direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C8.4Q8, C48.2C4, C16.1C12, C12.70D8, C24.20Q8, C8.5(C3×Q8), (C2×C16).5C6, C4.19(C3×D8), (C2×C6).6Q16, C24.77(C2×C4), (C2×C48).11C2, C8.15(C2×C12), C12.58(C4⋊C4), C8.C4.3C6, (C2×C12).410D4, C6.15(C2.D8), C22.1(C3×Q16), (C2×C24).409C22, C4.9(C3×C4⋊C4), C2.5(C3×C2.D8), (C2×C8).89(C2×C6), (C2×C4).64(C3×D4), (C3×C8.C4).6C2, SmallGroup(192,174)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.4Q8
G = < a,b,c,d | a3=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >
Smallest permutation representation of C3×C8.4Q8
►On 96 points
Generators in S96
(1 52 67)(2 53 68)(3 54 69)(4 55 70)(5 56 71)(6 57 72)(7 58 73)(8 59 74)(9 60 75)(10 61 76)(11 62 77)(12 63 78)(13 64 79)(14 49 80)(15 50 65)(16 51 66)(17 95 36)(18 96 37)(19 81 38)(20 82 39)(21 83 40)(22 84 41)(23 85 42)(24 86 43)(25 87 44)(26 88 45)(27 89 46)(28 90 47)(29 91 48)(30 92 33)(31 93 34)(32 94 35)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 56)(65 75 69 79 73 67 77 71)(66 76 70 80 74 68 78 72)(81 83 85 87 89 91 93 95)(82 84 86 88 90 92 94 96)
(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 18 13 22 9 26 5 30)(2 17 14 21 10 25 6 29)(3 32 15 20 11 24 7 28)(4 31 16 19 12 23 8 27)(33 67 37 79 41 75 45 71)(34 66 38 78 42 74 46 70)(35 65 39 77 43 73 47 69)(36 80 40 76 44 72 48 68)(49 83 61 87 57 91 53 95)(50 82 62 86 58 90 54 94)(51 81 63 85 59 89 55 93)(52 96 64 84 60 88 56 92)
G:=sub<Sym(96)| (1,52,67)(2,53,68)(3,54,69)(4,55,70)(5,56,71)(6,57,72)(7,58,73)(8,59,74)(9,60,75)(10,61,76)(11,62,77)(12,63,78)(13,64,79)(14,49,80)(15,50,65)(16,51,66)(17,95,36)(18,96,37)(19,81,38)(20,82,39)(21,83,40)(22,84,41)(23,85,42)(24,86,43)(25,87,44)(26,88,45)(27,89,46)(28,90,47)(29,91,48)(30,92,33)(31,93,34)(32,94,35), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,59,53,63,57,51,61,55)(50,60,54,64,58,52,62,56)(65,75,69,79,73,67,77,71)(66,76,70,80,74,68,78,72)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96), (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), (1,18,13,22,9,26,5,30)(2,17,14,21,10,25,6,29)(3,32,15,20,11,24,7,28)(4,31,16,19,12,23,8,27)(33,67,37,79,41,75,45,71)(34,66,38,78,42,74,46,70)(35,65,39,77,43,73,47,69)(36,80,40,76,44,72,48,68)(49,83,61,87,57,91,53,95)(50,82,62,86,58,90,54,94)(51,81,63,85,59,89,55,93)(52,96,64,84,60,88,56,92)>;
G:=Group( (1,52,67)(2,53,68)(3,54,69)(4,55,70)(5,56,71)(6,57,72)(7,58,73)(8,59,74)(9,60,75)(10,61,76)(11,62,77)(12,63,78)(13,64,79)(14,49,80)(15,50,65)(16,51,66)(17,95,36)(18,96,37)(19,81,38)(20,82,39)(21,83,40)(22,84,41)(23,85,42)(24,86,43)(25,87,44)(26,88,45)(27,89,46)(28,90,47)(29,91,48)(30,92,33)(31,93,34)(32,94,35), (1,11,5,15,9,3,13,7)(2,12,6,16,10,4,14,8)(17,19,21,23,25,27,29,31)(18,20,22,24,26,28,30,32)(33,35,37,39,41,43,45,47)(34,36,38,40,42,44,46,48)(49,59,53,63,57,51,61,55)(50,60,54,64,58,52,62,56)(65,75,69,79,73,67,77,71)(66,76,70,80,74,68,78,72)(81,83,85,87,89,91,93,95)(82,84,86,88,90,92,94,96), (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), (1,18,13,22,9,26,5,30)(2,17,14,21,10,25,6,29)(3,32,15,20,11,24,7,28)(4,31,16,19,12,23,8,27)(33,67,37,79,41,75,45,71)(34,66,38,78,42,74,46,70)(35,65,39,77,43,73,47,69)(36,80,40,76,44,72,48,68)(49,83,61,87,57,91,53,95)(50,82,62,86,58,90,54,94)(51,81,63,85,59,89,55,93)(52,96,64,84,60,88,56,92) );
G=PermutationGroup([[(1,52,67),(2,53,68),(3,54,69),(4,55,70),(5,56,71),(6,57,72),(7,58,73),(8,59,74),(9,60,75),(10,61,76),(11,62,77),(12,63,78),(13,64,79),(14,49,80),(15,50,65),(16,51,66),(17,95,36),(18,96,37),(19,81,38),(20,82,39),(21,83,40),(22,84,41),(23,85,42),(24,86,43),(25,87,44),(26,88,45),(27,89,46),(28,90,47),(29,91,48),(30,92,33),(31,93,34),(32,94,35)], [(1,11,5,15,9,3,13,7),(2,12,6,16,10,4,14,8),(17,19,21,23,25,27,29,31),(18,20,22,24,26,28,30,32),(33,35,37,39,41,43,45,47),(34,36,38,40,42,44,46,48),(49,59,53,63,57,51,61,55),(50,60,54,64,58,52,62,56),(65,75,69,79,73,67,77,71),(66,76,70,80,74,68,78,72),(81,83,85,87,89,91,93,95),(82,84,86,88,90,92,94,96)], [(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)], [(1,18,13,22,9,26,5,30),(2,17,14,21,10,25,6,29),(3,32,15,20,11,24,7,28),(4,31,16,19,12,23,8,27),(33,67,37,79,41,75,45,71),(34,66,38,78,42,74,46,70),(35,65,39,77,43,73,47,69),(36,80,40,76,44,72,48,68),(49,83,61,87,57,91,53,95),(50,82,62,86,58,90,54,94),(51,81,63,85,59,89,55,93),(52,96,64,84,60,88,56,92)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 16A | ··· | 16H | 24A | ··· | 24H | 24I | ··· | 24P | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | Q8 | D4 | D8 | Q16 | C3×Q8 | C3×D4 | C3×D8 | C3×Q16 | C8.4Q8 | C3×C8.4Q8 |
| kernel | C3×C8.4Q8 | C3×C8.C4 | C2×C48 | C8.4Q8 | C48 | C8.C4 | C2×C16 | C16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C3×C8.4Q8 ►in GL2(𝔽97) generated by
| 61 | 0 |
| 0 | 61 |
| 47 | 0 |
| 0 | 33 |
| 70 | 0 |
| 0 | 79 |
| 0 | 1 |
| 22 | 0 |
G:=sub<GL(2,GF(97))| [61,0,0,61],[47,0,0,33],[70,0,0,79],[0,22,1,0] >;
C3×C8.4Q8 in GAP, Magma, Sage, TeX
C_3\times C_8._4Q_8
% in TeX
G:=Group("C3xC8.4Q8"); // GroupNames label
G:=SmallGroup(192,174);
// by ID
G=gap.SmallGroup(192,174);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,428,1683,360,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations
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