direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q16×F5, Dic20⋊6C4, C5⋊(C4×Q16), (C5×Q16)⋊5C4, C5⋊Q16⋊3C4, (C8×F5).1C2, C8.15(C2×F5), C2.23(D4×F5), Q8.3(C2×F5), (Q8×F5).1C2, C40.13(C2×C4), C10.22(C4×D4), (C2×F5).12D4, D5.D8.2C2, D5.2(C2×Q16), (D5×Q16).5C2, C4⋊F5.5C22, C4.9(C22×F5), D5.4(C4○D8), D10.67(C2×D4), C20.9(C22×C4), Q8⋊F5.1C2, (Q8×D5).6C22, D5⋊C8.13C22, Dic10.5(C2×C4), (C4×D5).31C23, (C8×D5).23C22, (C4×F5).13C22, Dic5.5(C4○D4), (C5×Q8).3(C2×C4), C5⋊2C8.11(C2×C4), SmallGroup(320,1076)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 418 in 110 conjugacy classes, 44 normal (26 characteristic)
C1, C2, C2 [×2], C4, C4 [×10], C22, C5, C8, C8 [×2], C2×C4 [×7], Q8 [×2], Q8 [×4], D5 [×2], C10, C42 [×3], C4⋊C4 [×4], C2×C8 [×2], Q16, Q16 [×3], C2×Q8 [×2], Dic5, Dic5 [×2], C20, C20 [×2], F5 [×2], F5 [×3], D10, C4×C8, Q8⋊C4 [×2], C2.D8, C4×Q8 [×2], C2×Q16, C5⋊2C8, C40, C5⋊C8, Dic10 [×2], Dic10 [×2], C4×D5, C4×D5 [×2], C5×Q8 [×2], C2×F5 [×2], C2×F5 [×2], C4×Q16, C8×D5, Dic20, C5⋊Q16 [×2], C5×Q16, D5⋊C8, C4×F5, C4×F5 [×2], C4⋊F5 [×2], C4⋊F5 [×2], Q8×D5 [×2], C8×F5, D5.D8, Q8⋊F5 [×2], D5×Q16, Q8×F5 [×2], Q16×F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, Q16 [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×Q16, C4○D8, C2×F5 [×3], C4×Q16, C22×F5, D4×F5, Q16×F5
Generators and relations
G = < a,b,c,d | a8=c5=d4=1, b2=a4, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►On 80 points
(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 75 5 79)(2 74 6 78)(3 73 7 77)(4 80 8 76)(9 27 13 31)(10 26 14 30)(11 25 15 29)(12 32 16 28)(17 35 21 39)(18 34 22 38)(19 33 23 37)(20 40 24 36)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)(49 71 53 67)(50 70 54 66)(51 69 55 65)(52 68 56 72)
(1 15 39 62 66)(2 16 40 63 67)(3 9 33 64 68)(4 10 34 57 69)(5 11 35 58 70)(6 12 36 59 71)(7 13 37 60 72)(8 14 38 61 65)(17 45 50 75 29)(18 46 51 76 30)(19 47 52 77 31)(20 48 53 78 32)(21 41 54 79 25)(22 42 55 80 26)(23 43 56 73 27)(24 44 49 74 28)
(1 75 5 79)(2 76 6 80)(3 77 7 73)(4 78 8 74)(9 19 72 43)(10 20 65 44)(11 21 66 45)(12 22 67 46)(13 23 68 47)(14 24 69 48)(15 17 70 41)(16 18 71 42)(25 39 50 58)(26 40 51 59)(27 33 52 60)(28 34 53 61)(29 35 54 62)(30 36 55 63)(31 37 56 64)(32 38 49 57)
G:=sub<Sym(80)| (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), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,27,13,31)(10,26,14,30)(11,25,15,29)(12,32,16,28)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,15,39,62,66)(2,16,40,63,67)(3,9,33,64,68)(4,10,34,57,69)(5,11,35,58,70)(6,12,36,59,71)(7,13,37,60,72)(8,14,38,61,65)(17,45,50,75,29)(18,46,51,76,30)(19,47,52,77,31)(20,48,53,78,32)(21,41,54,79,25)(22,42,55,80,26)(23,43,56,73,27)(24,44,49,74,28), (1,75,5,79)(2,76,6,80)(3,77,7,73)(4,78,8,74)(9,19,72,43)(10,20,65,44)(11,21,66,45)(12,22,67,46)(13,23,68,47)(14,24,69,48)(15,17,70,41)(16,18,71,42)(25,39,50,58)(26,40,51,59)(27,33,52,60)(28,34,53,61)(29,35,54,62)(30,36,55,63)(31,37,56,64)(32,38,49,57)>;
G:=Group( (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), (1,75,5,79)(2,74,6,78)(3,73,7,77)(4,80,8,76)(9,27,13,31)(10,26,14,30)(11,25,15,29)(12,32,16,28)(17,35,21,39)(18,34,22,38)(19,33,23,37)(20,40,24,36)(41,62,45,58)(42,61,46,57)(43,60,47,64)(44,59,48,63)(49,71,53,67)(50,70,54,66)(51,69,55,65)(52,68,56,72), (1,15,39,62,66)(2,16,40,63,67)(3,9,33,64,68)(4,10,34,57,69)(5,11,35,58,70)(6,12,36,59,71)(7,13,37,60,72)(8,14,38,61,65)(17,45,50,75,29)(18,46,51,76,30)(19,47,52,77,31)(20,48,53,78,32)(21,41,54,79,25)(22,42,55,80,26)(23,43,56,73,27)(24,44,49,74,28), (1,75,5,79)(2,76,6,80)(3,77,7,73)(4,78,8,74)(9,19,72,43)(10,20,65,44)(11,21,66,45)(12,22,67,46)(13,23,68,47)(14,24,69,48)(15,17,70,41)(16,18,71,42)(25,39,50,58)(26,40,51,59)(27,33,52,60)(28,34,53,61)(29,35,54,62)(30,36,55,63)(31,37,56,64)(32,38,49,57) );
G=PermutationGroup([(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)], [(1,75,5,79),(2,74,6,78),(3,73,7,77),(4,80,8,76),(9,27,13,31),(10,26,14,30),(11,25,15,29),(12,32,16,28),(17,35,21,39),(18,34,22,38),(19,33,23,37),(20,40,24,36),(41,62,45,58),(42,61,46,57),(43,60,47,64),(44,59,48,63),(49,71,53,67),(50,70,54,66),(51,69,55,65),(52,68,56,72)], [(1,15,39,62,66),(2,16,40,63,67),(3,9,33,64,68),(4,10,34,57,69),(5,11,35,58,70),(6,12,36,59,71),(7,13,37,60,72),(8,14,38,61,65),(17,45,50,75,29),(18,46,51,76,30),(19,47,52,77,31),(20,48,53,78,32),(21,41,54,79,25),(22,42,55,80,26),(23,43,56,73,27),(24,44,49,74,28)], [(1,75,5,79),(2,76,6,80),(3,77,7,73),(4,78,8,74),(9,19,72,43),(10,20,65,44),(11,21,66,45),(12,22,67,46),(13,23,68,47),(14,24,69,48),(15,17,70,41),(16,18,71,42),(25,39,50,58),(26,40,51,59),(27,33,52,60),(28,34,53,61),(29,35,54,62),(30,36,55,63),(31,37,56,64),(32,38,49,57)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 0 | 17 | 0 | 0 | 0 | 0 |
| 12 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 9 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 | 0 | 40 |
| 0 | 0 | 0 | 1 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [0,12,0,0,0,0,17,17,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,9,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,40,40,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0] >;
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5 | 8A | 8B | 8C | ··· | 8H | 10 | 20A | 20B | 20C | 40A | 40B |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | ··· | 8 | 10 | 20 | 20 | 20 | 40 | 40 |
| size | 1 | 1 | 5 | 5 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 2 | 2 | 10 | ··· | 10 | 4 | 8 | 16 | 16 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | + | + | + | - | + | + | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4○D4 | Q16 | C4○D8 | F5 | C2×F5 | C2×F5 | D4×F5 | Q16×F5 |
| kernel | Q16×F5 | C8×F5 | D5.D8 | Q8⋊F5 | D5×Q16 | Q8×F5 | Dic20 | C5⋊Q16 | C5×Q16 | C2×F5 | Dic5 | F5 | D5 | Q16 | C8 | Q8 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 1 | 2 |
In GAP, Magma, Sage, TeX
Q_{16}\times F_5 % in TeX
G:=Group("Q16xF5"); // GroupNames label
G:=SmallGroup(320,1076);
// by ID
G=gap.SmallGroup(320,1076);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,219,184,851,438,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=c^5=d^4=1,b^2=a^4,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations