metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.2F5, Dic5.23D4, C10.5M4(2), C22⋊(C5⋊C8), (C2×C10)⋊1C8, C5⋊2(C22⋊C8), C10.9(C2×C8), (C22×C10).3C4, C2.3(C22⋊F5), C22.14(C2×F5), (C2×Dic5).12C4, C10.10(C22⋊C4), C2.3(C22.F5), (C22×Dic5).7C2, (C2×Dic5).53C22, (C2×C5⋊C8)⋊2C2, C2.5(C2×C5⋊C8), (C2×C10).11(C2×C4), SmallGroup(160,87)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.2F5
G = < a,b,c,d,e | a2=b2=c2=d5=1, e4=c, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d3 >
Character table of C23.2F5
| 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 | i | i | -i | -i | -i | -i | i | i | -1 | 1 | -1 | -1 | 1 | 1 | -1 | linear of order 4 |
| ρ6 | 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 |
| ρ7 | 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 |
| ρ8 | 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 |
| ρ9 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | 1 | ζ85 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | ζ87 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ10 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | i | 1 | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 8 |
| ρ11 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | -i | -i | -i | i | 1 | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 8 |
| ρ12 | 1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | i | -i | 1 | ζ8 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | ζ83 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ13 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | 1 | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 8 |
| ρ14 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | 1 | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | 1 | -1 | -1 | -1 | 1 | -1 | 1 | linear of order 8 |
| ρ15 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | 1 | ζ87 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | ζ85 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ16 | 1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | -i | i | 1 | ζ83 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | ζ8 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ17 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | -2 | -2 | 0 | 0 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | 0 | -2 | 2 | 0 | orthogonal lifted from D4 |
| 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 | -2 | 0 | 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 | -2 | 0 | 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 F5 |
| ρ23 | 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 | -1 | √5 | orthogonal lifted from C22⋊F5 |
| ρ24 | 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 | -1 | -√5 | orthogonal lifted from C22⋊F5 |
| ρ25 | 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 | symplectic lifted from C5⋊C8, Schur index 2 |
| ρ26 | 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 | symplectic lifted from C5⋊C8, 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 | 1 | √5 | 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 | 1 | -√5 | symplectic lifted from C22.F5, Schur index 2 |
►On 80 points
Generators in S80
(2 59)(4 61)(6 63)(8 57)(9 26)(11 28)(13 30)(15 32)(17 37)(19 39)(21 33)(23 35)(42 55)(44 49)(46 51)(48 53)(66 80)(68 74)(70 76)(72 78)
(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)(65 79)(66 80)(67 73)(68 74)(69 75)(70 76)(71 77)(72 78)
(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 12 18 41 79)(2 42 13 80 19)(3 73 43 20 14)(4 21 74 15 44)(5 16 22 45 75)(6 46 9 76 23)(7 77 47 24 10)(8 17 78 11 48)(25 34 50 69 62)(26 70 35 63 51)(27 64 71 52 36)(28 53 57 37 72)(29 38 54 65 58)(30 66 39 59 55)(31 60 67 56 40)(32 49 61 33 68)
(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)| (2,59)(4,61)(6,63)(8,57)(9,26)(11,28)(13,30)(15,32)(17,37)(19,39)(21,33)(23,35)(42,55)(44,49)(46,51)(48,53)(66,80)(68,74)(70,76)(72,78), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (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), (1,12,18,41,79)(2,42,13,80,19)(3,73,43,20,14)(4,21,74,15,44)(5,16,22,45,75)(6,46,9,76,23)(7,77,47,24,10)(8,17,78,11,48)(25,34,50,69,62)(26,70,35,63,51)(27,64,71,52,36)(28,53,57,37,72)(29,38,54,65,58)(30,66,39,59,55)(31,60,67,56,40)(32,49,61,33,68), (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( (2,59)(4,61)(6,63)(8,57)(9,26)(11,28)(13,30)(15,32)(17,37)(19,39)(21,33)(23,35)(42,55)(44,49)(46,51)(48,53)(66,80)(68,74)(70,76)(72,78), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,25)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,36)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53)(65,79)(66,80)(67,73)(68,74)(69,75)(70,76)(71,77)(72,78), (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), (1,12,18,41,79)(2,42,13,80,19)(3,73,43,20,14)(4,21,74,15,44)(5,16,22,45,75)(6,46,9,76,23)(7,77,47,24,10)(8,17,78,11,48)(25,34,50,69,62)(26,70,35,63,51)(27,64,71,52,36)(28,53,57,37,72)(29,38,54,65,58)(30,66,39,59,55)(31,60,67,56,40)(32,49,61,33,68), (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([[(2,59),(4,61),(6,63),(8,57),(9,26),(11,28),(13,30),(15,32),(17,37),(19,39),(21,33),(23,35),(42,55),(44,49),(46,51),(48,53),(66,80),(68,74),(70,76),(72,78)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,25),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,36),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53),(65,79),(66,80),(67,73),(68,74),(69,75),(70,76),(71,77),(72,78)], [(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)], [(1,12,18,41,79),(2,42,13,80,19),(3,73,43,20,14),(4,21,74,15,44),(5,16,22,45,75),(6,46,9,76,23),(7,77,47,24,10),(8,17,78,11,48),(25,34,50,69,62),(26,70,35,63,51),(27,64,71,52,36),(28,53,57,37,72),(29,38,54,65,58),(30,66,39,59,55),(31,60,67,56,40),(32,49,61,33,68)], [(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)]])
C23.2F5 is a maximal subgroup of
(C2×C20)⋊1C8 (C22×C4).F5 C5⋊(C23⋊C8) C24.F5 C5⋊C8⋊8D4 C5⋊C8⋊D4 D10⋊M4(2) Dic5⋊M4(2) C23.(C2×F5) Dic5.12M4(2) D10.11M4(2) C20.34M4(2) D10⋊10M4(2) Dic5.13M4(2) C20.30M4(2) D4×C5⋊C8 C5⋊C8⋊7D4 C20⋊2M4(2) (C2×D4).7F5 (C2×D4).8F5 C24.4F5 Dic5.22D12 C30.22M4(2) Dic5.S4
C23.2F5 is a maximal quotient of
C10.6M5(2) C20.29M4(2) (C2×C20)⋊1C8 C10.(C4⋊C8) Dic5.23D8 Dic5.12Q16 D4.(C5⋊C8) C24.F5 Dic5.22D12 C30.22M4(2)
Matrix representation of C23.2F5 ►in GL6(𝔽41)
| 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 | 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 | 7 | 40 | 0 | 0 |
| 0 | 0 | 8 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 7 | 34 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 21 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 14 | 31 | 0 | 0 |
| 0 | 0 | 32 | 27 | 0 | 0 |
G:=sub<GL(6,GF(41))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,8,0,0,0,0,40,40,0,0,0,0,0,0,0,7,0,0,0,0,35,34],[0,21,0,0,0,0,2,0,0,0,0,0,0,0,0,0,14,32,0,0,0,0,31,27,0,0,1,0,0,0,0,0,0,1,0,0] >;
C23.2F5 in GAP, Magma, Sage, TeX
C_2^3._2F_5
% in TeX
G:=Group("C2^3.2F5"); // GroupNames label
G:=SmallGroup(160,87);
// by ID
G=gap.SmallGroup(160,87);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,2309,1169]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^5=1,e^4=c,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
// generators/relations
Export
Subgroup lattice of C23.2F5 in TeX
Character table of C23.2F5 in TeX