direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C8×F5, C20.29C42, D10.11C42, Dic5.9C42, C10⋊1(C4×C8), C40⋊9(C2×C4), (C2×C40)⋊8C4, D5⋊C8⋊7C4, D5⋊2(C4×C8), (C8×D5)⋊11C4, (C4×F5).7C4, C4.22(C4×F5), D5.(C22×C8), D10.7(C2×C8), (C22×F5).5C4, C4.48(C22×F5), C22.18(C4×F5), C20.88(C22×C4), C10.11(C2×C42), (C2×C10).16C42, D5⋊C8.21C22, (C4×D5).85C23, (C8×D5).64C22, (C4×F5).20C22, D10.31(C22×C4), Dic5.30(C22×C4), C5⋊2(C2×C4×C8), (C2×C5⋊C8)⋊9C4, C5⋊C8⋊8(C2×C4), C2.3(C2×C4×F5), (D5×C2×C8).32C2, (C2×C5⋊2C8)⋊22C4, C5⋊2C8⋊36(C2×C4), (C2×C4×F5).14C2, (C2×F5).9(C2×C4), (C2×D5⋊C8).12C2, (C4×D5).94(C2×C4), (C2×C4).165(C2×F5), (C2×C20).174(C2×C4), (C2×C4×D5).412C22, (C22×D5).86(C2×C4), (C2×Dic5).124(C2×C4), SmallGroup(320,1054)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C2×C8×F5 |
Subgroups: 442 in 162 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×10], C22, C22 [×6], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×17], C23, D5 [×4], C10, C10 [×2], C42 [×4], C2×C8, C2×C8 [×11], C22×C4 [×3], Dic5 [×2], C20 [×2], F5 [×8], D10 [×2], D10 [×4], C2×C10, C4×C8 [×4], C2×C42, C22×C8 [×2], C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×12], C22×D5, C2×C4×C8, C8×D5 [×4], C2×C5⋊2C8, C2×C40, D5⋊C8 [×4], C4×F5 [×4], C2×C5⋊C8 [×2], C2×C4×D5, C22×F5 [×2], C8×F5 [×4], D5×C2×C8, C2×D5⋊C8, C2×C4×F5, C2×C8×F5
Quotients:
C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], F5, C4×C8 [×4], C2×C42, C22×C8 [×2], C2×F5 [×3], C2×C4×C8, C4×F5 [×2], C22×F5, C8×F5 [×2], C2×C4×F5, C2×C8×F5
Generators and relations
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
►On 80 points
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(49 76)(50 77)(51 78)(52 79)(53 80)(54 73)(55 74)(56 75)
(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 68 49 46 59)(2 69 50 47 60)(3 70 51 48 61)(4 71 52 41 62)(5 72 53 42 63)(6 65 54 43 64)(7 66 55 44 57)(8 67 56 45 58)(9 79 17 38 27)(10 80 18 39 28)(11 73 19 40 29)(12 74 20 33 30)(13 75 21 34 31)(14 76 22 35 32)(15 77 23 36 25)(16 78 24 37 26)
(1 5)(2 6)(3 7)(4 8)(9 75 38 21)(10 76 39 22)(11 77 40 23)(12 78 33 24)(13 79 34 17)(14 80 35 18)(15 73 36 19)(16 74 37 20)(25 29)(26 30)(27 31)(28 32)(41 67 52 58)(42 68 53 59)(43 69 54 60)(44 70 55 61)(45 71 56 62)(46 72 49 63)(47 65 50 64)(48 66 51 57)
G:=sub<Sym(80)| (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,68,49,46,59)(2,69,50,47,60)(3,70,51,48,61)(4,71,52,41,62)(5,72,53,42,63)(6,65,54,43,64)(7,66,55,44,57)(8,67,56,45,58)(9,79,17,38,27)(10,80,18,39,28)(11,73,19,40,29)(12,74,20,33,30)(13,75,21,34,31)(14,76,22,35,32)(15,77,23,36,25)(16,78,24,37,26), (1,5)(2,6)(3,7)(4,8)(9,75,38,21)(10,76,39,22)(11,77,40,23)(12,78,33,24)(13,79,34,17)(14,80,35,18)(15,73,36,19)(16,74,37,20)(25,29)(26,30)(27,31)(28,32)(41,67,52,58)(42,68,53,59)(43,69,54,60)(44,70,55,61)(45,71,56,62)(46,72,49,63)(47,65,50,64)(48,66,51,57)>;
G:=Group( (1,32)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(49,76)(50,77)(51,78)(52,79)(53,80)(54,73)(55,74)(56,75), (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), (1,68,49,46,59)(2,69,50,47,60)(3,70,51,48,61)(4,71,52,41,62)(5,72,53,42,63)(6,65,54,43,64)(7,66,55,44,57)(8,67,56,45,58)(9,79,17,38,27)(10,80,18,39,28)(11,73,19,40,29)(12,74,20,33,30)(13,75,21,34,31)(14,76,22,35,32)(15,77,23,36,25)(16,78,24,37,26), (1,5)(2,6)(3,7)(4,8)(9,75,38,21)(10,76,39,22)(11,77,40,23)(12,78,33,24)(13,79,34,17)(14,80,35,18)(15,73,36,19)(16,74,37,20)(25,29)(26,30)(27,31)(28,32)(41,67,52,58)(42,68,53,59)(43,69,54,60)(44,70,55,61)(45,71,56,62)(46,72,49,63)(47,65,50,64)(48,66,51,57) );
G=PermutationGroup([(1,32),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(49,76),(50,77),(51,78),(52,79),(53,80),(54,73),(55,74),(56,75)], [(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)], [(1,68,49,46,59),(2,69,50,47,60),(3,70,51,48,61),(4,71,52,41,62),(5,72,53,42,63),(6,65,54,43,64),(7,66,55,44,57),(8,67,56,45,58),(9,79,17,38,27),(10,80,18,39,28),(11,73,19,40,29),(12,74,20,33,30),(13,75,21,34,31),(14,76,22,35,32),(15,77,23,36,25),(16,78,24,37,26)], [(1,5),(2,6),(3,7),(4,8),(9,75,38,21),(10,76,39,22),(11,77,40,23),(12,78,33,24),(13,79,34,17),(14,80,35,18),(15,73,36,19),(16,74,37,20),(25,29),(26,30),(27,31),(28,32),(41,67,52,58),(42,68,53,59),(43,69,54,60),(44,70,55,61),(45,71,56,62),(46,72,49,63),(47,65,50,64),(48,66,51,57)])
Matrix representation ►G ⊆ GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 40 | 40 | 40 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[9,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,40,1,0,0,0,40,0,1,0,0,40,0,0,1,0,40,0,0,0],[9,0,0,0,0,0,40,0,0,1,0,0,0,40,1,0,0,0,0,1,0,0,40,0,1] >;
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4X | 5 | 8A | ··· | 8H | 8I | ··· | 8AF | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | ··· | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C4 | C4 | C4 | C8 | F5 | C2×F5 | C2×F5 | C4×F5 | C4×F5 | C8×F5 |
| kernel | C2×C8×F5 | C8×F5 | D5×C2×C8 | C2×D5⋊C8 | C2×C4×F5 | C8×D5 | C2×C5⋊2C8 | C2×C40 | D5⋊C8 | C4×F5 | C2×C5⋊C8 | C22×F5 | C2×F5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 4 | 4 | 4 | 4 | 32 | 1 | 2 | 1 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_8\times F_5
% in TeX
G:=Group("C2xC8xF5"); // GroupNames label
G:=SmallGroup(320,1054);
// by ID
G=gap.SmallGroup(320,1054);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations