direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×SD32, C80⋊6C2, D8.C10, C16⋊2C10, Q16⋊1C10, C10.16D8, C20.37D4, C40.25C22, C2.4(C5×D8), C4.2(C5×D4), C8.3(C2×C10), (C5×Q16)⋊5C2, (C5×D8).2C2, SmallGroup(160,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×SD32
G = < a,b,c | a5=b16=c2=1, ab=ba, ac=ca, cbc=b7 >
Smallest permutation representation of C5×SD32
►On 80 points
Generators in S80
(1 42 74 21 55)(2 43 75 22 56)(3 44 76 23 57)(4 45 77 24 58)(5 46 78 25 59)(6 47 79 26 60)(7 48 80 27 61)(8 33 65 28 62)(9 34 66 29 63)(10 35 67 30 64)(11 36 68 31 49)(12 37 69 32 50)(13 38 70 17 51)(14 39 71 18 52)(15 40 72 19 53)(16 41 73 20 54)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 25)(18 32)(19 23)(20 30)(22 28)(24 26)(27 31)(33 43)(35 41)(36 48)(37 39)(38 46)(40 44)(45 47)(49 61)(50 52)(51 59)(53 57)(54 64)(56 62)(58 60)(65 75)(67 73)(68 80)(69 71)(70 78)(72 76)(77 79)
G:=sub<Sym(80)| (1,42,74,21,55)(2,43,75,22,56)(3,44,76,23,57)(4,45,77,24,58)(5,46,78,25,59)(6,47,79,26,60)(7,48,80,27,61)(8,33,65,28,62)(9,34,66,29,63)(10,35,67,30,64)(11,36,68,31,49)(12,37,69,32,50)(13,38,70,17,51)(14,39,71,18,52)(15,40,72,19,53)(16,41,73,20,54), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79)>;
G:=Group( (1,42,74,21,55)(2,43,75,22,56)(3,44,76,23,57)(4,45,77,24,58)(5,46,78,25,59)(6,47,79,26,60)(7,48,80,27,61)(8,33,65,28,62)(9,34,66,29,63)(10,35,67,30,64)(11,36,68,31,49)(12,37,69,32,50)(13,38,70,17,51)(14,39,71,18,52)(15,40,72,19,53)(16,41,73,20,54), (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), (2,8)(3,15)(4,6)(5,13)(7,11)(10,16)(12,14)(17,25)(18,32)(19,23)(20,30)(22,28)(24,26)(27,31)(33,43)(35,41)(36,48)(37,39)(38,46)(40,44)(45,47)(49,61)(50,52)(51,59)(53,57)(54,64)(56,62)(58,60)(65,75)(67,73)(68,80)(69,71)(70,78)(72,76)(77,79) );
G=PermutationGroup([[(1,42,74,21,55),(2,43,75,22,56),(3,44,76,23,57),(4,45,77,24,58),(5,46,78,25,59),(6,47,79,26,60),(7,48,80,27,61),(8,33,65,28,62),(9,34,66,29,63),(10,35,67,30,64),(11,36,68,31,49),(12,37,69,32,50),(13,38,70,17,51),(14,39,71,18,52),(15,40,72,19,53),(16,41,73,20,54)], [(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)], [(2,8),(3,15),(4,6),(5,13),(7,11),(10,16),(12,14),(17,25),(18,32),(19,23),(20,30),(22,28),(24,26),(27,31),(33,43),(35,41),(36,48),(37,39),(38,46),(40,44),(45,47),(49,61),(50,52),(51,59),(53,57),(54,64),(56,62),(58,60),(65,75),(67,73),(68,80),(69,71),(70,78),(72,76),(77,79)]])
C5×SD32 is a maximal subgroup of
C16⋊D10 SD32⋊D5 SD32⋊3D5
55 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H | 80A | ··· | 80P |
| order | 1 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 8 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | D4 | D8 | SD32 | C5×D4 | C5×D8 | C5×SD32 |
| kernel | C5×SD32 | C80 | C5×D8 | C5×Q16 | SD32 | C16 | D8 | Q16 | C20 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 1 | 2 | 4 | 4 | 8 | 16 |
Matrix representation of C5×SD32 ►in GL2(𝔽241) generated by
| 98 | 0 |
| 0 | 98 |
| 103 | 41 |
| 200 | 103 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(241))| [98,0,0,98],[103,200,41,103],[0,1,1,0] >;
C5×SD32 in GAP, Magma, Sage, TeX
C_5\times {\rm SD}_{32} % in TeX
G:=Group("C5xSD32"); // GroupNames label
G:=SmallGroup(160,62);
// by ID
G=gap.SmallGroup(160,62);
# by ID
G:=PCGroup([6,-2,-2,-5,-2,-2,-2,480,265,1443,729,165,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c|a^5=b^16=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^7>;
// generators/relations
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