direct product, metabelian, supersoluble, monomial
Aliases: S3×C3⋊Q16, Dic6.8D6, Q8.8S32, C3⋊C8.15D6, C3⋊5(S3×Q16), (C3×S3)⋊2Q16, (S3×Q8).2S3, C32⋊7(C2×Q16), (S3×C6).36D4, (C4×S3).23D6, C6.156(S3×D4), (C3×Q8).41D6, (S3×Dic6).2C2, C32⋊7Q16⋊1C2, C32⋊3Q16⋊9C2, C32⋊2Q16⋊8C2, D6.22(C3⋊D4), (C3×C12).19C23, C12.19(C22×S3), (C3×Dic3).16D4, (S3×C12).20C22, Dic3.6(C3⋊D4), C32⋊4C8.9C22, (Q8×C32).1C22, (C3×Dic6).15C22, C32⋊4Q8.10C22, (S3×C3⋊C8).C2, C4.19(C2×S32), (C3×S3×Q8).1C2, C3⋊2(C2×C3⋊Q16), (C3×C3⋊Q16)⋊1C2, C2.30(S3×C3⋊D4), C6.52(C2×C3⋊D4), (C3×C3⋊C8).6C22, (C3×C6).134(C2×D4), SmallGroup(288,590)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C3⋊Q16
G = < a,b,c,d,e | a3=b2=c3=d8=1, e2=d4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
Subgroups: 434 in 129 conjugacy classes, 44 normal (40 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, Q8, Q8, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C8, Q16, C2×Q8, C3×S3, C3×C6, C3⋊C8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Q8, C2×Q16, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, S3×C8, Dic12, C2×C3⋊C8, C3⋊Q16, C3⋊Q16, C3×Q16, C2×Dic6, S3×Q8, S3×Q8, C6×Q8, C3×C3⋊C8, C32⋊4C8, S3×Dic3, C32⋊2Q8, C3×Dic6, C3×Dic6, S3×C12, S3×C12, C32⋊4Q8, Q8×C32, S3×Q16, C2×C3⋊Q16, S3×C3⋊C8, C32⋊2Q16, C32⋊3Q16, C3×C3⋊Q16, C32⋊7Q16, S3×Dic6, C3×S3×Q8, S3×C3⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C2×Q16, S32, C3⋊Q16, S3×D4, C2×C3⋊D4, C2×S32, S3×Q16, C2×C3⋊Q16, S3×C3⋊D4, S3×C3⋊Q16
►On 96 points
Generators in S96
(1 29 54)(2 30 55)(3 31 56)(4 32 49)(5 25 50)(6 26 51)(7 27 52)(8 28 53)(9 48 87)(10 41 88)(11 42 81)(12 43 82)(13 44 83)(14 45 84)(15 46 85)(16 47 86)(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 58 96)(34 59 89)(35 60 90)(36 61 91)(37 62 92)(38 63 93)(39 64 94)(40 57 95)
(1 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 57)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 72)(34 65)(35 66)(36 67)(37 68)(38 69)(39 70)(40 71)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(73 96)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)
(1 54 29)(2 30 55)(3 56 31)(4 32 49)(5 50 25)(6 26 51)(7 52 27)(8 28 53)(9 48 87)(10 88 41)(11 42 81)(12 82 43)(13 44 83)(14 84 45)(15 46 85)(16 86 47)(17 73 72)(18 65 74)(19 75 66)(20 67 76)(21 77 68)(22 69 78)(23 79 70)(24 71 80)(33 58 96)(34 89 59)(35 60 90)(36 91 61)(37 62 92)(38 93 63)(39 64 94)(40 95 57)
(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 58 13 62)(10 57 14 61)(11 64 15 60)(12 63 16 59)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(33 83 37 87)(34 82 38 86)(35 81 39 85)(36 88 40 84)(41 95 45 91)(42 94 46 90)(43 93 47 89)(44 92 48 96)(49 80 53 76)(50 79 54 75)(51 78 55 74)(52 77 56 73)
G:=sub<Sym(96)| (1,29,54)(2,30,55)(3,31,56)(4,32,49)(5,25,50)(6,26,51)(7,27,52)(8,28,53)(9,48,87)(10,41,88)(11,42,81)(12,43,82)(13,44,83)(14,45,84)(15,46,85)(16,47,86)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,58,96)(34,59,89)(35,60,90)(36,61,91)(37,62,92)(38,63,93)(39,64,94)(40,57,95), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95), (1,54,29)(2,30,55)(3,56,31)(4,32,49)(5,50,25)(6,26,51)(7,52,27)(8,28,53)(9,48,87)(10,88,41)(11,42,81)(12,82,43)(13,44,83)(14,84,45)(15,46,85)(16,86,47)(17,73,72)(18,65,74)(19,75,66)(20,67,76)(21,77,68)(22,69,78)(23,79,70)(24,71,80)(33,58,96)(34,89,59)(35,60,90)(36,91,61)(37,62,92)(38,93,63)(39,64,94)(40,95,57), (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)(9,58,13,62)(10,57,14,61)(11,64,15,60)(12,63,16,59)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(49,80,53,76)(50,79,54,75)(51,78,55,74)(52,77,56,73)>;
G:=Group( (1,29,54)(2,30,55)(3,31,56)(4,32,49)(5,25,50)(6,26,51)(7,27,52)(8,28,53)(9,48,87)(10,41,88)(11,42,81)(12,43,82)(13,44,83)(14,45,84)(15,46,85)(16,47,86)(17,72,73)(18,65,74)(19,66,75)(20,67,76)(21,68,77)(22,69,78)(23,70,79)(24,71,80)(33,58,96)(34,59,89)(35,60,90)(36,61,91)(37,62,92)(38,63,93)(39,64,94)(40,57,95), (1,15)(2,16)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,57)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,72)(34,65)(35,66)(36,67)(37,68)(38,69)(39,70)(40,71)(41,49)(42,50)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(73,96)(74,89)(75,90)(76,91)(77,92)(78,93)(79,94)(80,95), (1,54,29)(2,30,55)(3,56,31)(4,32,49)(5,50,25)(6,26,51)(7,52,27)(8,28,53)(9,48,87)(10,88,41)(11,42,81)(12,82,43)(13,44,83)(14,84,45)(15,46,85)(16,86,47)(17,73,72)(18,65,74)(19,75,66)(20,67,76)(21,77,68)(22,69,78)(23,79,70)(24,71,80)(33,58,96)(34,89,59)(35,60,90)(36,91,61)(37,62,92)(38,93,63)(39,64,94)(40,95,57), (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)(9,58,13,62)(10,57,14,61)(11,64,15,60)(12,63,16,59)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(33,83,37,87)(34,82,38,86)(35,81,39,85)(36,88,40,84)(41,95,45,91)(42,94,46,90)(43,93,47,89)(44,92,48,96)(49,80,53,76)(50,79,54,75)(51,78,55,74)(52,77,56,73) );
G=PermutationGroup([[(1,29,54),(2,30,55),(3,31,56),(4,32,49),(5,25,50),(6,26,51),(7,27,52),(8,28,53),(9,48,87),(10,41,88),(11,42,81),(12,43,82),(13,44,83),(14,45,84),(15,46,85),(16,47,86),(17,72,73),(18,65,74),(19,66,75),(20,67,76),(21,68,77),(22,69,78),(23,70,79),(24,71,80),(33,58,96),(34,59,89),(35,60,90),(36,61,91),(37,62,92),(38,63,93),(39,64,94),(40,57,95)], [(1,15),(2,16),(3,9),(4,10),(5,11),(6,12),(7,13),(8,14),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,57),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,72),(34,65),(35,66),(36,67),(37,68),(38,69),(39,70),(40,71),(41,49),(42,50),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(73,96),(74,89),(75,90),(76,91),(77,92),(78,93),(79,94),(80,95)], [(1,54,29),(2,30,55),(3,56,31),(4,32,49),(5,50,25),(6,26,51),(7,52,27),(8,28,53),(9,48,87),(10,88,41),(11,42,81),(12,82,43),(13,44,83),(14,84,45),(15,46,85),(16,86,47),(17,73,72),(18,65,74),(19,75,66),(20,67,76),(21,77,68),(22,69,78),(23,79,70),(24,71,80),(33,58,96),(34,89,59),(35,60,90),(36,91,61),(37,62,92),(38,93,63),(39,64,94),(40,95,57)], [(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),(9,58,13,62),(10,57,14,61),(11,64,15,60),(12,63,16,59),(25,70,29,66),(26,69,30,65),(27,68,31,72),(28,67,32,71),(33,83,37,87),(34,82,38,86),(35,81,39,85),(36,88,40,84),(41,95,45,91),(42,94,46,90),(43,93,47,89),(44,92,48,96),(49,80,53,76),(50,79,54,75),(51,78,55,74),(52,77,56,73)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 24A | 24B |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 2 | 4 | 6 | 12 | 12 | 36 | 2 | 2 | 4 | 6 | 6 | 6 | 6 | 18 | 18 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 24 | 12 | 12 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D6 | Q16 | C3⋊D4 | C3⋊D4 | S32 | C3⋊Q16 | S3×D4 | C2×S32 | S3×Q16 | S3×C3⋊D4 | S3×C3⋊Q16 |
| kernel | S3×C3⋊Q16 | S3×C3⋊C8 | C32⋊2Q16 | C32⋊3Q16 | C3×C3⋊Q16 | C32⋊7Q16 | S3×Dic6 | C3×S3×Q8 | C3⋊Q16 | S3×Q8 | C3×Dic3 | S3×C6 | C3⋊C8 | Dic6 | C4×S3 | C3×Q8 | C3×S3 | Dic3 | D6 | Q8 | S3 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 |
Matrix representation of S3×C3⋊Q16 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 57 | 57 | 0 | 0 | 0 | 0 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 62 | 43 | 0 | 0 | 0 | 0 |
| 43 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[57,16,0,0,0,0,57,57,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[62,43,0,0,0,0,43,11,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
S3×C3⋊Q16 in GAP, Magma, Sage, TeX
S_3\times C_3\rtimes Q_{16} % in TeX
G:=Group("S3xC3:Q16"); // GroupNames label
G:=SmallGroup(288,590);
// by ID
G=gap.SmallGroup(288,590);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,135,100,346,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^3=d^8=1,e^2=d^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations