direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C8.C22, Q16⋊3D10, SD16⋊5D10, C40.4C23, C20.23C24, M4(2)⋊11D10, Dic20⋊3C22, D20.16C23, Dic10.16C23, (D5×Q16)⋊1C2, (C2×Q8)⋊22D10, (D5×SD16)⋊3C2, C4.191(D4×D5), Q8⋊D5⋊5C22, C8.4(C22×D5), Q16⋊D5⋊1C2, C4○D4.29D10, (C4×D5).100D4, C8.D10⋊3C2, C20.244(C2×D4), SD16⋊D5⋊3C2, C40⋊C2⋊5C22, C8⋊D5⋊5C22, (D5×M4(2))⋊3C2, D4.D5⋊6C22, (Q8×D5)⋊11C22, (C5×Q16)⋊1C22, C5⋊Q16⋊4C22, (C8×D5).1C22, C22.48(D4×D5), C4.23(C23×D5), D10.116(C2×D4), C20.C23⋊9C2, C5⋊2C8.11C23, (C2×Dic5).90D4, (Q8×C10)⋊20C22, (C5×SD16)⋊5C22, D4.16(C22×D5), (C5×D4).16C23, (D4×D5).11C22, (C4×D5).15C23, D4.9D10⋊10C2, Q8.16(C22×D5), (C5×Q8).16C23, (C2×C20).114C23, Dic5.101(C2×D4), C4○D20.30C22, (C22×D5).139D4, C10.124(C22×D4), (C5×M4(2))⋊5C22, C4.Dic5⋊14C22, (C2×Dic10)⋊41C22, D4⋊2D5.10C22, Q8⋊2D5.10C22, (C2×Q8×D5)⋊17C2, C2.97(C2×D4×D5), C5⋊4(C2×C8.C22), (D5×C4○D4).4C2, (C2×C10).69(C2×D4), (C5×C8.C22)⋊1C2, (C2×C4×D5).171C22, (C2×C4).98(C22×D5), (C5×C4○D4).25C22, SmallGroup(320,1448)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C8.C22
G = < a,b,c,d,e | a5=b2=c8=d2=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >
Subgroups: 942 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), SD16, SD16, Q16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C8.C22, C22×Q8, C2×C4○D4, C5⋊2C8, C40, Dic10, 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, C5×Q8, C22×D5, C22×D5, C2×C8.C22, C8×D5, C8⋊D5, C40⋊C2, Dic20, C4.Dic5, D4.D5, Q8⋊D5, C5⋊Q16, C5×M4(2), C5×SD16, C5×Q16, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8×D5, Q8×D5, Q8⋊2D5, Q8×C10, C5×C4○D4, D5×M4(2), C8.D10, D5×SD16, SD16⋊D5, D5×Q16, Q16⋊D5, C20.C23, D4.9D10, C5×C8.C22, C2×Q8×D5, D5×C4○D4, D5×C8.C22
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8.C22, C22×D4, C22×D5, C2×C8.C22, D4×D5, C23×D5, C2×D4×D5, D5×C8.C22
►On 80 points
(1 75 67 37 14)(2 76 68 38 15)(3 77 69 39 16)(4 78 70 40 9)(5 79 71 33 10)(6 80 72 34 11)(7 73 65 35 12)(8 74 66 36 13)(17 26 64 52 47)(18 27 57 53 48)(19 28 58 54 41)(20 29 59 55 42)(21 30 60 56 43)(22 31 61 49 44)(23 32 62 50 45)(24 25 63 51 46)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 63)(26 64)(27 57)(28 58)(29 59)(30 60)(31 61)(32 62)(33 79)(34 80)(35 73)(36 74)(37 75)(38 76)(39 77)(40 78)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(34 36)(35 39)(38 40)(41 43)(42 46)(45 47)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)
(1 63)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(9 32)(10 29)(11 26)(12 31)(13 28)(14 25)(15 30)(16 27)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(41 66)(42 71)(43 68)(44 65)(45 70)(46 67)(47 72)(48 69)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)
G:=sub<Sym(80)| (1,75,67,37,14)(2,76,68,38,15)(3,77,69,39,16)(4,78,70,40,9)(5,79,71,33,10)(6,80,72,34,11)(7,73,65,35,12)(8,74,66,36,13)(17,26,64,52,47)(18,27,57,53,48)(19,28,58,54,41)(20,29,59,55,42)(21,30,60,56,43)(22,31,61,49,44)(23,32,62,50,45)(24,25,63,51,46), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,79)(34,80)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76)>;
G:=Group( (1,75,67,37,14)(2,76,68,38,15)(3,77,69,39,16)(4,78,70,40,9)(5,79,71,33,10)(6,80,72,34,11)(7,73,65,35,12)(8,74,66,36,13)(17,26,64,52,47)(18,27,57,53,48)(19,28,58,54,41)(20,29,59,55,42)(21,30,60,56,43)(22,31,61,49,44)(23,32,62,50,45)(24,25,63,51,46), (1,14)(2,15)(3,16)(4,9)(5,10)(6,11)(7,12)(8,13)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,63)(26,64)(27,57)(28,58)(29,59)(30,60)(31,61)(32,62)(33,79)(34,80)(35,73)(36,74)(37,75)(38,76)(39,77)(40,78), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(34,36)(35,39)(38,40)(41,43)(42,46)(45,47)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(9,32)(10,29)(11,26)(12,31)(13,28)(14,25)(15,30)(16,27)(17,34)(18,39)(19,36)(20,33)(21,38)(22,35)(23,40)(24,37)(41,66)(42,71)(43,68)(44,65)(45,70)(46,67)(47,72)(48,69)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76) );
G=PermutationGroup([[(1,75,67,37,14),(2,76,68,38,15),(3,77,69,39,16),(4,78,70,40,9),(5,79,71,33,10),(6,80,72,34,11),(7,73,65,35,12),(8,74,66,36,13),(17,26,64,52,47),(18,27,57,53,48),(19,28,58,54,41),(20,29,59,55,42),(21,30,60,56,43),(22,31,61,49,44),(23,32,62,50,45),(24,25,63,51,46)], [(1,14),(2,15),(3,16),(4,9),(5,10),(6,11),(7,12),(8,13),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(25,63),(26,64),(27,57),(28,58),(29,59),(30,60),(31,61),(32,62),(33,79),(34,80),(35,73),(36,74),(37,75),(38,76),(39,77),(40,78)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,29),(26,32),(28,30),(34,36),(35,39),(38,40),(41,43),(42,46),(45,47),(50,52),(51,55),(54,56),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70),(73,77),(74,80),(76,78)], [(1,63),(2,60),(3,57),(4,62),(5,59),(6,64),(7,61),(8,58),(9,32),(10,29),(11,26),(12,31),(13,28),(14,25),(15,30),(16,27),(17,34),(18,39),(19,36),(20,33),(21,38),(22,35),(23,40),(24,37),(41,66),(42,71),(43,68),(44,65),(45,70),(46,67),(47,72),(48,69),(49,73),(50,78),(51,75),(52,80),(53,77),(54,74),(55,79),(56,76)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 20 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D4 | D5 | D10 | D10 | D10 | D10 | D10 | C8.C22 | D4×D5 | D4×D5 | D5×C8.C22 |
| kernel | D5×C8.C22 | D5×M4(2) | C8.D10 | D5×SD16 | SD16⋊D5 | D5×Q16 | Q16⋊D5 | C20.C23 | D4.9D10 | C5×C8.C22 | C2×Q8×D5 | D5×C4○D4 | C4×D5 | C2×Dic5 | C22×D5 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | D5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of D5×C8.C22 ►in GL8(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 31 | 39 | 4 |
| 0 | 0 | 0 | 0 | 9 | 26 | 0 | 10 |
| 0 | 0 | 0 | 0 | 2 | 30 | 0 | 21 |
| 0 | 0 | 0 | 0 | 36 | 3 | 10 | 36 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 38 | 40 | 2 |
| 0 | 0 | 0 | 0 | 28 | 38 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 39 | 20 | 21 | 20 |
| 0 | 0 | 0 | 0 | 31 | 16 | 32 | 18 |
| 0 | 0 | 0 | 0 | 20 | 31 | 39 | 4 |
| 0 | 0 | 0 | 0 | 36 | 17 | 5 | 29 |
G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,20,9,2,36,0,0,0,0,31,26,30,3,0,0,0,0,39,0,0,10,0,0,0,0,4,10,21,36],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,5,0,28,0,0,0,0,0,40,38,38,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,39,31,20,36,0,0,0,0,20,16,31,17,0,0,0,0,21,32,39,5,0,0,0,0,20,18,4,29] >;
D5×C8.C22 in GAP, Magma, Sage, TeX
D_5\times C_8.C_2^2
% in TeX
G:=Group("D5xC8.C2^2"); // GroupNames label
G:=SmallGroup(320,1448);
// by ID
G=gap.SmallGroup(320,1448);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,570,185,438,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^8=d^2=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
// generators/relations