direct product, metabelian, supersoluble, monomial
Aliases: C3×C6.Q16, C12.16Dic6, C62.102D4, C3⋊C8⋊1C12, C6.6(C3×D8), (C3×C6).28D8, C6.3(C3×Q16), C12.1(C3×Q8), (C3×C12).9Q8, C4.11(S3×C12), C12.1(C2×C12), C4⋊Dic3.8C6, (C3×C6).12Q16, C4.1(C3×Dic6), C12.102(C4×S3), C6.28(D4⋊S3), C32⋊6(C2.D8), (C2×C12).311D6, (C6×C12).39C22, C6.13(C3⋊Q16), C6.21(Dic3⋊C4), (C3×C3⋊C8)⋊3C4, C6.2(C3×C4⋊C4), (C2×C3⋊C8).1C6, C3⋊1(C3×C2.D8), (C3×C4⋊C4).1C6, (C6×C3⋊C8).13C2, C4⋊C4.1(C3×S3), C2.1(C3×D4⋊S3), (C3×C4⋊C4).24S3, (C2×C4).32(S3×C6), (C2×C12).9(C2×C6), (C2×C6).37(C3×D4), C2.1(C3×C3⋊Q16), (C3×C6).28(C4⋊C4), (C3×C12).37(C2×C4), (C32×C4⋊C4).1C2, (C3×C4⋊Dic3).7C2, C2.3(C3×Dic3⋊C4), C22.12(C3×C3⋊D4), (C2×C6).105(C3⋊D4), SmallGroup(288,241)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C6.Q16
G = < a,b,c,d | a3=b12=c4=1, d2=b9c2, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=b9c-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, C2.D8, 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, C6.Q16, C3×C2.D8, C6×C3⋊C8, C3×C4⋊Dic3, C32×C4⋊C4, C3×C6.Q16
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C12, D6, C2×C6, C4⋊C4, D8, Q16, C3×S3, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C2.D8, S3×C6, Dic3⋊C4, D4⋊S3, C3⋊Q16, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic6, S3×C12, C3×C3⋊D4, C6.Q16, C3×C2.D8, C3×Dic3⋊C4, C3×D4⋊S3, C3×C3⋊Q16, C3×C6.Q16
►On 96 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(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 81 77)(74 82 78)(75 83 79)(76 84 80)(85 93 89)(86 94 90)(87 95 91)(88 96 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 31 20 45)(2 26 21 40)(3 33 22 47)(4 28 23 42)(5 35 24 37)(6 30 13 44)(7 25 14 39)(8 32 15 46)(9 27 16 41)(10 34 17 48)(11 29 18 43)(12 36 19 38)(49 78 72 92)(50 73 61 87)(51 80 62 94)(52 75 63 89)(53 82 64 96)(54 77 65 91)(55 84 66 86)(56 79 67 93)(57 74 68 88)(58 81 69 95)(59 76 70 90)(60 83 71 85)
(1 56 17 64 7 50 23 70)(2 49 18 69 8 55 24 63)(3 54 19 62 9 60 13 68)(4 59 20 67 10 53 14 61)(5 52 21 72 11 58 15 66)(6 57 22 65 12 51 16 71)(25 96 48 79 31 90 42 73)(26 89 37 84 32 95 43 78)(27 94 38 77 33 88 44 83)(28 87 39 82 34 93 45 76)(29 92 40 75 35 86 46 81)(30 85 41 80 36 91 47 74)
G:=sub<Sym(96)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(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)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,31,20,45)(2,26,21,40)(3,33,22,47)(4,28,23,42)(5,35,24,37)(6,30,13,44)(7,25,14,39)(8,32,15,46)(9,27,16,41)(10,34,17,48)(11,29,18,43)(12,36,19,38)(49,78,72,92)(50,73,61,87)(51,80,62,94)(52,75,63,89)(53,82,64,96)(54,77,65,91)(55,84,66,86)(56,79,67,93)(57,74,68,88)(58,81,69,95)(59,76,70,90)(60,83,71,85), (1,56,17,64,7,50,23,70)(2,49,18,69,8,55,24,63)(3,54,19,62,9,60,13,68)(4,59,20,67,10,53,14,61)(5,52,21,72,11,58,15,66)(6,57,22,65,12,51,16,71)(25,96,48,79,31,90,42,73)(26,89,37,84,32,95,43,78)(27,94,38,77,33,88,44,83)(28,87,39,82,34,93,45,76)(29,92,40,75,35,86,46,81)(30,85,41,80,36,91,47,74)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(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)(73,81,77)(74,82,78)(75,83,79)(76,84,80)(85,93,89)(86,94,90)(87,95,91)(88,96,92), (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), (1,31,20,45)(2,26,21,40)(3,33,22,47)(4,28,23,42)(5,35,24,37)(6,30,13,44)(7,25,14,39)(8,32,15,46)(9,27,16,41)(10,34,17,48)(11,29,18,43)(12,36,19,38)(49,78,72,92)(50,73,61,87)(51,80,62,94)(52,75,63,89)(53,82,64,96)(54,77,65,91)(55,84,66,86)(56,79,67,93)(57,74,68,88)(58,81,69,95)(59,76,70,90)(60,83,71,85), (1,56,17,64,7,50,23,70)(2,49,18,69,8,55,24,63)(3,54,19,62,9,60,13,68)(4,59,20,67,10,53,14,61)(5,52,21,72,11,58,15,66)(6,57,22,65,12,51,16,71)(25,96,48,79,31,90,42,73)(26,89,37,84,32,95,43,78)(27,94,38,77,33,88,44,83)(28,87,39,82,34,93,45,76)(29,92,40,75,35,86,46,81)(30,85,41,80,36,91,47,74) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(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),(73,81,77),(74,82,78),(75,83,79),(76,84,80),(85,93,89),(86,94,90),(87,95,91),(88,96,92)], [(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)], [(1,31,20,45),(2,26,21,40),(3,33,22,47),(4,28,23,42),(5,35,24,37),(6,30,13,44),(7,25,14,39),(8,32,15,46),(9,27,16,41),(10,34,17,48),(11,29,18,43),(12,36,19,38),(49,78,72,92),(50,73,61,87),(51,80,62,94),(52,75,63,89),(53,82,64,96),(54,77,65,91),(55,84,66,86),(56,79,67,93),(57,74,68,88),(58,81,69,95),(59,76,70,90),(60,83,71,85)], [(1,56,17,64,7,50,23,70),(2,49,18,69,8,55,24,63),(3,54,19,62,9,60,13,68),(4,59,20,67,10,53,14,61),(5,52,21,72,11,58,15,66),(6,57,22,65,12,51,16,71),(25,96,48,79,31,90,42,73),(26,89,37,84,32,95,43,78),(27,94,38,77,33,88,44,83),(28,87,39,82,34,93,45,76),(29,92,40,75,35,86,46,81),(30,85,41,80,36,91,47,74)]])
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 | Q8 | D4 | D6 | D8 | Q16 | C3×S3 | Dic6 | C4×S3 | C3×Q8 | C3⋊D4 | C3×D4 | S3×C6 | C3×D8 | C3×Q16 | C3×Dic6 | S3×C12 | C3×C3⋊D4 | D4⋊S3 | C3⋊Q16 | C3×D4⋊S3 | C3×C3⋊Q16 |
| kernel | C3×C6.Q16 | C6×C3⋊C8 | C3×C4⋊Dic3 | C32×C4⋊C4 | C6.Q16 | C3×C3⋊C8 | C2×C3⋊C8 | C4⋊Dic3 | C3×C4⋊C4 | C3⋊C8 | 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.Q16 ►in GL5(𝔽73)
| 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 41 | 65 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 27 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 |
| 0 | 51 | 1 | 0 | 0 |
| 0 | 0 | 0 | 47 | 36 |
| 0 | 0 | 0 | 36 | 26 |
| 72 | 0 | 0 | 0 | 0 |
| 0 | 69 | 7 | 0 | 0 |
| 0 | 8 | 4 | 0 | 0 |
| 0 | 0 | 0 | 57 | 57 |
| 0 | 0 | 0 | 16 | 57 |
G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,1,0,0,0,0,0,1],[72,0,0,0,0,0,9,41,0,0,0,0,65,0,0,0,0,0,0,72,0,0,0,1,0],[27,0,0,0,0,0,72,51,0,0,0,0,1,0,0,0,0,0,47,36,0,0,0,36,26],[72,0,0,0,0,0,69,8,0,0,0,7,4,0,0,0,0,0,57,16,0,0,0,57,57] >;
C3×C6.Q16 in GAP, Magma, Sage, TeX
C_3\times C_6.Q_{16} % in TeX
G:=Group("C3xC6.Q16"); // GroupNames label
G:=SmallGroup(288,241);
// by ID
G=gap.SmallGroup(288,241);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,168,365,92,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^9*c^2,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=b^9*c^-1>;
// generators/relations