direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C8○D4, C20.71C24, C40.48C23, M4(2)⋊27D10, (C2×C8)⋊30D10, (D4×D5).4C4, (Q8×D5).4C4, D4.12(C4×D5), Q8.13(C4×D5), (C2×C40)⋊25C22, C4○D4.42D10, D20.34(C2×C4), D4⋊2D5.4C4, (C8×D5)⋊20C22, Q8⋊2D5.4C4, C4.70(C23×D5), C8.66(C22×D5), C8⋊D5⋊20C22, (D5×M4(2))⋊12C2, D4.Dic5⋊14C2, C10.55(C23×C4), C20.73(C22×C4), C5⋊2C8.43C23, (C4×D5).96C23, D20.2C4⋊14C2, D20.3C4⋊16C2, (C2×C20).513C23, Dic10.36(C2×C4), C4○D20.51C22, D10.24(C22×C4), C4.Dic5⋊26C22, (C5×M4(2))⋊27C22, Dic5.23(C22×C4), C5⋊6(C2×C8○D4), (D5×C2×C8)⋊30C2, C4.38(C2×C4×D5), (C5×C8○D4)⋊8C2, C22.4(C2×C4×D5), C5⋊D4.5(C2×C4), C2.35(D5×C22×C4), (D5×C4○D4).13C2, (C5×D4).30(C2×C4), (C4×D5).54(C2×C4), (C5×Q8).32(C2×C4), (C2×C5⋊2C8)⋊34C22, (C2×C4×D5).324C22, (C2×C10).11(C22×C4), (C5×C4○D4).43C22, (C22×D5).84(C2×C4), (C2×C4).606(C22×D5), (C2×Dic5).118(C2×C4), SmallGroup(320,1421)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C8○D4
G = < a,b,c,d,e | a5=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >
Subgroups: 734 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C22×C8, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, C5⋊2C8, C5⋊2C8, C40, C40, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×C8○D4, C8×D5, C8×D5, C8⋊D5, C2×C5⋊2C8, C4.Dic5, C2×C40, C5×M4(2), C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D5×C2×C8, D20.3C4, D5×M4(2), D20.2C4, D4.Dic5, C5×C8○D4, D5×C4○D4, D5×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C8○D4, C23×C4, C4×D5, C22×D5, C2×C8○D4, C2×C4×D5, C23×D5, D5×C22×C4, D5×C8○D4
►On 80 points
(1 59 56 48 65)(2 60 49 41 66)(3 61 50 42 67)(4 62 51 43 68)(5 63 52 44 69)(6 64 53 45 70)(7 57 54 46 71)(8 58 55 47 72)(9 17 33 30 74)(10 18 34 31 75)(11 19 35 32 76)(12 20 36 25 77)(13 21 37 26 78)(14 22 38 27 79)(15 23 39 28 80)(16 24 40 29 73)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 73)(41 60)(42 61)(43 62)(44 63)(45 64)(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)
(1 19 5 23)(2 20 6 24)(3 21 7 17)(4 22 8 18)(9 67 13 71)(10 68 14 72)(11 69 15 65)(12 70 16 66)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)(33 61 37 57)(34 62 38 58)(35 63 39 59)(36 64 40 60)(41 77 45 73)(42 78 46 74)(43 79 47 75)(44 80 48 76)
(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(73 77)(74 78)(75 79)(76 80)
G:=sub<Sym(80)| (1,59,56,48,65)(2,60,49,41,66)(3,61,50,42,67)(4,62,51,43,68)(5,63,52,44,69)(6,64,53,45,70)(7,57,54,46,71)(8,58,55,47,72)(9,17,33,30,74)(10,18,34,31,75)(11,19,35,32,76)(12,20,36,25,77)(13,21,37,26,78)(14,22,38,27,79)(15,23,39,28,80)(16,24,40,29,73), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(41,60)(42,61)(43,62)(44,63)(45,64)(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), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,67,13,71)(10,68,14,72)(11,69,15,65)(12,70,16,66)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(33,61,37,57)(34,62,38,58)(35,63,39,59)(36,64,40,60)(41,77,45,73)(42,78,46,74)(43,79,47,75)(44,80,48,76), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(73,77)(74,78)(75,79)(76,80)>;
G:=Group( (1,59,56,48,65)(2,60,49,41,66)(3,61,50,42,67)(4,62,51,43,68)(5,63,52,44,69)(6,64,53,45,70)(7,57,54,46,71)(8,58,55,47,72)(9,17,33,30,74)(10,18,34,31,75)(11,19,35,32,76)(12,20,36,25,77)(13,21,37,26,78)(14,22,38,27,79)(15,23,39,28,80)(16,24,40,29,73), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,73)(41,60)(42,61)(43,62)(44,63)(45,64)(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), (1,19,5,23)(2,20,6,24)(3,21,7,17)(4,22,8,18)(9,67,13,71)(10,68,14,72)(11,69,15,65)(12,70,16,66)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(33,61,37,57)(34,62,38,58)(35,63,39,59)(36,64,40,60)(41,77,45,73)(42,78,46,74)(43,79,47,75)(44,80,48,76), (9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(73,77)(74,78)(75,79)(76,80) );
G=PermutationGroup([[(1,59,56,48,65),(2,60,49,41,66),(3,61,50,42,67),(4,62,51,43,68),(5,63,52,44,69),(6,64,53,45,70),(7,57,54,46,71),(8,58,55,47,72),(9,17,33,30,74),(10,18,34,31,75),(11,19,35,32,76),(12,20,36,25,77),(13,21,37,26,78),(14,22,38,27,79),(15,23,39,28,80),(16,24,40,29,73)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,73),(41,60),(42,61),(43,62),(44,63),(45,64),(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)], [(1,19,5,23),(2,20,6,24),(3,21,7,17),(4,22,8,18),(9,67,13,71),(10,68,14,72),(11,69,15,65),(12,70,16,66),(25,53,29,49),(26,54,30,50),(27,55,31,51),(28,56,32,52),(33,61,37,57),(34,62,38,58),(35,63,39,59),(36,64,40,60),(41,77,45,73),(42,78,46,74),(43,79,47,75),(44,80,48,76)], [(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(73,77),(74,78),(75,79),(76,80)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 8K | 8L | 8M | 8N | 8O | ··· | 8T | 10A | 10B | 10C | ··· | 10H | 20A | 20B | 20C | 20D | 20E | ··· | 20J | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 1 | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 10 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D5 | D10 | D10 | D10 | C8○D4 | C4×D5 | C4×D5 | D5×C8○D4 |
| kernel | D5×C8○D4 | D5×C2×C8 | D20.3C4 | D5×M4(2) | D20.2C4 | D4.Dic5 | C5×C8○D4 | D5×C4○D4 | D4×D5 | D4⋊2D5 | Q8×D5 | Q8⋊2D5 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D5 | D4 | Q8 | C1 |
| # reps | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 6 | 6 | 2 | 8 | 12 | 4 | 8 |
Matrix representation of D5×C8○D4 ►in GL4(𝔽41) generated by
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 2 |
| 0 | 0 | 40 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 40 |
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,0,0,0,0,3],[1,0,0,0,0,1,0,0,0,0,40,40,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,40] >;
D5×C8○D4 in GAP, Magma, Sage, TeX
D_5\times C_8\circ D_4
% in TeX
G:=Group("D5xC8oD4"); // GroupNames label
G:=SmallGroup(320,1421);
// by ID
G=gap.SmallGroup(320,1421);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,80,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^8=e^2=1,d^2=c^4,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,c*d=d*c,c*e=e*c,e*d*e=c^4*d>;
// generators/relations