direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C8.17D4, C24.91D4, C12.63D8, Q16.2C12, M5(2).3C6, C8.3(C2×C12), C8.17(C3×D4), C4.12(C3×D8), C24.38(C2×C4), (C3×Q16).4C4, (C2×Q16).5C6, C8.C4.1C6, (C2×C12).281D4, (C6×Q16).12C2, (C2×C6).25SD16, (C3×M5(2)).7C2, C6.42(D4⋊C4), C22.4(C3×SD16), C12.74(C22⋊C4), (C2×C24).195C22, (C2×C8).14(C2×C6), (C2×C4).12(C3×D4), C4.6(C3×C22⋊C4), (C3×C8.C4).4C2, C2.11(C3×D4⋊C4), SmallGroup(192,168)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.17D4
G = < a,b,c,d | a3=b8=1, c4=b4, d2=cbc-1=b-1, ab=ba, ac=ca, ad=da, bd=db, dcd-1=b-1c3 >
Smallest permutation representation of C3×C8.17D4
►On 96 points
Generators in S96
(1 96 45)(2 81 46)(3 82 47)(4 83 48)(5 84 33)(6 85 34)(7 86 35)(8 87 36)(9 88 37)(10 89 38)(11 90 39)(12 91 40)(13 92 41)(14 93 42)(15 94 43)(16 95 44)(17 61 66)(18 62 67)(19 63 68)(20 64 69)(21 49 70)(22 50 71)(23 51 72)(24 52 73)(25 53 74)(26 54 75)(27 55 76)(28 56 77)(29 57 78)(30 58 79)(31 59 80)(32 60 65)
(1 15 13 11 9 7 5 3)(2 16 14 12 10 8 6 4)(17 31 29 27 25 23 21 19)(18 32 30 28 26 24 22 20)(33 47 45 43 41 39 37 35)(34 48 46 44 42 40 38 36)(49 63 61 59 57 55 53 51)(50 64 62 60 58 56 54 52)(65 79 77 75 73 71 69 67)(66 80 78 76 74 72 70 68)(81 95 93 91 89 87 85 83)(82 96 94 92 90 88 86 84)
(1 63 13 51 9 55 5 59)(2 50 6 62 10 58 14 54)(3 61 15 49 11 53 7 57)(4 64 8 60 12 56 16 52)(17 43 21 39 25 35 29 47)(18 38 30 42 26 46 22 34)(19 41 23 37 27 33 31 45)(20 36 32 40 28 44 24 48)(65 91 77 95 73 83 69 87)(66 94 70 90 74 86 78 82)(67 89 79 93 75 81 71 85)(68 92 72 88 76 84 80 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)
G:=sub<Sym(96)| (1,96,45)(2,81,46)(3,82,47)(4,83,48)(5,84,33)(6,85,34)(7,86,35)(8,87,36)(9,88,37)(10,89,38)(11,90,39)(12,91,40)(13,92,41)(14,93,42)(15,94,43)(16,95,44)(17,61,66)(18,62,67)(19,63,68)(20,64,69)(21,49,70)(22,50,71)(23,51,72)(24,52,73)(25,53,74)(26,54,75)(27,55,76)(28,56,77)(29,57,78)(30,58,79)(31,59,80)(32,60,65), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84), (1,63,13,51,9,55,5,59)(2,50,6,62,10,58,14,54)(3,61,15,49,11,53,7,57)(4,64,8,60,12,56,16,52)(17,43,21,39,25,35,29,47)(18,38,30,42,26,46,22,34)(19,41,23,37,27,33,31,45)(20,36,32,40,28,44,24,48)(65,91,77,95,73,83,69,87)(66,94,70,90,74,86,78,82)(67,89,79,93,75,81,71,85)(68,92,72,88,76,84,80,96), (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)>;
G:=Group( (1,96,45)(2,81,46)(3,82,47)(4,83,48)(5,84,33)(6,85,34)(7,86,35)(8,87,36)(9,88,37)(10,89,38)(11,90,39)(12,91,40)(13,92,41)(14,93,42)(15,94,43)(16,95,44)(17,61,66)(18,62,67)(19,63,68)(20,64,69)(21,49,70)(22,50,71)(23,51,72)(24,52,73)(25,53,74)(26,54,75)(27,55,76)(28,56,77)(29,57,78)(30,58,79)(31,59,80)(32,60,65), (1,15,13,11,9,7,5,3)(2,16,14,12,10,8,6,4)(17,31,29,27,25,23,21,19)(18,32,30,28,26,24,22,20)(33,47,45,43,41,39,37,35)(34,48,46,44,42,40,38,36)(49,63,61,59,57,55,53,51)(50,64,62,60,58,56,54,52)(65,79,77,75,73,71,69,67)(66,80,78,76,74,72,70,68)(81,95,93,91,89,87,85,83)(82,96,94,92,90,88,86,84), (1,63,13,51,9,55,5,59)(2,50,6,62,10,58,14,54)(3,61,15,49,11,53,7,57)(4,64,8,60,12,56,16,52)(17,43,21,39,25,35,29,47)(18,38,30,42,26,46,22,34)(19,41,23,37,27,33,31,45)(20,36,32,40,28,44,24,48)(65,91,77,95,73,83,69,87)(66,94,70,90,74,86,78,82)(67,89,79,93,75,81,71,85)(68,92,72,88,76,84,80,96), (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) );
G=PermutationGroup([[(1,96,45),(2,81,46),(3,82,47),(4,83,48),(5,84,33),(6,85,34),(7,86,35),(8,87,36),(9,88,37),(10,89,38),(11,90,39),(12,91,40),(13,92,41),(14,93,42),(15,94,43),(16,95,44),(17,61,66),(18,62,67),(19,63,68),(20,64,69),(21,49,70),(22,50,71),(23,51,72),(24,52,73),(25,53,74),(26,54,75),(27,55,76),(28,56,77),(29,57,78),(30,58,79),(31,59,80),(32,60,65)], [(1,15,13,11,9,7,5,3),(2,16,14,12,10,8,6,4),(17,31,29,27,25,23,21,19),(18,32,30,28,26,24,22,20),(33,47,45,43,41,39,37,35),(34,48,46,44,42,40,38,36),(49,63,61,59,57,55,53,51),(50,64,62,60,58,56,54,52),(65,79,77,75,73,71,69,67),(66,80,78,76,74,72,70,68),(81,95,93,91,89,87,85,83),(82,96,94,92,90,88,86,84)], [(1,63,13,51,9,55,5,59),(2,50,6,62,10,58,14,54),(3,61,15,49,11,53,7,57),(4,64,8,60,12,56,16,52),(17,43,21,39,25,35,29,47),(18,38,30,42,26,46,22,34),(19,41,23,37,27,33,31,45),(20,36,32,40,28,44,24,48),(65,91,77,95,73,83,69,87),(66,94,70,90,74,86,78,82),(67,89,79,93,75,81,71,85),(68,92,72,88,76,84,80,96)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 24I | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D4 | D8 | SD16 | C3×D4 | C3×D4 | C3×D8 | C3×SD16 | C8.17D4 | C3×C8.17D4 |
| kernel | C3×C8.17D4 | C3×C8.C4 | C3×M5(2) | C6×Q16 | C8.17D4 | C3×Q16 | C8.C4 | M5(2) | C2×Q16 | Q16 | C24 | C2×C12 | C12 | C2×C6 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 |
Matrix representation of C3×C8.17D4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 5 | 0 | 2 | 6 |
| 6 | 5 | 5 | 6 |
| 6 | 1 | 1 | 2 |
| 2 | 2 | 6 | 2 |
| 1 | 1 | 5 | 0 |
| 4 | 5 | 3 | 3 |
| 6 | 6 | 3 | 2 |
| 6 | 3 | 0 | 5 |
| 4 | 5 | 1 | 6 |
| 4 | 1 | 4 | 3 |
| 2 | 3 | 0 | 6 |
| 5 | 0 | 5 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[5,6,6,2,0,5,1,2,2,5,1,6,6,6,2,2],[1,4,6,6,1,5,6,3,5,3,3,0,0,3,2,5],[4,4,2,5,5,1,3,0,1,4,0,5,6,3,6,2] >;
C3×C8.17D4 in GAP, Magma, Sage, TeX
C_3\times C_8._{17}D_4 % in TeX
G:=Group("C3xC8.17D4"); // GroupNames label
G:=SmallGroup(192,168);
// by ID
G=gap.SmallGroup(192,168);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,680,1683,2194,136,2111,172,6053,3036,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^4=b^4,d^2=c*b*c^-1=b^-1,a*b=b*a,a*c=c*a,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations
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