direct product, metabelian, supersoluble, monomial
Aliases: C3×C6.SD16, Dic6⋊3C12, C12.85D12, C62.105D4, C4.2(S3×C12), C12.2(C3×D4), C6.4(C3×Q16), C12.50(C4×S3), C12.4(C2×C12), (C3×Dic6)⋊5C4, C4.10(C3×D12), (C3×C12).39D4, (C3×C6).13Q16, C6.4(C3×SD16), C6.45(D6⋊C4), (C2×C12).314D6, (C2×Dic6).5C6, (C6×Dic6).4C2, (C3×C6).21SD16, (C6×C12).42C22, C6.15(D4.S3), C6.14(C3⋊Q16), C32⋊8(Q8⋊C4), (C6×C3⋊C8).6C2, (C2×C3⋊C8).3C6, (C3×C4⋊C4).3C6, C4⋊C4.3(C3×S3), C2.6(C3×D6⋊C4), (C3×C4⋊C4).26S3, (C2×C4).35(S3×C6), C3⋊1(C3×Q8⋊C4), (C2×C6).40(C3×D4), C6.4(C3×C22⋊C4), C2.2(C3×D4.S3), C2.2(C3×C3⋊Q16), (C2×C12).12(C2×C6), (C3×C12).40(C2×C4), (C32×C4⋊C4).3C2, C22.15(C3×C3⋊D4), (C2×C6).108(C3⋊D4), (C3×C6).44(C22⋊C4), SmallGroup(288,244)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6.SD16
G = < a,b,c,d | a3=b6=c8=1, d2=b3c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b3c-1 >
Subgroups: 210 in 97 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, C24, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, Q8⋊C4, C3×Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, C6×Q8, C3×C3⋊C8, C3×Dic6, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C6.SD16, C3×Q8⋊C4, C6×C3⋊C8, C32×C4⋊C4, C6×Dic6, C3×C6.SD16
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, SD16, Q16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, Q8⋊C4, S3×C6, D6⋊C4, D4.S3, C3⋊Q16, C3×C22⋊C4, C3×SD16, C3×Q16, S3×C12, C3×D12, C3×C3⋊D4, C6.SD16, C3×Q8⋊C4, C3×D6⋊C4, C3×D4.S3, C3×C3⋊Q16, C3×C6.SD16
►On 96 points
(1 25 93)(2 26 94)(3 27 95)(4 28 96)(5 29 89)(6 30 90)(7 31 91)(8 32 92)(9 67 34)(10 68 35)(11 69 36)(12 70 37)(13 71 38)(14 72 39)(15 65 40)(16 66 33)(17 77 63)(18 78 64)(19 79 57)(20 80 58)(21 73 59)(22 74 60)(23 75 61)(24 76 62)(41 84 56)(42 85 49)(43 86 50)(44 87 51)(45 88 52)(46 81 53)(47 82 54)(48 83 55)
(1 61 25 23 93 75)(2 76 94 24 26 62)(3 63 27 17 95 77)(4 78 96 18 28 64)(5 57 29 19 89 79)(6 80 90 20 30 58)(7 59 31 21 91 73)(8 74 92 22 32 60)(9 83 67 55 34 48)(10 41 35 56 68 84)(11 85 69 49 36 42)(12 43 37 50 70 86)(13 87 71 51 38 44)(14 45 39 52 72 88)(15 81 65 53 40 46)(16 47 33 54 66 82)
(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 19 13)(2 16 20 50)(3 53 21 11)(4 14 22 56)(5 51 23 9)(6 12 24 54)(7 49 17 15)(8 10 18 52)(25 48 79 71)(26 66 80 43)(27 46 73 69)(28 72 74 41)(29 44 75 67)(30 70 76 47)(31 42 77 65)(32 68 78 45)(33 58 86 94)(34 89 87 61)(35 64 88 92)(36 95 81 59)(37 62 82 90)(38 93 83 57)(39 60 84 96)(40 91 85 63)
G:=sub<Sym(96)| (1,25,93)(2,26,94)(3,27,95)(4,28,96)(5,29,89)(6,30,90)(7,31,91)(8,32,92)(9,67,34)(10,68,35)(11,69,36)(12,70,37)(13,71,38)(14,72,39)(15,65,40)(16,66,33)(17,77,63)(18,78,64)(19,79,57)(20,80,58)(21,73,59)(22,74,60)(23,75,61)(24,76,62)(41,84,56)(42,85,49)(43,86,50)(44,87,51)(45,88,52)(46,81,53)(47,82,54)(48,83,55), (1,61,25,23,93,75)(2,76,94,24,26,62)(3,63,27,17,95,77)(4,78,96,18,28,64)(5,57,29,19,89,79)(6,80,90,20,30,58)(7,59,31,21,91,73)(8,74,92,22,32,60)(9,83,67,55,34,48)(10,41,35,56,68,84)(11,85,69,49,36,42)(12,43,37,50,70,86)(13,87,71,51,38,44)(14,45,39,52,72,88)(15,81,65,53,40,46)(16,47,33,54,66,82), (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), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,48,79,71)(26,66,80,43)(27,46,73,69)(28,72,74,41)(29,44,75,67)(30,70,76,47)(31,42,77,65)(32,68,78,45)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63)>;
G:=Group( (1,25,93)(2,26,94)(3,27,95)(4,28,96)(5,29,89)(6,30,90)(7,31,91)(8,32,92)(9,67,34)(10,68,35)(11,69,36)(12,70,37)(13,71,38)(14,72,39)(15,65,40)(16,66,33)(17,77,63)(18,78,64)(19,79,57)(20,80,58)(21,73,59)(22,74,60)(23,75,61)(24,76,62)(41,84,56)(42,85,49)(43,86,50)(44,87,51)(45,88,52)(46,81,53)(47,82,54)(48,83,55), (1,61,25,23,93,75)(2,76,94,24,26,62)(3,63,27,17,95,77)(4,78,96,18,28,64)(5,57,29,19,89,79)(6,80,90,20,30,58)(7,59,31,21,91,73)(8,74,92,22,32,60)(9,83,67,55,34,48)(10,41,35,56,68,84)(11,85,69,49,36,42)(12,43,37,50,70,86)(13,87,71,51,38,44)(14,45,39,52,72,88)(15,81,65,53,40,46)(16,47,33,54,66,82), (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), (1,55,19,13)(2,16,20,50)(3,53,21,11)(4,14,22,56)(5,51,23,9)(6,12,24,54)(7,49,17,15)(8,10,18,52)(25,48,79,71)(26,66,80,43)(27,46,73,69)(28,72,74,41)(29,44,75,67)(30,70,76,47)(31,42,77,65)(32,68,78,45)(33,58,86,94)(34,89,87,61)(35,64,88,92)(36,95,81,59)(37,62,82,90)(38,93,83,57)(39,60,84,96)(40,91,85,63) );
G=PermutationGroup([[(1,25,93),(2,26,94),(3,27,95),(4,28,96),(5,29,89),(6,30,90),(7,31,91),(8,32,92),(9,67,34),(10,68,35),(11,69,36),(12,70,37),(13,71,38),(14,72,39),(15,65,40),(16,66,33),(17,77,63),(18,78,64),(19,79,57),(20,80,58),(21,73,59),(22,74,60),(23,75,61),(24,76,62),(41,84,56),(42,85,49),(43,86,50),(44,87,51),(45,88,52),(46,81,53),(47,82,54),(48,83,55)], [(1,61,25,23,93,75),(2,76,94,24,26,62),(3,63,27,17,95,77),(4,78,96,18,28,64),(5,57,29,19,89,79),(6,80,90,20,30,58),(7,59,31,21,91,73),(8,74,92,22,32,60),(9,83,67,55,34,48),(10,41,35,56,68,84),(11,85,69,49,36,42),(12,43,37,50,70,86),(13,87,71,51,38,44),(14,45,39,52,72,88),(15,81,65,53,40,46),(16,47,33,54,66,82)], [(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)], [(1,55,19,13),(2,16,20,50),(3,53,21,11),(4,14,22,56),(5,51,23,9),(6,12,24,54),(7,49,17,15),(8,10,18,52),(25,48,79,71),(26,66,80,43),(27,46,73,69),(28,72,74,41),(29,44,75,67),(30,70,76,47),(31,42,77,65),(32,68,78,45),(33,58,86,94),(34,89,87,61),(35,64,88,92),(36,95,81,59),(37,62,82,90),(38,93,83,57),(39,60,84,96),(40,91,85,63)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12Z | 12AA | 12AB | 12AC | 12AD | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 | 6 | ··· | 6 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | - | ||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D4 | D6 | SD16 | Q16 | C3×S3 | C4×S3 | D12 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | C3×Q16 | S3×C12 | C3×D12 | C3×C3⋊D4 | D4.S3 | C3⋊Q16 | C3×D4.S3 | C3×C3⋊Q16 |
| kernel | C3×C6.SD16 | C6×C3⋊C8 | C32×C4⋊C4 | C6×Dic6 | C6.SD16 | C3×Dic6 | C2×C3⋊C8 | C3×C4⋊C4 | C2×Dic6 | Dic6 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C4⋊C4 | C12 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C6.SD16 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 27 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 41 |
| 0 | 0 | 16 | 41 |
| 27 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 72 | 57 |
| 0 | 0 | 64 | 1 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,65,0,0,0,0,1,0,0,0,0,1],[0,46,0,0,27,0,0,0,0,0,0,16,0,0,41,41],[27,0,0,0,0,46,0,0,0,0,72,64,0,0,57,1] >;
C3×C6.SD16 in GAP, Magma, Sage, TeX
C_3\times C_6.{\rm SD}_{16} % in TeX
G:=Group("C3xC6.SD16"); // GroupNames label
G:=SmallGroup(288,244);
// by ID
G=gap.SmallGroup(288,244);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,336,365,92,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^6=c^8=1,d^2=b^3*c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c^-1>;
// generators/relations