direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C8×F5, C20.29C42, D10.11C42, Dic5.9C42, (C2×C40)⋊8C4, C40⋊9(C2×C4), C10⋊1(C4×C8), 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), (C2×C10).16C42, C10.11(C2×C42), C20.88(C22×C4), D5⋊C8.21C22, (C8×D5).64C22, (C4×D5).85C23, (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 |
Generators and relations for C2×C8×F5
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 442 in 162 conjugacy classes, 92 normal (30 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, C23, D5, C10, C10, C42, C2×C8, C2×C8, C22×C4, Dic5, C20, F5, D10, D10, C2×C10, C4×C8, C2×C42, C22×C8, C5⋊2C8, C40, C5⋊C8, C4×D5, C2×Dic5, C2×C20, C2×F5, C22×D5, C2×C4×C8, C8×D5, C2×C5⋊2C8, C2×C40, D5⋊C8, C4×F5, C2×C5⋊C8, C2×C4×D5, C22×F5, C8×F5, D5×C2×C8, C2×D5⋊C8, C2×C4×F5, C2×C8×F5
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, F5, C4×C8, C2×C42, C22×C8, C2×F5, C2×C4×C8, C4×F5, C22×F5, C8×F5, C2×C4×F5, C2×C8×F5
►On 80 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 71)(10 72)(11 65)(12 66)(13 67)(14 68)(15 69)(16 70)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)
(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 65 48 56 59)(2 66 41 49 60)(3 67 42 50 61)(4 68 43 51 62)(5 69 44 52 63)(6 70 45 53 64)(7 71 46 54 57)(8 72 47 55 58)(9 74 30 33 17)(10 75 31 34 18)(11 76 32 35 19)(12 77 25 36 20)(13 78 26 37 21)(14 79 27 38 22)(15 80 28 39 23)(16 73 29 40 24)
(1 5)(2 6)(3 7)(4 8)(9 78 33 26)(10 79 34 27)(11 80 35 28)(12 73 36 29)(13 74 37 30)(14 75 38 31)(15 76 39 32)(16 77 40 25)(17 21)(18 22)(19 23)(20 24)(41 64 49 70)(42 57 50 71)(43 58 51 72)(44 59 52 65)(45 60 53 66)(46 61 54 67)(47 62 55 68)(48 63 56 69)
G:=sub<Sym(80)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76), (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), (1,65,48,56,59)(2,66,41,49,60)(3,67,42,50,61)(4,68,43,51,62)(5,69,44,52,63)(6,70,45,53,64)(7,71,46,54,57)(8,72,47,55,58)(9,74,30,33,17)(10,75,31,34,18)(11,76,32,35,19)(12,77,25,36,20)(13,78,26,37,21)(14,79,27,38,22)(15,80,28,39,23)(16,73,29,40,24), (1,5)(2,6)(3,7)(4,8)(9,78,33,26)(10,79,34,27)(11,80,35,28)(12,73,36,29)(13,74,37,30)(14,75,38,31)(15,76,39,32)(16,77,40,25)(17,21)(18,22)(19,23)(20,24)(41,64,49,70)(42,57,50,71)(43,58,51,72)(44,59,52,65)(45,60,53,66)(46,61,54,67)(47,62,55,68)(48,63,56,69)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,71)(10,72)(11,65)(12,66)(13,67)(14,68)(15,69)(16,70)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76), (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), (1,65,48,56,59)(2,66,41,49,60)(3,67,42,50,61)(4,68,43,51,62)(5,69,44,52,63)(6,70,45,53,64)(7,71,46,54,57)(8,72,47,55,58)(9,74,30,33,17)(10,75,31,34,18)(11,76,32,35,19)(12,77,25,36,20)(13,78,26,37,21)(14,79,27,38,22)(15,80,28,39,23)(16,73,29,40,24), (1,5)(2,6)(3,7)(4,8)(9,78,33,26)(10,79,34,27)(11,80,35,28)(12,73,36,29)(13,74,37,30)(14,75,38,31)(15,76,39,32)(16,77,40,25)(17,21)(18,22)(19,23)(20,24)(41,64,49,70)(42,57,50,71)(43,58,51,72)(44,59,52,65)(45,60,53,66)(46,61,54,67)(47,62,55,68)(48,63,56,69) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,71),(10,72),(11,65),(12,66),(13,67),(14,68),(15,69),(16,70),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76)], [(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)], [(1,65,48,56,59),(2,66,41,49,60),(3,67,42,50,61),(4,68,43,51,62),(5,69,44,52,63),(6,70,45,53,64),(7,71,46,54,57),(8,72,47,55,58),(9,74,30,33,17),(10,75,31,34,18),(11,76,32,35,19),(12,77,25,36,20),(13,78,26,37,21),(14,79,27,38,22),(15,80,28,39,23),(16,73,29,40,24)], [(1,5),(2,6),(3,7),(4,8),(9,78,33,26),(10,79,34,27),(11,80,35,28),(12,73,36,29),(13,74,37,30),(14,75,38,31),(15,76,39,32),(16,77,40,25),(17,21),(18,22),(19,23),(20,24),(41,64,49,70),(42,57,50,71),(43,58,51,72),(44,59,52,65),(45,60,53,66),(46,61,54,67),(47,62,55,68),(48,63,56,69)]])
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 |
Matrix representation of C2×C8×F5 ►in 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] >;
C2×C8×F5 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