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, C20.244(C2×D4), C8.D10⋊3C2, SD16⋊D5⋊3C2, C40⋊C2⋊5C22, C8⋊D5⋊5C22, (D5×M4(2))⋊3C2, D4.D5⋊6C22, (C5×Q16)⋊1C22, (Q8×D5)⋊11C22, (C8×D5).1C22, C5⋊Q16⋊4C22, C4.23(C23×D5), C22.48(D4×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
Subgroups: 942 in 258 conjugacy classes, 101 normal (51 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×8], C22, C22 [×8], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×16], D4, D4 [×6], Q8, Q8 [×2], Q8 [×10], C23 [×2], D5 [×2], D5 [×2], C10, C10 [×2], C2×C8 [×2], M4(2), M4(2) [×3], SD16 [×2], SD16 [×6], Q16 [×2], Q16 [×6], C22×C4 [×3], C2×D4 [×2], C2×Q8, C2×Q8 [×9], C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10 [×5], C2×C10, C2×C10, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C8.C22, C8.C22 [×7], C22×Q8, C2×C4○D4, C5⋊2C8 [×2], C40 [×2], Dic10, Dic10 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×7], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, C2×C8.C22, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C4.Dic5, D4.D5 [×2], Q8⋊D5 [×2], C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5 [×2], C4○D20, C4○D20, D4×D5, D4×D5, D4⋊2D5, D4⋊2D5, Q8×D5, Q8×D5 [×4], Q8×D5 [×2], Q8⋊2D5, Q8×C10, C5×C4○D4, D5×M4(2), C8.D10, D5×SD16 [×2], SD16⋊D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C20.C23, D4.9D10, C5×C8.C22, C2×Q8×D5, D5×C4○D4, D5×C8.C22
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C8.C22 [×2], C22×D4, C22×D5 [×7], C2×C8.C22, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C8.C22
Generators and relations
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 >
►On 80 points
(1 75 12 42 64)(2 76 13 43 57)(3 77 14 44 58)(4 78 15 45 59)(5 79 16 46 60)(6 80 9 47 61)(7 73 10 48 62)(8 74 11 41 63)(17 26 69 52 34)(18 27 70 53 35)(19 28 71 54 36)(20 29 72 55 37)(21 30 65 56 38)(22 31 66 49 39)(23 32 67 50 40)(24 25 68 51 33)
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(17 52)(18 53)(19 54)(20 55)(21 56)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 65)(31 66)(32 67)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 73)
(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 11)(10 14)(13 15)(17 23)(19 21)(20 24)(25 29)(26 32)(28 30)(33 37)(34 40)(36 38)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(57 59)(58 62)(61 63)(65 71)(67 69)(68 72)(73 77)(74 80)(76 78)
(1 68)(2 65)(3 70)(4 67)(5 72)(6 69)(7 66)(8 71)(9 34)(10 39)(11 36)(12 33)(13 38)(14 35)(15 40)(16 37)(17 47)(18 44)(19 41)(20 46)(21 43)(22 48)(23 45)(24 42)(25 64)(26 61)(27 58)(28 63)(29 60)(30 57)(31 62)(32 59)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)
G:=sub<Sym(80)| (1,75,12,42,64)(2,76,13,43,57)(3,77,14,44,58)(4,78,15,45,59)(5,79,16,46,60)(6,80,9,47,61)(7,73,10,48,62)(8,74,11,41,63)(17,26,69,52,34)(18,27,70,53,35)(19,28,71,54,36)(20,29,72,55,37)(21,30,65,56,38)(22,31,66,49,39)(23,32,67,50,40)(24,25,68,51,33), (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,68)(26,69)(27,70)(28,71)(29,72)(30,65)(31,66)(32,67)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73), (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)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,47)(43,45)(44,48)(50,52)(51,55)(54,56)(57,59)(58,62)(61,63)(65,71)(67,69)(68,72)(73,77)(74,80)(76,78), (1,68)(2,65)(3,70)(4,67)(5,72)(6,69)(7,66)(8,71)(9,34)(10,39)(11,36)(12,33)(13,38)(14,35)(15,40)(16,37)(17,47)(18,44)(19,41)(20,46)(21,43)(22,48)(23,45)(24,42)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76)>;
G:=Group( (1,75,12,42,64)(2,76,13,43,57)(3,77,14,44,58)(4,78,15,45,59)(5,79,16,46,60)(6,80,9,47,61)(7,73,10,48,62)(8,74,11,41,63)(17,26,69,52,34)(18,27,70,53,35)(19,28,71,54,36)(20,29,72,55,37)(21,30,65,56,38)(22,31,66,49,39)(23,32,67,50,40)(24,25,68,51,33), (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(17,52)(18,53)(19,54)(20,55)(21,56)(22,49)(23,50)(24,51)(25,68)(26,69)(27,70)(28,71)(29,72)(30,65)(31,66)(32,67)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,73), (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)(9,11)(10,14)(13,15)(17,23)(19,21)(20,24)(25,29)(26,32)(28,30)(33,37)(34,40)(36,38)(41,47)(43,45)(44,48)(50,52)(51,55)(54,56)(57,59)(58,62)(61,63)(65,71)(67,69)(68,72)(73,77)(74,80)(76,78), (1,68)(2,65)(3,70)(4,67)(5,72)(6,69)(7,66)(8,71)(9,34)(10,39)(11,36)(12,33)(13,38)(14,35)(15,40)(16,37)(17,47)(18,44)(19,41)(20,46)(21,43)(22,48)(23,45)(24,42)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76) );
G=PermutationGroup([(1,75,12,42,64),(2,76,13,43,57),(3,77,14,44,58),(4,78,15,45,59),(5,79,16,46,60),(6,80,9,47,61),(7,73,10,48,62),(8,74,11,41,63),(17,26,69,52,34),(18,27,70,53,35),(19,28,71,54,36),(20,29,72,55,37),(21,30,65,56,38),(22,31,66,49,39),(23,32,67,50,40),(24,25,68,51,33)], [(1,64),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(17,52),(18,53),(19,54),(20,55),(21,56),(22,49),(23,50),(24,51),(25,68),(26,69),(27,70),(28,71),(29,72),(30,65),(31,66),(32,67),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,73)], [(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),(9,11),(10,14),(13,15),(17,23),(19,21),(20,24),(25,29),(26,32),(28,30),(33,37),(34,40),(36,38),(41,47),(43,45),(44,48),(50,52),(51,55),(54,56),(57,59),(58,62),(61,63),(65,71),(67,69),(68,72),(73,77),(74,80),(76,78)], [(1,68),(2,65),(3,70),(4,67),(5,72),(6,69),(7,66),(8,71),(9,34),(10,39),(11,36),(12,33),(13,38),(14,35),(15,40),(16,37),(17,47),(18,44),(19,41),(20,46),(21,43),(22,48),(23,45),(24,42),(25,64),(26,61),(27,58),(28,63),(29,60),(30,57),(31,62),(32,59),(49,73),(50,78),(51,75),(52,80),(53,77),(54,74),(55,79),(56,76)])
Matrix representation ►G ⊆ 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] >;
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 |
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