direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4○SD16, C40.53C23, C20.86C24, 2+ (1+4)⋊4C10, 2- (1+4)⋊3C10, C4○D8⋊5C10, C8○D4⋊4C10, D8⋊5(C2×C10), C8⋊C22⋊5C10, Q16⋊5(C2×C10), D4.12(C5×D4), (C5×D4).46D4, C4.46(D4×C10), Q8.12(C5×D4), (C5×Q8).46D4, (C2×C40)⋊32C22, SD16⋊6(C2×C10), (C2×SD16)⋊6C10, C20.407(C2×D4), (C5×D8)⋊22C22, C8.C22⋊4C10, C4.9(C23×C10), C22.8(D4×C10), (C10×SD16)⋊17C2, M4(2)⋊7(C2×C10), C8.13(C22×C10), (Q8×C10)⋊32C22, (C5×Q16)⋊19C22, (C5×D4).39C23, D4.6(C22×C10), Q8.6(C22×C10), (C5×Q8).40C23, (C2×C20).688C23, (C5×SD16)⋊21C22, C10.207(C22×D4), (C5×2- (1+4))⋊7C2, (D4×C10).226C22, (C5×M4(2))⋊33C22, (C5×2+ (1+4))⋊10C2, (C2×C8)⋊5(C2×C10), C2.31(D4×C2×C10), C4○D4⋊2(C2×C10), (C5×C8○D4)⋊13C2, (C5×C4○D8)⋊12C2, (C2×Q8)⋊7(C2×C10), (C5×C8⋊C22)⋊12C2, (C2×D4).39(C2×C10), (C2×C10).185(C2×D4), (C5×C4○D4)⋊15C22, (C5×C8.C22)⋊11C2, (C2×C4).49(C22×C10), SmallGroup(320,1579)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2 [×7], C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C5, C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4, D4 [×6], D4 [×9], Q8 [×2], Q8 [×3], Q8 [×3], C23 [×3], C10, C10 [×7], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16, SD16 [×9], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×6], C4○D4 [×4], C20, C20 [×3], C20 [×4], C2×C10 [×3], C2×C10 [×7], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C40, C40 [×3], C2×C20 [×3], C2×C20 [×9], C5×D4, C5×D4 [×6], C5×D4 [×9], C5×Q8 [×2], C5×Q8 [×3], C5×Q8 [×3], C22×C10 [×3], D4○SD16, C2×C40 [×3], C5×M4(2) [×3], C5×D8 [×3], C5×SD16, C5×SD16 [×9], C5×Q16 [×3], D4×C10 [×3], D4×C10 [×3], Q8×C10 [×3], Q8×C10, C5×C4○D4, C5×C4○D4 [×6], C5×C4○D4 [×4], C5×C8○D4, C10×SD16 [×3], C5×C4○D8 [×3], C5×C8⋊C22 [×3], C5×C8.C22 [×3], C5×2+ (1+4), C5×2- (1+4), C5×D4○SD16
Quotients:
C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, C5×D4 [×4], C22×C10 [×15], D4○SD16, D4×C10 [×6], C23×C10, D4×C2×C10, C5×D4○SD16
Generators and relations
G = < a,b,c,d,e | a5=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >
►On 80 points
(1 20 67 53 26)(2 21 68 54 27)(3 22 69 55 28)(4 23 70 56 29)(5 24 71 49 30)(6 17 72 50 31)(7 18 65 51 32)(8 19 66 52 25)(9 80 58 39 43)(10 73 59 40 44)(11 74 60 33 45)(12 75 61 34 46)(13 76 62 35 47)(14 77 63 36 48)(15 78 64 37 41)(16 79 57 38 42)
(1 38 5 34)(2 39 6 35)(3 40 7 36)(4 33 8 37)(9 72 13 68)(10 65 14 69)(11 66 15 70)(12 67 16 71)(17 47 21 43)(18 48 22 44)(19 41 23 45)(20 42 24 46)(25 64 29 60)(26 57 30 61)(27 58 31 62)(28 59 32 63)(49 75 53 79)(50 76 54 80)(51 77 55 73)(52 78 56 74)
(1 34)(2 35)(3 36)(4 37)(5 38)(6 39)(7 40)(8 33)(9 72)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 41)(24 42)(25 60)(26 61)(27 62)(28 63)(29 64)(30 57)(31 58)(32 59)(49 79)(50 80)(51 73)(52 74)(53 75)(54 76)(55 77)(56 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 11)(10 14)(13 15)(17 19)(18 22)(21 23)(25 31)(27 29)(28 32)(33 39)(35 37)(36 40)(41 47)(43 45)(44 48)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(65 69)(66 72)(68 70)(73 77)(74 80)(76 78)
G:=sub<Sym(80)| (1,20,67,53,26)(2,21,68,54,27)(3,22,69,55,28)(4,23,70,56,29)(5,24,71,49,30)(6,17,72,50,31)(7,18,65,51,32)(8,19,66,52,25)(9,80,58,39,43)(10,73,59,40,44)(11,74,60,33,45)(12,75,61,34,46)(13,76,62,35,47)(14,77,63,36,48)(15,78,64,37,41)(16,79,57,38,42), (1,38,5,34)(2,39,6,35)(3,40,7,36)(4,33,8,37)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,47,21,43)(18,48,22,44)(19,41,23,45)(20,42,24,46)(25,64,29,60)(26,57,30,61)(27,58,31,62)(28,59,32,63)(49,75,53,79)(50,76,54,80)(51,77,55,73)(52,78,56,74), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,72)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78), (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)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78)>;
G:=Group( (1,20,67,53,26)(2,21,68,54,27)(3,22,69,55,28)(4,23,70,56,29)(5,24,71,49,30)(6,17,72,50,31)(7,18,65,51,32)(8,19,66,52,25)(9,80,58,39,43)(10,73,59,40,44)(11,74,60,33,45)(12,75,61,34,46)(13,76,62,35,47)(14,77,63,36,48)(15,78,64,37,41)(16,79,57,38,42), (1,38,5,34)(2,39,6,35)(3,40,7,36)(4,33,8,37)(9,72,13,68)(10,65,14,69)(11,66,15,70)(12,67,16,71)(17,47,21,43)(18,48,22,44)(19,41,23,45)(20,42,24,46)(25,64,29,60)(26,57,30,61)(27,58,31,62)(28,59,32,63)(49,75,53,79)(50,76,54,80)(51,77,55,73)(52,78,56,74), (1,34)(2,35)(3,36)(4,37)(5,38)(6,39)(7,40)(8,33)(9,72)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,41)(24,42)(25,60)(26,61)(27,62)(28,63)(29,64)(30,57)(31,58)(32,59)(49,79)(50,80)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78), (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)(9,11)(10,14)(13,15)(17,19)(18,22)(21,23)(25,31)(27,29)(28,32)(33,39)(35,37)(36,40)(41,47)(43,45)(44,48)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(65,69)(66,72)(68,70)(73,77)(74,80)(76,78) );
G=PermutationGroup([(1,20,67,53,26),(2,21,68,54,27),(3,22,69,55,28),(4,23,70,56,29),(5,24,71,49,30),(6,17,72,50,31),(7,18,65,51,32),(8,19,66,52,25),(9,80,58,39,43),(10,73,59,40,44),(11,74,60,33,45),(12,75,61,34,46),(13,76,62,35,47),(14,77,63,36,48),(15,78,64,37,41),(16,79,57,38,42)], [(1,38,5,34),(2,39,6,35),(3,40,7,36),(4,33,8,37),(9,72,13,68),(10,65,14,69),(11,66,15,70),(12,67,16,71),(17,47,21,43),(18,48,22,44),(19,41,23,45),(20,42,24,46),(25,64,29,60),(26,57,30,61),(27,58,31,62),(28,59,32,63),(49,75,53,79),(50,76,54,80),(51,77,55,73),(52,78,56,74)], [(1,34),(2,35),(3,36),(4,37),(5,38),(6,39),(7,40),(8,33),(9,72),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,41),(24,42),(25,60),(26,61),(27,62),(28,63),(29,64),(30,57),(31,58),(32,59),(49,79),(50,80),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78)], [(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),(9,11),(10,14),(13,15),(17,19),(18,22),(21,23),(25,31),(27,29),(28,32),(33,39),(35,37),(36,40),(41,47),(43,45),(44,48),(50,52),(51,55),(54,56),(58,60),(59,63),(62,64),(65,69),(66,72),(68,70),(73,77),(74,80),(76,78)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 40 | 0 | 40 | 40 |
| 0 | 0 | 0 | 1 |
| 2 | 1 | 1 | 1 |
| 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 39 | 40 | 40 | 40 |
| 0 | 1 | 0 | 0 |
| 11 | 26 | 0 | 26 |
| 30 | 0 | 15 | 15 |
| 0 | 0 | 26 | 15 |
| 0 | 0 | 26 | 26 |
| 1 | 1 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[40,0,2,0,0,0,1,40,40,0,1,0,40,1,1,0],[1,0,39,0,0,0,40,1,0,0,40,0,0,1,40,0],[11,30,0,0,26,0,0,0,0,15,26,26,26,15,15,26],[1,0,0,0,1,40,0,0,0,0,1,0,1,0,0,40] >;
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10AF | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40H | 40I | ··· | 40T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | D4○SD16 | C5×D4○SD16 |
| kernel | C5×D4○SD16 | C5×C8○D4 | C10×SD16 | C5×C4○D8 | C5×C8⋊C22 | C5×C8.C22 | C5×2+ (1+4) | C5×2- (1+4) | D4○SD16 | C8○D4 | C2×SD16 | C4○D8 | C8⋊C22 | C8.C22 | 2+ (1+4) | 2- (1+4) | C5×D4 | C5×Q8 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 3 | 1 | 12 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_5\times D_4\circ SD_{16} % in TeX
G:=Group("C5xD4oSD16"); // GroupNames label
G:=SmallGroup(320,1579);
// by ID
G=gap.SmallGroup(320,1579);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,1193,10085,5052,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=e^2=1,d^4=b^2,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