direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D5×SD16, C40⋊6C23, C20.5C24, D20.3C23, Dic10⋊2C23, (C2×C8)⋊28D10, C4.42(D4×D5), C8⋊6(C22×D5), C5⋊2C8⋊7C23, (C2×Q8)⋊20D10, (C4×D5).67D4, C10⋊2(C2×SD16), C20.80(C2×D4), Q8⋊D5⋊7C22, (Q8×D5)⋊5C22, Q8⋊1(C22×D5), (C5×Q8)⋊1C23, C4.5(C23×D5), C5⋊2(C22×SD16), (C2×C40)⋊18C22, (C8×D5)⋊17C22, D4.D5⋊9C22, D4.3(C22×D5), (D4×D5).9C22, (C5×D4).3C23, (C10×SD16)⋊10C2, D10.112(C2×D4), (C2×D4).181D10, C40⋊C2⋊17C22, Dic5.24(C2×D4), (Q8×C10)⋊17C22, (C4×D5).61C23, C22.138(D4×D5), (C2×C20).522C23, (C2×Dic5).167D4, (C5×SD16)⋊12C22, (C22×D5).159D4, C10.106(C22×D4), (C2×Dic10)⋊37C22, (D4×C10).163C22, (C2×D20).183C22, (D5×C2×C8)⋊9C2, (C2×Q8×D5)⋊14C2, C2.79(C2×D4×D5), (C2×D4×D5).12C2, (C2×Q8⋊D5)⋊25C2, (C2×C40⋊C2)⋊31C2, (C2×D4.D5)⋊27C2, (C2×C5⋊2C8)⋊36C22, (C2×C10).395(C2×D4), (C2×C4×D5).327C22, (C2×C4).611(C22×D5), SmallGroup(320,1430)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D5×SD16
G = < a,b,c,d,e | a2=b5=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
Subgroups: 1278 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C10, C2×C8, C2×C8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C22×C8, C2×SD16, C2×SD16, C22×D4, C22×Q8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C4×D5, D20, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×D5, C22×D5, C22×C10, C22×SD16, C8×D5, C40⋊C2, C2×C5⋊2C8, D4.D5, Q8⋊D5, C2×C40, C5×SD16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5, D4×D5, Q8×D5, Q8×D5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, D5×C2×C8, C2×C40⋊C2, D5×SD16, C2×D4.D5, C2×Q8⋊D5, C10×SD16, C2×D4×D5, C2×Q8×D5, C2×D5×SD16
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C24, D10, C2×SD16, C22×D4, C22×D5, C22×SD16, D4×D5, C23×D5, D5×SD16, C2×D4×D5, C2×D5×SD16
►On 80 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(57 78)(58 79)(59 80)(60 73)(61 74)(62 75)(63 76)(64 77)
(1 49 46 63 68)(2 50 47 64 69)(3 51 48 57 70)(4 52 41 58 71)(5 53 42 59 72)(6 54 43 60 65)(7 55 44 61 66)(8 56 45 62 67)(9 24 38 26 73)(10 17 39 27 74)(11 18 40 28 75)(12 19 33 29 76)(13 20 34 30 77)(14 21 35 31 78)(15 22 36 32 79)(16 23 37 25 80)
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 66)(18 67)(19 68)(20 69)(21 70)(22 71)(23 72)(24 65)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 41)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)
(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 19)(2 22)(3 17)(4 20)(5 23)(6 18)(7 21)(8 24)(9 67)(10 70)(11 65)(12 68)(13 71)(14 66)(15 69)(16 72)(25 42)(26 45)(27 48)(28 43)(29 46)(30 41)(31 44)(32 47)(33 49)(34 52)(35 55)(36 50)(37 53)(38 56)(39 51)(40 54)(57 74)(58 77)(59 80)(60 75)(61 78)(62 73)(63 76)(64 79)
G:=sub<Sym(80)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(57,78)(58,79)(59,80)(60,73)(61,74)(62,75)(63,76)(64,77), (1,49,46,63,68)(2,50,47,64,69)(3,51,48,57,70)(4,52,41,58,71)(5,53,42,59,72)(6,54,43,60,65)(7,55,44,61,66)(8,56,45,62,67)(9,24,38,26,73)(10,17,39,27,74)(11,18,40,28,75)(12,19,33,29,76)(13,20,34,30,77)(14,21,35,31,78)(15,22,36,32,79)(16,23,37,25,80), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,67)(10,70)(11,65)(12,68)(13,71)(14,66)(15,69)(16,72)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(57,74)(58,77)(59,80)(60,75)(61,78)(62,73)(63,76)(64,79)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,49)(34,50)(35,51)(36,52)(37,53)(38,54)(39,55)(40,56)(57,78)(58,79)(59,80)(60,73)(61,74)(62,75)(63,76)(64,77), (1,49,46,63,68)(2,50,47,64,69)(3,51,48,57,70)(4,52,41,58,71)(5,53,42,59,72)(6,54,43,60,65)(7,55,44,61,66)(8,56,45,62,67)(9,24,38,26,73)(10,17,39,27,74)(11,18,40,28,75)(12,19,33,29,76)(13,20,34,30,77)(14,21,35,31,78)(15,22,36,32,79)(16,23,37,25,80), (1,12)(2,13)(3,14)(4,15)(5,16)(6,9)(7,10)(8,11)(17,66)(18,67)(19,68)(20,69)(21,70)(22,71)(23,72)(24,65)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,41)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,19)(2,22)(3,17)(4,20)(5,23)(6,18)(7,21)(8,24)(9,67)(10,70)(11,65)(12,68)(13,71)(14,66)(15,69)(16,72)(25,42)(26,45)(27,48)(28,43)(29,46)(30,41)(31,44)(32,47)(33,49)(34,52)(35,55)(36,50)(37,53)(38,56)(39,51)(40,54)(57,74)(58,77)(59,80)(60,75)(61,78)(62,73)(63,76)(64,79) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41),(33,49),(34,50),(35,51),(36,52),(37,53),(38,54),(39,55),(40,56),(57,78),(58,79),(59,80),(60,73),(61,74),(62,75),(63,76),(64,77)], [(1,49,46,63,68),(2,50,47,64,69),(3,51,48,57,70),(4,52,41,58,71),(5,53,42,59,72),(6,54,43,60,65),(7,55,44,61,66),(8,56,45,62,67),(9,24,38,26,73),(10,17,39,27,74),(11,18,40,28,75),(12,19,33,29,76),(13,20,34,30,77),(14,21,35,31,78),(15,22,36,32,79),(16,23,37,25,80)], [(1,12),(2,13),(3,14),(4,15),(5,16),(6,9),(7,10),(8,11),(17,66),(18,67),(19,68),(20,69),(21,70),(22,71),(23,72),(24,65),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,41),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75)], [(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)], [(1,19),(2,22),(3,17),(4,20),(5,23),(6,18),(7,21),(8,24),(9,67),(10,70),(11,65),(12,68),(13,71),(14,66),(15,69),(16,72),(25,42),(26,45),(27,48),(28,43),(29,46),(30,41),(31,44),(32,47),(33,49),(34,52),(35,55),(36,50),(37,53),(38,56),(39,51),(40,54),(57,74),(58,77),(59,80),(60,75),(61,78),(62,73),(63,76),(64,79)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 5 | 5 | 5 | 5 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 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 | C2 | D4 | D4 | D4 | D5 | SD16 | D10 | D10 | D10 | D10 | D4×D5 | D4×D5 | D5×SD16 |
| kernel | C2×D5×SD16 | D5×C2×C8 | C2×C40⋊C2 | D5×SD16 | C2×D4.D5 | C2×Q8⋊D5 | C10×SD16 | C2×D4×D5 | C2×Q8×D5 | C4×D5 | C2×Dic5 | C22×D5 | C2×SD16 | D10 | C2×C8 | SD16 | C2×D4 | C2×Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 8 | 2 | 8 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C2×D5×SD16 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 |
| 40 | 34 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 30 | 12 |
| 0 | 0 | 24 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 23 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,30,24,0,0,12,0],[40,0,0,0,0,40,0,0,0,0,40,23,0,0,0,1] >;
C2×D5×SD16 in GAP, Magma, Sage, TeX
C_2\times D_5\times {\rm SD}_{16} % in TeX
G:=Group("C2xD5xSD16"); // GroupNames label
G:=SmallGroup(320,1430);
// by ID
G=gap.SmallGroup(320,1430);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,185,136,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^5=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations