direct product, metabelian, supersoluble, monomial
Aliases: C3×Q8⋊2Dic3, C62.108D4, (C3×Q8)⋊3C12, C12.9(C3×D4), (C6×Q8).9C6, C6.5(C3×Q16), C12.8(C2×C12), (C3×Q8)⋊6Dic3, Q8⋊3(C3×Dic3), (C3×C12).47D4, (C6×Q8).26S3, (C3×C6).14Q16, (Q8×C32)⋊4C4, C4.2(C6×Dic3), C6.8(C3×SD16), (C2×C12).318D6, C4⋊Dic3.10C6, (C3×C6).27SD16, (C6×C12).49C22, C12.37(C2×Dic3), C6.15(C3⋊Q16), C12.101(C3⋊D4), C6.16(Q8⋊2S3), C32⋊12(Q8⋊C4), C6.34(C6.D4), (C6×C3⋊C8).8C2, (C2×C3⋊C8).5C6, (Q8×C3×C6).1C2, (C2×C4).39(S3×C6), C3⋊3(C3×Q8⋊C4), (C2×C6).43(C3×D4), C4.14(C3×C3⋊D4), (C2×Q8).5(C3×S3), C2.3(C3×C3⋊Q16), (C3×C12).44(C2×C4), (C2×C12).19(C2×C6), C6.16(C3×C22⋊C4), (C3×C4⋊Dic3).9C2, C2.3(C3×Q8⋊2S3), C2.6(C3×C6.D4), C22.18(C3×C3⋊D4), (C2×C6).111(C3⋊D4), (C3×C6).67(C22⋊C4), SmallGroup(288,269)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×Q8⋊2Dic3
G = < a,b,c,d,e | a3=b4=d6=1, c2=b2, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d-1 >
Subgroups: 202 in 103 conjugacy classes, 50 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×Q8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C3×Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C2×C24, C6×Q8, C6×Q8, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, Q8×C32, Q8×C32, Q8⋊2Dic3, C3×Q8⋊C4, C6×C3⋊C8, C3×C4⋊Dic3, Q8×C3×C6, C3×Q8⋊2Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, SD16, Q16, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, Q8⋊C4, C3×Dic3, S3×C6, Q8⋊2S3, C3⋊Q16, C6.D4, C3×C22⋊C4, C3×SD16, C3×Q16, C6×Dic3, C3×C3⋊D4, Q8⋊2Dic3, C3×Q8⋊C4, C3×Q8⋊2S3, C3×C3⋊Q16, C3×C6.D4, C3×Q8⋊2Dic3
►On 96 points
Generators in S96
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 57 59)(56 58 60)(61 63 65)(62 64 66)(67 69 71)(68 70 72)(73 75 77)(74 76 78)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)
(1 29 19 7)(2 30 20 8)(3 25 21 9)(4 26 22 10)(5 27 23 11)(6 28 24 12)(13 95 90 80)(14 96 85 81)(15 91 86 82)(16 92 87 83)(17 93 88 84)(18 94 89 79)(31 42 43 54)(32 37 44 49)(33 38 45 50)(34 39 46 51)(35 40 47 52)(36 41 48 53)(55 65 70 75)(56 66 71 76)(57 61 72 77)(58 62 67 78)(59 63 68 73)(60 64 69 74)
(1 46 19 34)(2 47 20 35)(3 48 21 36)(4 43 22 31)(5 44 23 32)(6 45 24 33)(7 51 29 39)(8 52 30 40)(9 53 25 41)(10 54 26 42)(11 49 27 37)(12 50 28 38)(13 68 90 59)(14 69 85 60)(15 70 86 55)(16 71 87 56)(17 72 88 57)(18 67 89 58)(61 84 77 93)(62 79 78 94)(63 80 73 95)(64 81 74 96)(65 82 75 91)(66 83 76 92)
(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 59 4 56)(2 58 5 55)(3 57 6 60)(7 63 10 66)(8 62 11 65)(9 61 12 64)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 83 34 80)(32 82 35 79)(33 81 36 84)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
G:=sub<Sym(96)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,95,90,80)(14,96,85,81)(15,91,86,82)(16,92,87,83)(17,93,88,84)(18,94,89,79)(31,42,43,54)(32,37,44,49)(33,38,45,50)(34,39,46,51)(35,40,47,52)(36,41,48,53)(55,65,70,75)(56,66,71,76)(57,61,72,77)(58,62,67,78)(59,63,68,73)(60,64,69,74), (1,46,19,34)(2,47,20,35)(3,48,21,36)(4,43,22,31)(5,44,23,32)(6,45,24,33)(7,51,29,39)(8,52,30,40)(9,53,25,41)(10,54,26,42)(11,49,27,37)(12,50,28,38)(13,68,90,59)(14,69,85,60)(15,70,86,55)(16,71,87,56)(17,72,88,57)(18,67,89,58)(61,84,77,93)(62,79,78,94)(63,80,73,95)(64,81,74,96)(65,82,75,91)(66,83,76,92), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,63,10,66)(8,62,11,65)(9,61,12,64)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;
G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,57,59)(56,58,60)(61,63,65)(62,64,66)(67,69,71)(68,70,72)(73,75,77)(74,76,78)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96), (1,29,19,7)(2,30,20,8)(3,25,21,9)(4,26,22,10)(5,27,23,11)(6,28,24,12)(13,95,90,80)(14,96,85,81)(15,91,86,82)(16,92,87,83)(17,93,88,84)(18,94,89,79)(31,42,43,54)(32,37,44,49)(33,38,45,50)(34,39,46,51)(35,40,47,52)(36,41,48,53)(55,65,70,75)(56,66,71,76)(57,61,72,77)(58,62,67,78)(59,63,68,73)(60,64,69,74), (1,46,19,34)(2,47,20,35)(3,48,21,36)(4,43,22,31)(5,44,23,32)(6,45,24,33)(7,51,29,39)(8,52,30,40)(9,53,25,41)(10,54,26,42)(11,49,27,37)(12,50,28,38)(13,68,90,59)(14,69,85,60)(15,70,86,55)(16,71,87,56)(17,72,88,57)(18,67,89,58)(61,84,77,93)(62,79,78,94)(63,80,73,95)(64,81,74,96)(65,82,75,91)(66,83,76,92), (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), (1,59,4,56)(2,58,5,55)(3,57,6,60)(7,63,10,66)(8,62,11,65)(9,61,12,64)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,83,34,80)(32,82,35,79)(33,81,36,84)(37,86,40,89)(38,85,41,88)(39,90,42,87)(43,92,46,95)(44,91,47,94)(45,96,48,93) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46),(49,53,51),(50,54,52),(55,57,59),(56,58,60),(61,63,65),(62,64,66),(67,69,71),(68,70,72),(73,75,77),(74,76,78),(79,81,83),(80,82,84),(85,87,89),(86,88,90),(91,93,95),(92,94,96)], [(1,29,19,7),(2,30,20,8),(3,25,21,9),(4,26,22,10),(5,27,23,11),(6,28,24,12),(13,95,90,80),(14,96,85,81),(15,91,86,82),(16,92,87,83),(17,93,88,84),(18,94,89,79),(31,42,43,54),(32,37,44,49),(33,38,45,50),(34,39,46,51),(35,40,47,52),(36,41,48,53),(55,65,70,75),(56,66,71,76),(57,61,72,77),(58,62,67,78),(59,63,68,73),(60,64,69,74)], [(1,46,19,34),(2,47,20,35),(3,48,21,36),(4,43,22,31),(5,44,23,32),(6,45,24,33),(7,51,29,39),(8,52,30,40),(9,53,25,41),(10,54,26,42),(11,49,27,37),(12,50,28,38),(13,68,90,59),(14,69,85,60),(15,70,86,55),(16,71,87,56),(17,72,88,57),(18,67,89,58),(61,84,77,93),(62,79,78,94),(63,80,73,95),(64,81,74,96),(65,82,75,91),(66,83,76,92)], [(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)], [(1,59,4,56),(2,58,5,55),(3,57,6,60),(7,63,10,66),(8,62,11,65),(9,61,12,64),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,83,34,80),(32,82,35,79),(33,81,36,84),(37,86,40,89),(38,85,41,88),(39,90,42,87),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])
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 | Dic3 | SD16 | Q16 | C3×S3 | C3⋊D4 | C3×D4 | C3⋊D4 | C3×D4 | S3×C6 | C3×Dic3 | C3×SD16 | C3×Q16 | C3×C3⋊D4 | C3×C3⋊D4 | Q8⋊2S3 | C3⋊Q16 | C3×Q8⋊2S3 | C3×C3⋊Q16 |
| kernel | C3×Q8⋊2Dic3 | C6×C3⋊C8 | C3×C4⋊Dic3 | Q8×C3×C6 | Q8⋊2Dic3 | Q8×C32 | C2×C3⋊C8 | C4⋊Dic3 | C6×Q8 | C3×Q8 | C6×Q8 | C3×C12 | C62 | C2×C12 | C3×Q8 | C3×C6 | C3×C6 | C2×Q8 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | Q8 | C6 | C6 | 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×Q8⋊2Dic3 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 60 | 7 |
| 0 | 0 | 7 | 13 |
| 65 | 0 | 0 | 0 |
| 30 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 63 | 0 | 0 |
| 15 | 52 | 0 | 0 |
| 0 | 0 | 36 | 26 |
| 0 | 0 | 26 | 37 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[72,0,0,0,0,72,0,0,0,0,60,7,0,0,7,13],[65,30,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[21,15,0,0,63,52,0,0,0,0,36,26,0,0,26,37] >;
C3×Q8⋊2Dic3 in GAP, Magma, Sage, TeX
C_3\times Q_8\rtimes_2{\rm Dic}_3 % in TeX
G:=Group("C3xQ8:2Dic3"); // GroupNames label
G:=SmallGroup(288,269);
// by ID
G=gap.SmallGroup(288,269);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,344,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=d^6=1,c^2=b^2,e^2=d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations