direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4.F5, C10⋊1M4(2), Dic5.10C23, C5⋊C8⋊1C22, (C2×C4).7F5, (C2×C20).6C4, (C4×D5).4C4, C5⋊1(C2×M4(2)), C4.12(C2×F5), C20.12(C2×C4), C2.4(C22×F5), D10.13(C2×C4), C10.2(C22×C4), (C22×D5).8C4, C22.16(C2×F5), Dic5.15(C2×C4), (C4×D5).29C22, (C2×Dic5).55C22, (C2×C5⋊C8)⋊3C2, (C2×C4×D5).13C2, (C2×C10).14(C2×C4), SmallGroup(160,201)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4.F5
G = < a,b,c,d | a2=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >
Subgroups: 196 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C2×C8, M4(2), C22×C4, Dic5, C20, D10, D10, C2×C10, C2×M4(2), C5⋊C8, C4×D5, C2×Dic5, C2×C20, C22×D5, C4.F5, C2×C5⋊C8, C2×C4×D5, C2×C4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C4.F5, C22×F5, C2×C4.F5
Character table of C2×C4.F5
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | 2 | 5 | 5 | 5 | 5 | 4 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 2 |
| ρ9 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -i | -i | i | i | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -i | i | -i | i | -i | i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ11 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | i | i | -i | -i | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | i | -i | i | -i | i | -i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ13 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | -i | i | -i | -i | i | -i | i | i | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | -i | i | i | -i | i | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ15 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | i | -i | i | i | -i | i | -i | -i | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 4 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | i | -i | -i | i | -i | 1 | -1 | -1 | 1 | -1 | 1 | -1 | linear of order 4 |
| ρ17 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | -2i | 2i | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | -2i | 2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 2i | -2i | 2i | -2i | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | -4 | -4 | 0 | 0 | -4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | -4 | -4 | 0 | 0 | 4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ24 | 4 | 4 | 4 | 4 | 0 | 0 | -4 | -4 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from C2×F5 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -√-5 | -√-5 | √-5 | √-5 | complex lifted from C4.F5 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | √-5 | √-5 | -√-5 | -√-5 | complex lifted from C4.F5 |
| ρ27 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | √-5 | -√-5 | -√-5 | √-5 | complex lifted from C4.F5 |
| ρ28 | 4 | -4 | -4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1 | 1 | -√-5 | √-5 | √-5 | -√-5 | complex lifted from C4.F5 |
►On 80 points
Generators in S80
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 65)(9 62)(10 63)(11 64)(12 57)(13 58)(14 59)(15 60)(16 61)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 76)(42 77)(43 78)(44 79)(45 80)(46 73)(47 74)(48 75)
(1 7 5 3)(2 4 6 8)(9 52 13 56)(10 49 14 53)(11 54 15 50)(12 51 16 55)(17 61 21 57)(18 58 22 62)(19 63 23 59)(20 60 24 64)(25 43 29 47)(26 48 30 44)(27 45 31 41)(28 42 32 46)(33 79 37 75)(34 76 38 80)(35 73 39 77)(36 78 40 74)(65 67 69 71)(66 72 70 68)
(1 57 35 75 19)(2 76 58 20 36)(3 21 77 37 59)(4 38 22 60 78)(5 61 39 79 23)(6 80 62 24 40)(7 17 73 33 63)(8 34 18 64 74)(9 50 29 71 45)(10 72 51 46 30)(11 47 65 31 52)(12 32 48 53 66)(13 54 25 67 41)(14 68 55 42 26)(15 43 69 27 56)(16 28 44 49 70)
(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)
G:=sub<Sym(80)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (1,7,5,3)(2,4,6,8)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,43,29,47)(26,48,30,44)(27,45,31,41)(28,42,32,46)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(65,67,69,71)(66,72,70,68), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,65)(9,62)(10,63)(11,64)(12,57)(13,58)(14,59)(15,60)(16,61)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,76)(42,77)(43,78)(44,79)(45,80)(46,73)(47,74)(48,75), (1,7,5,3)(2,4,6,8)(9,52,13,56)(10,49,14,53)(11,54,15,50)(12,51,16,55)(17,61,21,57)(18,58,22,62)(19,63,23,59)(20,60,24,64)(25,43,29,47)(26,48,30,44)(27,45,31,41)(28,42,32,46)(33,79,37,75)(34,76,38,80)(35,73,39,77)(36,78,40,74)(65,67,69,71)(66,72,70,68), (1,57,35,75,19)(2,76,58,20,36)(3,21,77,37,59)(4,38,22,60,78)(5,61,39,79,23)(6,80,62,24,40)(7,17,73,33,63)(8,34,18,64,74)(9,50,29,71,45)(10,72,51,46,30)(11,47,65,31,52)(12,32,48,53,66)(13,54,25,67,41)(14,68,55,42,26)(15,43,69,27,56)(16,28,44,49,70), (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) );
G=PermutationGroup([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,65),(9,62),(10,63),(11,64),(12,57),(13,58),(14,59),(15,60),(16,61),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,76),(42,77),(43,78),(44,79),(45,80),(46,73),(47,74),(48,75)], [(1,7,5,3),(2,4,6,8),(9,52,13,56),(10,49,14,53),(11,54,15,50),(12,51,16,55),(17,61,21,57),(18,58,22,62),(19,63,23,59),(20,60,24,64),(25,43,29,47),(26,48,30,44),(27,45,31,41),(28,42,32,46),(33,79,37,75),(34,76,38,80),(35,73,39,77),(36,78,40,74),(65,67,69,71),(66,72,70,68)], [(1,57,35,75,19),(2,76,58,20,36),(3,21,77,37,59),(4,38,22,60,78),(5,61,39,79,23),(6,80,62,24,40),(7,17,73,33,63),(8,34,18,64,74),(9,50,29,71,45),(10,72,51,46,30),(11,47,65,31,52),(12,32,48,53,66),(13,54,25,67,41),(14,68,55,42,26),(15,43,69,27,56),(16,28,44,49,70)], [(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)]])
C2×C4.F5 is a maximal subgroup of
C20.C42 C20.10C42 M4(2)⋊4F5 C20⋊3M4(2) C42.14F5 C5⋊C8⋊D4 Dic5⋊M4(2) D10.C42 D10⋊2M4(2) C20⋊M4(2) C4⋊C4.9F5 M4(2).1F5 D10⋊10M4(2) (C2×D4).8F5 (C2×Q8).5F5 D4⋊F5⋊C2 Dic5.22C24
C2×C4.F5 is a maximal quotient of
C42.12F5 C20⋊3M4(2) C42.15F5 C5⋊C8⋊D4 C20⋊M4(2) C20.M4(2) D10⋊10M4(2) C20.30M4(2)
Matrix representation of C2×C4.F5 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 27 | 0 | 14 |
| 0 | 0 | 0 | 34 | 27 | 14 |
| 0 | 0 | 14 | 27 | 34 | 0 |
| 0 | 0 | 14 | 0 | 27 | 7 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 40 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 9 | 39 | 0 | 0 | 0 | 0 |
| 36 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 | 38 | 24 |
| 0 | 0 | 29 | 15 | 35 | 15 |
| 0 | 0 | 26 | 6 | 26 | 12 |
| 0 | 0 | 17 | 3 | 9 | 9 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,1,0,0,0,0,0,9,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,36,0,0,0,0,39,32,0,0,0,0,0,0,32,29,26,17,0,0,32,15,6,3,0,0,38,35,26,9,0,0,24,15,12,9] >;
C2×C4.F5 in GAP, Magma, Sage, TeX
C_2\times C_4.F_5
% in TeX
G:=Group("C2xC4.F5"); // GroupNames label
G:=SmallGroup(160,201);
// by ID
G=gap.SmallGroup(160,201);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,69,2309,599]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^5=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^3>;
// generators/relations
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