direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.Q8, C12.17Dic6, C62.103D4, C3⋊C8⋊2C12, C12.2(C3×Q8), C4.12(S3×C12), C12.2(C2×C12), C4⋊Dic3.9C6, (C3×C12).10Q8, C4.2(C3×Dic6), C6.6(C3×SD16), C12.103(C4×S3), C32⋊7(C4.Q8), (C2×C12).312D6, (C3×C6).25SD16, (C6×C12).40C22, C6.14(D4.S3), C6.22(Dic3⋊C4), C6.14(Q8⋊2S3), (C3×C3⋊C8)⋊8C4, C6.3(C3×C4⋊C4), (C2×C3⋊C8).2C6, C3⋊1(C3×C4.Q8), (C3×C4⋊C4).2C6, (C6×C3⋊C8).20C2, C4⋊C4.2(C3×S3), (C3×C4⋊C4).25S3, (C2×C4).33(S3×C6), (C2×C6).38(C3×D4), C2.1(C3×D4.S3), (C3×C6).29(C4⋊C4), (C3×C12).38(C2×C4), (C2×C12).10(C2×C6), (C32×C4⋊C4).2C2, C2.4(C3×Dic3⋊C4), (C3×C4⋊Dic3).8C2, C2.1(C3×Q8⋊2S3), C22.13(C3×C3⋊D4), (C2×C6).106(C3⋊D4), SmallGroup(288,242)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.Q8
G = < a,b,c,d | a3=b4=c12=1, d2=bc6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 178 in 85 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C3×C6, C3⋊C8, C24, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3×C12, C3×C12, C62, C2×C3⋊C8, C4⋊Dic3, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×C3⋊C8, C6×Dic3, C6×C12, C6×C12, C12.Q8, C3×C4.Q8, C6×C3⋊C8, C3×C4⋊Dic3, C32×C4⋊C4, C3×C12.Q8
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, SD16, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C4.Q8, S3×C6, Dic3⋊C4, D4.S3, Q8⋊2S3, C3×C4⋊C4, C3×SD16, C3×Dic6, S3×C12, C3×C3⋊D4, C12.Q8, C3×C4.Q8, C3×Dic3⋊C4, C3×D4.S3, C3×Q8⋊2S3, C3×C12.Q8
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)(49 57 53)(50 58 54)(51 59 55)(52 60 56)(61 69 65)(62 70 66)(63 71 67)(64 72 68)(73 77 81)(74 78 82)(75 79 83)(76 80 84)(85 93 89)(86 94 90)(87 95 91)(88 96 92)
(1 42 83 31)(2 32 84 43)(3 44 73 33)(4 34 74 45)(5 46 75 35)(6 36 76 47)(7 48 77 25)(8 26 78 37)(9 38 79 27)(10 28 80 39)(11 40 81 29)(12 30 82 41)(13 60 87 72)(14 61 88 49)(15 50 89 62)(16 63 90 51)(17 52 91 64)(18 65 92 53)(19 54 93 66)(20 67 94 55)(21 56 95 68)(22 69 96 57)(23 58 85 70)(24 71 86 59)
(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 48 52 83 85 25 64)(2 57 26 16 84 69 37 90)(3 21 38 50 73 95 27 62)(4 55 28 14 74 67 39 88)(5 19 40 60 75 93 29 72)(6 53 30 24 76 65 41 86)(7 17 42 58 77 91 31 70)(8 51 32 22 78 63 43 96)(9 15 44 56 79 89 33 68)(10 49 34 20 80 61 45 94)(11 13 46 54 81 87 35 66)(12 59 36 18 82 71 47 92)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,42,83,31)(2,32,84,43)(3,44,73,33)(4,34,74,45)(5,46,75,35)(6,36,76,47)(7,48,77,25)(8,26,78,37)(9,38,79,27)(10,28,80,39)(11,40,81,29)(12,30,82,41)(13,60,87,72)(14,61,88,49)(15,50,89,62)(16,63,90,51)(17,52,91,64)(18,65,92,53)(19,54,93,66)(20,67,94,55)(21,56,95,68)(22,69,96,57)(23,58,85,70)(24,71,86,59), (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), (1,23,48,52,83,85,25,64)(2,57,26,16,84,69,37,90)(3,21,38,50,73,95,27,62)(4,55,28,14,74,67,39,88)(5,19,40,60,75,93,29,72)(6,53,30,24,76,65,41,86)(7,17,42,58,77,91,31,70)(8,51,32,22,78,63,43,96)(9,15,44,56,79,89,33,68)(10,49,34,20,80,61,45,94)(11,13,46,54,81,87,35,66)(12,59,36,18,82,71,47,92)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48)(49,57,53)(50,58,54)(51,59,55)(52,60,56)(61,69,65)(62,70,66)(63,71,67)(64,72,68)(73,77,81)(74,78,82)(75,79,83)(76,80,84)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (1,42,83,31)(2,32,84,43)(3,44,73,33)(4,34,74,45)(5,46,75,35)(6,36,76,47)(7,48,77,25)(8,26,78,37)(9,38,79,27)(10,28,80,39)(11,40,81,29)(12,30,82,41)(13,60,87,72)(14,61,88,49)(15,50,89,62)(16,63,90,51)(17,52,91,64)(18,65,92,53)(19,54,93,66)(20,67,94,55)(21,56,95,68)(22,69,96,57)(23,58,85,70)(24,71,86,59), (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), (1,23,48,52,83,85,25,64)(2,57,26,16,84,69,37,90)(3,21,38,50,73,95,27,62)(4,55,28,14,74,67,39,88)(5,19,40,60,75,93,29,72)(6,53,30,24,76,65,41,86)(7,17,42,58,77,91,31,70)(8,51,32,22,78,63,43,96)(9,15,44,56,79,89,33,68)(10,49,34,20,80,61,45,94)(11,13,46,54,81,87,35,66)(12,59,36,18,82,71,47,92) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48),(49,57,53),(50,58,54),(51,59,55),(52,60,56),(61,69,65),(62,70,66),(63,71,67),(64,72,68),(73,77,81),(74,78,82),(75,79,83),(76,80,84),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(1,42,83,31),(2,32,84,43),(3,44,73,33),(4,34,74,45),(5,46,75,35),(6,36,76,47),(7,48,77,25),(8,26,78,37),(9,38,79,27),(10,28,80,39),(11,40,81,29),(12,30,82,41),(13,60,87,72),(14,61,88,49),(15,50,89,62),(16,63,90,51),(17,52,91,64),(18,65,92,53),(19,54,93,66),(20,67,94,55),(21,56,95,68),(22,69,96,57),(23,58,85,70),(24,71,86,59)], [(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)], [(1,23,48,52,83,85,25,64),(2,57,26,16,84,69,37,90),(3,21,38,50,73,95,27,62),(4,55,28,14,74,67,39,88),(5,19,40,60,75,93,29,72),(6,53,30,24,76,65,41,86),(7,17,42,58,77,91,31,70),(8,51,32,22,78,63,43,96),(9,15,44,56,79,89,33,68),(10,49,34,20,80,61,45,94),(11,13,46,54,81,87,35,66),(12,59,36,18,82,71,47,92)]])
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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | - | + | + | - | - | + | |||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | Q8 | D4 | D6 | SD16 | C3×S3 | Dic6 | C4×S3 | C3×Q8 | C3⋊D4 | C3×D4 | S3×C6 | C3×SD16 | C3×Dic6 | S3×C12 | C3×C3⋊D4 | D4.S3 | Q8⋊2S3 | C3×D4.S3 | C3×Q8⋊2S3 |
| kernel | C3×C12.Q8 | C6×C3⋊C8 | C3×C4⋊Dic3 | C32×C4⋊C4 | C12.Q8 | C3×C3⋊C8 | C2×C3⋊C8 | C4⋊Dic3 | C3×C4⋊C4 | C3⋊C8 | C3×C4⋊C4 | C3×C12 | C62 | C2×C12 | C3×C6 | C4⋊C4 | C12 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C4 | C4 | C22 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 4 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C12.Q8 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 71 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 19 | 69 | 0 | 0 |
| 17 | 54 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 24 | 24 |
| 0 | 12 | 0 | 0 |
| 67 | 12 | 0 | 0 |
| 0 | 0 | 72 | 63 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,71,72,0,0,0,0,72,0,0,0,0,72],[19,17,0,0,69,54,0,0,0,0,3,24,0,0,0,24],[0,67,0,0,12,12,0,0,0,0,72,0,0,0,63,1] >;
C3×C12.Q8 in GAP, Magma, Sage, TeX
C_3\times C_{12}.Q_8 % in TeX
G:=Group("C3xC12.Q8"); // GroupNames label
G:=SmallGroup(288,242);
// by ID
G=gap.SmallGroup(288,242);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,1037,92,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=c^12=1,d^2=b*c^6,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^-1*c^-1>;
// generators/relations