(34 52 46)(35 53 47)(36 54 48)(37 55 41)(38 56 42)(39 49 43)(40 50 44)
(1 29 82)(2 30 83)(3 31 84)(4 32 85)(5 25 86)(6 26 87)(7 27 88)(8 28 81)(9 64 94)(10 57 95)(11 58 96)(12 59 89)(13 60 90)(14 61 91)(15 62 92)(16 63 93)(17 72 73)(18 65 74)(19 66 75)(20 67 76)(21 68 77)(22 69 78)(23 70 79)(24 71 80)(33 45 51)(34 46 52)(35 47 53)(36 48 54)(37 41 55)(38 42 56)(39 43 49)(40 44 50)
(1 38)(2 39)(3 40)(4 33)(5 34)(6 35)(7 36)(8 37)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 62)(18 63)(19 64)(20 57)(21 58)(22 59)(23 60)(24 61)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(73 92)(74 93)(75 94)(76 95)(77 96)(78 89)(79 90)(80 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 19 5 23)(2 18 6 22)(3 17 7 21)(4 24 8 20)(9 52 13 56)(10 51 14 55)(11 50 15 54)(12 49 16 53)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(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)(73 88 77 84)(74 87 78 83)(75 86 79 82)(76 85 80 81)
G:=sub<Sym(96)| (1,29,82)(2,30,83)(3,31,84)(4,32,85)(5,25,86)(6,26,87)(7,27,88)(8,28,81)(9,94,64)(10,95,57)(11,96,58)(12,89,59)(13,90,60)(14,91,61)(15,92,62)(16,93,63)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,51,45)(34,52,46)(35,53,47)(36,54,48)(37,55,41)(38,56,42)(39,49,43)(40,50,44), (1,29,82)(2,30,83)(3,31,84)(4,32,85)(5,25,86)(6,26,87)(7,27,88)(8,28,81)(9,64,94)(10,57,95)(11,58,96)(12,59,89)(13,60,90)(14,61,91)(15,62,92)(16,63,93)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,45,51)(34,46,52)(35,47,53)(36,48,54)(37,41,55)(38,42,56)(39,43,49)(40,44,50), (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(73,92)(74,93)(75,94)(76,95)(77,96)(78,89)(79,90)(80,91), (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), (1,19,5,23)(2,18,6,22)(3,17,7,21)(4,24,8,20)(9,52,13,56)(10,51,14,55)(11,50,15,54)(12,49,16,53)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(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)(73,88,77,84)(74,87,78,83)(75,86,79,82)(76,85,80,81)>;
G:=Group( (1,29,82)(2,30,83)(3,31,84)(4,32,85)(5,25,86)(6,26,87)(7,27,88)(8,28,81)(9,94,64)(10,95,57)(11,96,58)(12,89,59)(13,90,60)(14,91,61)(15,92,62)(16,93,63)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,51,45)(34,52,46)(35,53,47)(36,54,48)(37,55,41)(38,56,42)(39,49,43)(40,50,44), (1,29,82)(2,30,83)(3,31,84)(4,32,85)(5,25,86)(6,26,87)(7,27,88)(8,28,81)(9,64,94)(10,57,95)(11,58,96)(12,59,89)(13,60,90)(14,61,91)(15,62,92)(16,63,93)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,45,51)(34,46,52)(35,47,53)(36,48,54)(37,41,55)(38,42,56)(39,43,49)(40,44,50), (1,38)(2,39)(3,40)(4,33)(5,34)(6,35)(7,36)(8,37)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,62)(18,63)(19,64)(20,57)(21,58)(22,59)(23,60)(24,61)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88)(73,92)(74,93)(75,94)(76,95)(77,96)(78,89)(79,90)(80,91), (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), (1,19,5,23)(2,18,6,22)(3,17,7,21)(4,24,8,20)(9,52,13,56)(10,51,14,55)(11,50,15,54)(12,49,16,53)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(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)(73,88,77,84)(74,87,78,83)(75,86,79,82)(76,85,80,81) );
G=PermutationGroup([[(1,29,82),(2,30,83),(3,31,84),(4,32,85),(5,25,86),(6,26,87),(7,27,88),(8,28,81),(9,94,64),(10,95,57),(11,96,58),(12,89,59),(13,90,60),(14,91,61),(15,92,62),(16,93,63),(17,72,73),(18,65,74),(19,66,75),(20,67,76),(21,68,77),(22,69,78),(23,70,79),(24,71,80),(33,51,45),(34,52,46),(35,53,47),(36,54,48),(37,55,41),(38,56,42),(39,49,43),(40,50,44)], [(1,29,82),(2,30,83),(3,31,84),(4,32,85),(5,25,86),(6,26,87),(7,27,88),(8,28,81),(9,64,94),(10,57,95),(11,58,96),(12,59,89),(13,60,90),(14,61,91),(15,62,92),(16,63,93),(17,72,73),(18,65,74),(19,66,75),(20,67,76),(21,68,77),(22,69,78),(23,70,79),(24,71,80),(33,45,51),(34,46,52),(35,47,53),(36,48,54),(37,41,55),(38,42,56),(39,43,49),(40,44,50)], [(1,38),(2,39),(3,40),(4,33),(5,34),(6,35),(7,36),(8,37),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,62),(18,63),(19,64),(20,57),(21,58),(22,59),(23,60),(24,61),(25,52),(26,53),(27,54),(28,55),(29,56),(30,49),(31,50),(32,51),(41,81),(42,82),(43,83),(44,84),(45,85),(46,86),(47,87),(48,88),(73,92),(74,93),(75,94),(76,95),(77,96),(78,89),(79,90),(80,91)], [(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)], [(1,19,5,23),(2,18,6,22),(3,17,7,21),(4,24,8,20),(9,52,13,56),(10,51,14,55),(11,50,15,54),(12,49,16,53),(25,70,29,66),(26,69,30,65),(27,68,31,72),(28,67,32,71),(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),(73,88,77,84),(74,87,78,83),(75,86,79,82),(76,85,80,81)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 12A | 12B | 12C | ··· | 12I | 12J | 12K | 12L | ··· | 12Q | 12R | 12S | 12T | 12U | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 2 | 2 | 6 | 6 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
63 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 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | Q16 | C3×S3 | C3×D4 | C3×D4 | S3×C6 | S3×C6 | C3×Q16 | S3×D4 | S3×Q16 | C3×S3×D4 | C3×S3×Q16 |
| kernel | C3×S3×Q16 | S3×C24 | C3×Dic12 | C3×C3⋊Q16 | C32×Q16 | C3×S3×Q8 | S3×Q16 | S3×C8 | Dic12 | C3⋊Q16 | C3×Q16 | S3×Q8 | C3×Q16 | C3×Dic3 | S3×C6 | C24 | C3×Q8 | C3×S3 | Q16 | Dic3 | D6 | C8 | Q8 | S3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×S3×Q16 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 4 | 4 | 6 |
| 1 | 5 | 6 | 3 |
| 5 | 5 | 6 | 1 |
| 3 | 4 | 5 | 4 |
| 4 | 2 | 5 | 3 |
| 6 | 2 | 1 | 4 |
| 0 | 6 | 5 | 4 |
| 5 | 0 | 3 | 3 |
| 6 | 3 | 0 | 2 |
| 1 | 6 | 1 | 1 |
| 2 | 2 | 2 | 3 |
| 3 | 4 | 2 | 6 |
| 3 | 6 | 6 | 1 |
| 6 | 3 | 6 | 6 |
| 6 | 6 | 5 | 0 |
| 2 | 5 | 0 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,1,5,3,4,5,5,4,4,6,6,5,6,3,1,4],[4,6,0,5,2,2,6,0,5,1,5,3,3,4,4,3],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[3,6,6,2,6,3,6,5,6,6,5,0,1,6,0,3] >;
C3×S3×Q16 in GAP, Magma, Sage, TeX
C_3\times S_3\times Q_{16} % in TeX
G:=Group("C3xS3xQ16"); // GroupNames label
G:=SmallGroup(288,688);
// by ID
G=gap.SmallGroup(288,688);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,344,303,268,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations