direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C10×SD16, C20.42D4, C40⋊13C22, C20.45C23, (C2×C8)⋊5C10, C8⋊3(C2×C10), C4.7(C5×D4), (C2×C40)⋊13C2, (C2×Q8)⋊3C10, Q8⋊1(C2×C10), (Q8×C10)⋊10C2, D4.1(C2×C10), (C2×D4).6C10, C10.75(C2×D4), (C2×C10).53D4, C2.12(D4×C10), (C5×Q8)⋊9C22, (D4×C10).13C2, C4.2(C22×C10), C22.15(C5×D4), (C5×D4).11C22, (C2×C20).130C22, (C2×C4).26(C2×C10), SmallGroup(160,194)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×SD16
G = < a,b,c | a10=b8=c2=1, ab=ba, ac=ca, cbc=b3 >
Subgroups: 108 in 68 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C10, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×SD16, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, C2×C40, C5×SD16, D4×C10, Q8×C10, C10×SD16
Quotients: C1, C2, C22, C5, D4, C23, C10, SD16, C2×D4, C2×C10, C2×SD16, C5×D4, C22×C10, C5×SD16, D4×C10, C10×SD16
►On 80 points
(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)
(1 31 66 11 60 28 50 79)(2 32 67 12 51 29 41 80)(3 33 68 13 52 30 42 71)(4 34 69 14 53 21 43 72)(5 35 70 15 54 22 44 73)(6 36 61 16 55 23 45 74)(7 37 62 17 56 24 46 75)(8 38 63 18 57 25 47 76)(9 39 64 19 58 26 48 77)(10 40 65 20 59 27 49 78)
(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 71)(41 67)(42 68)(43 69)(44 70)(45 61)(46 62)(47 63)(48 64)(49 65)(50 66)
G:=sub<Sym(80)| (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), (1,31,66,11,60,28,50,79)(2,32,67,12,51,29,41,80)(3,33,68,13,52,30,42,71)(4,34,69,14,53,21,43,72)(5,35,70,15,54,22,44,73)(6,36,61,16,55,23,45,74)(7,37,62,17,56,24,46,75)(8,38,63,18,57,25,47,76)(9,39,64,19,58,26,48,77)(10,40,65,20,59,27,49,78), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66)>;
G:=Group( (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), (1,31,66,11,60,28,50,79)(2,32,67,12,51,29,41,80)(3,33,68,13,52,30,42,71)(4,34,69,14,53,21,43,72)(5,35,70,15,54,22,44,73)(6,36,61,16,55,23,45,74)(7,37,62,17,56,24,46,75)(8,38,63,18,57,25,47,76)(9,39,64,19,58,26,48,77)(10,40,65,20,59,27,49,78), (11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,71)(41,67)(42,68)(43,69)(44,70)(45,61)(46,62)(47,63)(48,64)(49,65)(50,66) );
G=PermutationGroup([[(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)], [(1,31,66,11,60,28,50,79),(2,32,67,12,51,29,41,80),(3,33,68,13,52,30,42,71),(4,34,69,14,53,21,43,72),(5,35,70,15,54,22,44,73),(6,36,61,16,55,23,45,74),(7,37,62,17,56,24,46,75),(8,38,63,18,57,25,47,76),(9,39,64,19,58,26,48,77),(10,40,65,20,59,27,49,78)], [(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,71),(41,67),(42,68),(43,69),(44,70),(45,61),(46,62),(47,63),(48,64),(49,65),(50,66)]])
C10×SD16 is a maximal subgroup of
Dic5⋊3SD16 Dic5⋊5SD16 SD16⋊Dic5 (C5×D4).D4 (C5×Q8).D4 C40.31D4 C40.43D4 D10⋊6SD16 D10⋊8SD16 C40⋊14D4 D20⋊7D4 Dic10.16D4 C40⋊8D4 C40⋊15D4 C40⋊9D4 C40.44D4 D20.29D4
70 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 10A | ··· | 10L | 10M | ··· | 10T | 20A | ··· | 20H | 20I | ··· | 20P | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
70 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | D4 | D4 | SD16 | C5×D4 | C5×D4 | C5×SD16 |
| kernel | C10×SD16 | C2×C40 | C5×SD16 | D4×C10 | Q8×C10 | C2×SD16 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C20 | C2×C10 | C10 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 16 | 4 | 4 | 1 | 1 | 4 | 4 | 4 | 16 |
Matrix representation of C10×SD16 ►in GL4(𝔽41) generated by
| 31 | 0 | 0 | 0 |
| 0 | 31 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 11 | 21 | 0 | 0 |
| 2 | 30 | 0 | 0 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 26 | 26 |
| 1 | 30 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [31,0,0,0,0,31,0,0,0,0,40,0,0,0,0,40],[11,2,0,0,21,30,0,0,0,0,26,26,0,0,15,26],[1,0,0,0,30,40,0,0,0,0,1,0,0,0,0,40] >;
C10×SD16 in GAP, Magma, Sage, TeX
C_{10}\times {\rm SD}_{16} % in TeX
G:=Group("C10xSD16"); // GroupNames label
G:=SmallGroup(160,194);
// by ID
G=gap.SmallGroup(160,194);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,480,505,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^3>;
// generators/relations