direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C6×Q16, C12.43D4, C24.22C22, C12.46C23, (C2×C8).4C6, C8.5(C2×C6), C4.8(C3×D4), C2.13(C6×D4), C6.76(C2×D4), (C2×C6).54D4, Q8.4(C2×C6), (C2×Q8).6C6, (C6×Q8).9C2, (C2×C24).10C2, C4.3(C22×C6), C22.16(C3×D4), (C3×Q8).12C22, (C2×C12).131C22, (C2×C4).27(C2×C6), SmallGroup(96,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6×Q16
G = < a,b,c | a6=b8=1, c2=b4, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 76 in 60 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, Q8, C12, C12, C2×C6, C2×C8, Q16, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C3×Q8, C2×Q16, C2×C24, C3×Q16, C6×Q8, C6×Q16
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C3×D4, C22×C6, C2×Q16, C3×Q16, C6×D4, C6×Q16
►Regular action on 96 points
(1 28 55 95 19 74)(2 29 56 96 20 75)(3 30 49 89 21 76)(4 31 50 90 22 77)(5 32 51 91 23 78)(6 25 52 92 24 79)(7 26 53 93 17 80)(8 27 54 94 18 73)(9 69 81 63 37 46)(10 70 82 64 38 47)(11 71 83 57 39 48)(12 72 84 58 40 41)(13 65 85 59 33 42)(14 66 86 60 34 43)(15 67 87 61 35 44)(16 68 88 62 36 45)
(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 15 5 11)(2 14 6 10)(3 13 7 9)(4 12 8 16)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(25 70 29 66)(26 69 30 65)(27 68 31 72)(28 67 32 71)(41 73 45 77)(42 80 46 76)(43 79 47 75)(44 78 48 74)(49 85 53 81)(50 84 54 88)(51 83 55 87)(52 82 56 86)(57 95 61 91)(58 94 62 90)(59 93 63 89)(60 92 64 96)
G:=sub<Sym(96)| (1,28,55,95,19,74)(2,29,56,96,20,75)(3,30,49,89,21,76)(4,31,50,90,22,77)(5,32,51,91,23,78)(6,25,52,92,24,79)(7,26,53,93,17,80)(8,27,54,94,18,73)(9,69,81,63,37,46)(10,70,82,64,38,47)(11,71,83,57,39,48)(12,72,84,58,40,41)(13,65,85,59,33,42)(14,66,86,60,34,43)(15,67,87,61,35,44)(16,68,88,62,36,45), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(41,73,45,77)(42,80,46,76)(43,79,47,75)(44,78,48,74)(49,85,53,81)(50,84,54,88)(51,83,55,87)(52,82,56,86)(57,95,61,91)(58,94,62,90)(59,93,63,89)(60,92,64,96)>;
G:=Group( (1,28,55,95,19,74)(2,29,56,96,20,75)(3,30,49,89,21,76)(4,31,50,90,22,77)(5,32,51,91,23,78)(6,25,52,92,24,79)(7,26,53,93,17,80)(8,27,54,94,18,73)(9,69,81,63,37,46)(10,70,82,64,38,47)(11,71,83,57,39,48)(12,72,84,58,40,41)(13,65,85,59,33,42)(14,66,86,60,34,43)(15,67,87,61,35,44)(16,68,88,62,36,45), (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), (1,15,5,11)(2,14,6,10)(3,13,7,9)(4,12,8,16)(17,37,21,33)(18,36,22,40)(19,35,23,39)(20,34,24,38)(25,70,29,66)(26,69,30,65)(27,68,31,72)(28,67,32,71)(41,73,45,77)(42,80,46,76)(43,79,47,75)(44,78,48,74)(49,85,53,81)(50,84,54,88)(51,83,55,87)(52,82,56,86)(57,95,61,91)(58,94,62,90)(59,93,63,89)(60,92,64,96) );
G=PermutationGroup([[(1,28,55,95,19,74),(2,29,56,96,20,75),(3,30,49,89,21,76),(4,31,50,90,22,77),(5,32,51,91,23,78),(6,25,52,92,24,79),(7,26,53,93,17,80),(8,27,54,94,18,73),(9,69,81,63,37,46),(10,70,82,64,38,47),(11,71,83,57,39,48),(12,72,84,58,40,41),(13,65,85,59,33,42),(14,66,86,60,34,43),(15,67,87,61,35,44),(16,68,88,62,36,45)], [(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)], [(1,15,5,11),(2,14,6,10),(3,13,7,9),(4,12,8,16),(17,37,21,33),(18,36,22,40),(19,35,23,39),(20,34,24,38),(25,70,29,66),(26,69,30,65),(27,68,31,72),(28,67,32,71),(41,73,45,77),(42,80,46,76),(43,79,47,75),(44,78,48,74),(49,85,53,81),(50,84,54,88),(51,83,55,87),(52,82,56,86),(57,95,61,91),(58,94,62,90),(59,93,63,89),(60,92,64,96)]])
C6×Q16 is a maximal subgroup of
C6.5Q32 Q16.Dic3 C24.27C23 Dic3⋊3Q16 C24.26D4 Q16⋊Dic3 (C2×Q16)⋊S3 D6⋊5Q16 D12.17D4 D6⋊3Q16 C24.36D4 C24.37D4 C24.28D4 C24.29D4 D12.30D4
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | Q16 | C3×D4 | C3×D4 | C3×Q16 |
| kernel | C6×Q16 | C2×C24 | C3×Q16 | C6×Q8 | C2×Q16 | C2×C8 | Q16 | C2×Q8 | C12 | C2×C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 1 | 1 | 4 | 2 | 2 | 8 |
Matrix representation of C6×Q16 ►in GL3(𝔽73) generated by
| 9 | 0 | 0 |
| 0 | 72 | 0 |
| 0 | 0 | 72 |
| 1 | 0 | 0 |
| 0 | 57 | 16 |
| 0 | 57 | 57 |
| 72 | 0 | 0 |
| 0 | 21 | 54 |
| 0 | 54 | 52 |
G:=sub<GL(3,GF(73))| [9,0,0,0,72,0,0,0,72],[1,0,0,0,57,57,0,16,57],[72,0,0,0,21,54,0,54,52] >;
C6×Q16 in GAP, Magma, Sage, TeX
C_6\times Q_{16} % in TeX
G:=Group("C6xQ16"); // GroupNames label
G:=SmallGroup(96,181);
// by ID
G=gap.SmallGroup(96,181);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,288,313,295,2164,1090,88]);
// Polycyclic
G:=Group<a,b,c|a^6=b^8=1,c^2=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations