direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×SD32, C16⋊5D10, C80⋊5C22, Q16⋊1D10, D8.2D10, D10.25D8, Dic5.8D8, C40.16C23, D40.2C22, Dic20⋊5C22, C4.4(D4×D5), C5⋊2(C2×SD32), (D5×C16)⋊4C2, D8.D5⋊3C2, (D5×Q16)⋊3C2, (D5×D8).1C2, C16⋊D5⋊5C2, C2.19(D5×D8), (C5×SD32)⋊3C2, (C4×D5).59D4, C10.35(C2×D8), C20.10(C2×D4), C5⋊SD32⋊1C2, C5⋊2C8.25D4, C5⋊2C16⋊6C22, (C5×Q16)⋊4C22, (C5×D8).2C22, C8.22(C22×D5), (C8×D5).40C22, SmallGroup(320,540)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×SD32
G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >
Subgroups: 518 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C23, D5, D5, C10, C10, C16, C16, C2×C8, D8, D8, Q16, Q16, C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C16, SD32, SD32, C2×D8, C2×Q16, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×SD32, C5⋊2C16, C80, C8×D5, D40, Dic20, D4⋊D5, C5⋊Q16, C5×D8, C5×Q16, D4×D5, Q8×D5, D5×C16, C16⋊D5, D8.D5, C5⋊SD32, C5×SD32, D5×D8, D5×Q16, D5×SD32
Quotients: C1, C2, C22, D4, C23, D5, D8, C2×D4, D10, SD32, C2×D8, C22×D5, C2×SD32, D4×D5, D5×D8, D5×SD32
►On 80 points
(1 56 30 37 77)(2 57 31 38 78)(3 58 32 39 79)(4 59 17 40 80)(5 60 18 41 65)(6 61 19 42 66)(7 62 20 43 67)(8 63 21 44 68)(9 64 22 45 69)(10 49 23 46 70)(11 50 24 47 71)(12 51 25 48 72)(13 52 26 33 73)(14 53 27 34 74)(15 54 28 35 75)(16 55 29 36 76)
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 65)(14 66)(15 67)(16 68)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 60)(34 61)(35 62)(36 63)(37 64)(38 49)(39 50)(40 51)(41 52)(42 53)(43 54)(44 55)(45 56)(46 57)(47 58)(48 59)
(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 19)(18 26)(20 24)(21 31)(23 29)(25 27)(28 32)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 73)(66 80)(67 71)(68 78)(70 76)(72 74)(75 79)
G:=sub<Sym(80)| (1,56,30,37,77)(2,57,31,38,78)(3,58,32,39,79)(4,59,17,40,80)(5,60,18,41,65)(6,61,19,42,66)(7,62,20,43,67)(8,63,21,44,68)(9,64,22,45,69)(10,49,23,46,70)(11,50,24,47,71)(12,51,25,48,72)(13,52,26,33,73)(14,53,27,34,74)(15,54,28,35,75)(16,55,29,36,76), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,65)(14,66)(15,67)(16,68)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,60)(34,61)(35,62)(36,63)(37,64)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (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)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,73)(66,80)(67,71)(68,78)(70,76)(72,74)(75,79)>;
G:=Group( (1,56,30,37,77)(2,57,31,38,78)(3,58,32,39,79)(4,59,17,40,80)(5,60,18,41,65)(6,61,19,42,66)(7,62,20,43,67)(8,63,21,44,68)(9,64,22,45,69)(10,49,23,46,70)(11,50,24,47,71)(12,51,25,48,72)(13,52,26,33,73)(14,53,27,34,74)(15,54,28,35,75)(16,55,29,36,76), (1,69)(2,70)(3,71)(4,72)(5,73)(6,74)(7,75)(8,76)(9,77)(10,78)(11,79)(12,80)(13,65)(14,66)(15,67)(16,68)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,60)(34,61)(35,62)(36,63)(37,64)(38,49)(39,50)(40,51)(41,52)(42,53)(43,54)(44,55)(45,56)(46,57)(47,58)(48,59), (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)(17,19)(18,26)(20,24)(21,31)(23,29)(25,27)(28,32)(33,41)(34,48)(35,39)(36,46)(38,44)(40,42)(43,47)(49,55)(50,62)(51,53)(52,60)(54,58)(57,63)(59,61)(65,73)(66,80)(67,71)(68,78)(70,76)(72,74)(75,79) );
G=PermutationGroup([[(1,56,30,37,77),(2,57,31,38,78),(3,58,32,39,79),(4,59,17,40,80),(5,60,18,41,65),(6,61,19,42,66),(7,62,20,43,67),(8,63,21,44,68),(9,64,22,45,69),(10,49,23,46,70),(11,50,24,47,71),(12,51,25,48,72),(13,52,26,33,73),(14,53,27,34,74),(15,54,28,35,75),(16,55,29,36,76)], [(1,69),(2,70),(3,71),(4,72),(5,73),(6,74),(7,75),(8,76),(9,77),(10,78),(11,79),(12,80),(13,65),(14,66),(15,67),(16,68),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,60),(34,61),(35,62),(36,63),(37,64),(38,49),(39,50),(40,51),(41,52),(42,53),(43,54),(44,55),(45,56),(46,57),(47,58),(48,59)], [(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),(17,19),(18,26),(20,24),(21,31),(23,29),(25,27),(28,32),(33,41),(34,48),(35,39),(36,46),(38,44),(40,42),(43,47),(49,55),(50,62),(51,53),(52,60),(54,58),(57,63),(59,61),(65,73),(66,80),(67,71),(68,78),(70,76),(72,74),(75,79)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 20A | 20B | 20C | 20D | 40A | 40B | 40C | 40D | 80A | ··· | 80H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 40 | 40 | 40 | 40 | 80 | ··· | 80 |
| size | 1 | 1 | 5 | 5 | 8 | 40 | 2 | 8 | 10 | 40 | 2 | 2 | 2 | 2 | 10 | 10 | 2 | 2 | 16 | 16 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 16 | 16 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D8 | D8 | D10 | D10 | D10 | SD32 | D4×D5 | D5×D8 | D5×SD32 |
| kernel | D5×SD32 | D5×C16 | C16⋊D5 | D8.D5 | C5⋊SD32 | C5×SD32 | D5×D8 | D5×Q16 | C5⋊2C8 | C4×D5 | SD32 | Dic5 | D10 | C16 | D8 | Q16 | D5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 8 |
Matrix representation of D5×SD32 ►in GL4(𝔽241) generated by
| 0 | 1 | 0 | 0 |
| 240 | 189 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 240 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 62 | 222 |
| 0 | 0 | 102 | 97 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 54 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(241))| [0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,62,102,0,0,222,97],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,54,1] >;
D5×SD32 in GAP, Magma, Sage, TeX
D_5\times {\rm SD}_{32} % in TeX
G:=Group("D5xSD32"); // GroupNames label
G:=SmallGroup(320,540);
// by ID
G=gap.SmallGroup(320,540);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,135,184,346,185,192,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations