direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22.F5, C23.3F5, C10⋊2M4(2), Dic5.15C23, C5⋊C8⋊3C22, C5⋊3(C2×M4(2)), (C22×C10).6C4, C2.12(C22×F5), C22.20(C2×F5), C10.12(C22×C4), Dic5.19(C2×C4), (C2×Dic5).14C4, (C22×Dic5).9C2, (C2×Dic5).58C22, (C2×C5⋊C8)⋊5C2, (C2×C10).20(C2×C4), SmallGroup(160,211)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C22.F5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e4=c, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d3 >
Subgroups: 164 in 68 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C8, C2×C4, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, Dic5, Dic5, C2×C10, C2×C10, C2×C10, C2×M4(2), C5⋊C8, C2×Dic5, C2×Dic5, C22×C10, C2×C5⋊C8, C22.F5, C22×Dic5, C2×C22.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, F5, C2×M4(2), C2×F5, C22.F5, C22×F5, C2×C22.F5
Character table of C2×C22.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 | 10D | 10E | 10F | 10G | |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 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 | 2i | -2i | -2i | 2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -2 | complex lifted from M4(2) |
| ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | complex lifted from M4(2) |
| ρ19 | 2 | -2 | 2 | -2 | 0 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | 2 | 0 | -2 | complex lifted from M4(2) |
| ρ20 | 2 | 2 | -2 | -2 | 0 | 0 | -2i | 2i | 2i | -2i | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | -2 | 0 | -2 | complex lifted from M4(2) |
| ρ21 | 4 | 4 | 4 | 4 | -4 | -4 | 0 | 0 | 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 | -4 | 4 | 0 | 0 | 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 | 4 | -4 | 0 | 0 | 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 |
| ρ24 | 4 | 4 | 4 | 4 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from F5 |
| ρ25 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √5 | -1 | -√5 | √5 | 1 | -√5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
| ρ26 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√5 | 1 | -√5 | √5 | -1 | √5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
| ρ27 | 4 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √5 | 1 | √5 | -√5 | -1 | -√5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
| ρ28 | 4 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√5 | -1 | √5 | -√5 | 1 | √5 | 1 | symplectic lifted from C22.F5, Schur index 2 |
►On 80 points
Generators in S80
(1 68)(2 69)(3 70)(4 71)(5 72)(6 65)(7 66)(8 67)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 80)(32 73)(33 48)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)(49 59)(50 60)(51 61)(52 62)(53 63)(54 64)(55 57)(56 58)
(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)
(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)| (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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( (1,68)(2,69)(3,70)(4,71)(5,72)(6,65)(7,66)(8,67)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,80)(32,73)(33,48)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47)(49,59)(50,60)(51,61)(52,62)(53,63)(54,64)(55,57)(56,58), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(41,45)(43,47)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80), (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([[(1,68),(2,69),(3,70),(4,71),(5,72),(6,65),(7,66),(8,67),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,80),(32,73),(33,48),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47),(49,59),(50,60),(51,61),(52,62),(53,63),(54,64),(55,57),(56,58)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(41,45),(43,47),(50,54),(52,56),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80)], [(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×C22.F5 is a maximal subgroup of
C22⋊C4.F5 C22.F5⋊C4 Dic5.C42 D10⋊M4(2) Dic5⋊M4(2) C20⋊C8⋊C2 D10⋊9M4(2) C20⋊8M4(2) C5⋊C8⋊7D4 C20⋊2M4(2) (C2×D4).7F5 (C2×D4).9F5 C24.4F5 Dic5.C24
C2×C22.F5 is a maximal quotient of
C20.34M4(2) Dic5.13M4(2) C20⋊8M4(2) C20.30M4(2) C5⋊C8⋊7D4 C20⋊2M4(2) C20.6M4(2) C24.4F5
Matrix representation of C2×C22.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 |
| 1 | 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 1 | 0 | 0 |
| 0 | 0 | 33 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 6 |
| 0 | 0 | 0 | 0 | 34 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 19 | 0 | 0 | 0 |
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],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,0,0,0,0,1,7,0,0,0,0,0,0,34,34,0,0,0,0,6,0],[0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,19,0,0,0,0,28,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C2×C22.F5 in GAP, Magma, Sage, TeX
C_2\times C_2^2.F_5
% in TeX
G:=Group("C2xC2^2.F5"); // GroupNames label
G:=SmallGroup(160,211);
// by ID
G=gap.SmallGroup(160,211);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,69,2309,599]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=1,e^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
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