direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C24.4C4, C24.4C20, C4.70(D4×C10), C22⋊C8⋊11C10, (C2×C40)⋊37C22, C20.475(C2×D4), (C2×C20).513D4, (C23×C4).8C10, (C2×M4(2))⋊6C10, (C2×C10)⋊13M4(2), C23.29(C2×C20), (C23×C20).23C2, (C23×C10).12C4, (C22×C20).48C4, (C22×C4).12C20, C2.6(C10×M4(2)), C22⋊2(C5×M4(2)), (C10×M4(2))⋊24C2, (C2×C20).982C23, C10.82(C2×M4(2)), C20.154(C22⋊C4), C22.41(C22×C20), (C22×C20).496C22, (C2×C8)⋊7(C2×C10), (C5×C22⋊C8)⋊28C2, (C2×C4).71(C2×C20), (C2×C4).118(C5×D4), C4.21(C5×C22⋊C4), (C2×C20).506(C2×C4), C2.10(C10×C22⋊C4), C10.139(C2×C22⋊C4), (C22×C4).92(C2×C10), C22.17(C5×C22⋊C4), (C2×C10).333(C22×C4), (C2×C4).150(C22×C10), (C22×C10).183(C2×C4), (C2×C10).144(C22⋊C4), SmallGroup(320,908)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C24.4C4
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 290 in 190 conjugacy classes, 90 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, C2×M4(2), C23×C4, C40, C2×C20, C2×C20, C2×C20, C22×C10, C22×C10, C22×C10, C24.4C4, C2×C40, C5×M4(2), C22×C20, C22×C20, C22×C20, C23×C10, C5×C22⋊C8, C10×M4(2), C23×C20, C5×C24.4C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, M4(2), C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C2×M4(2), C2×C20, C5×D4, C22×C10, C24.4C4, C5×C22⋊C4, C5×M4(2), C22×C20, D4×C10, C10×C22⋊C4, C10×M4(2), C5×C24.4C4
►On 80 points
(1 59 24 56 16)(2 60 17 49 9)(3 61 18 50 10)(4 62 19 51 11)(5 63 20 52 12)(6 64 21 53 13)(7 57 22 54 14)(8 58 23 55 15)(25 45 73 33 65)(26 46 74 34 66)(27 47 75 35 67)(28 48 76 36 68)(29 41 77 37 69)(30 42 78 38 70)(31 43 79 39 71)(32 44 80 40 72)
(2 32)(4 26)(6 28)(8 30)(9 72)(11 66)(13 68)(15 70)(17 80)(19 74)(21 76)(23 78)(34 51)(36 53)(38 55)(40 49)(42 58)(44 60)(46 62)(48 64)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(41 45)(43 47)(50 54)(52 56)(57 61)(59 63)(65 69)(67 71)(73 77)(75 79)
(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 72)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 80)(18 73)(19 74)(20 75)(21 76)(22 77)(23 78)(24 79)(33 50)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 49)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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 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)| (1,59,24,56,16)(2,60,17,49,9)(3,61,18,50,10)(4,62,19,51,11)(5,63,20,52,12)(6,64,21,53,13)(7,57,22,54,14)(8,58,23,55,15)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,72)(11,66)(13,68)(15,70)(17,80)(19,74)(21,76)(23,78)(34,51)(36,53)(38,55)(40,49)(42,58)(44,60)(46,62)(48,64), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,72)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,80)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,2,3,4,5,6,7,8)(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( (1,59,24,56,16)(2,60,17,49,9)(3,61,18,50,10)(4,62,19,51,11)(5,63,20,52,12)(6,64,21,53,13)(7,57,22,54,14)(8,58,23,55,15)(25,45,73,33,65)(26,46,74,34,66)(27,47,75,35,67)(28,48,76,36,68)(29,41,77,37,69)(30,42,78,38,70)(31,43,79,39,71)(32,44,80,40,72), (2,32)(4,26)(6,28)(8,30)(9,72)(11,66)(13,68)(15,70)(17,80)(19,74)(21,76)(23,78)(34,51)(36,53)(38,55)(40,49)(42,58)(44,60)(46,62)(48,64), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(41,45)(43,47)(50,54)(52,56)(57,61)(59,63)(65,69)(67,71)(73,77)(75,79), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,72)(10,65)(11,66)(12,67)(13,68)(14,69)(15,70)(16,71)(17,80)(18,73)(19,74)(20,75)(21,76)(22,77)(23,78)(24,79)(33,50)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,49)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (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), (1,2,3,4,5,6,7,8)(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([[(1,59,24,56,16),(2,60,17,49,9),(3,61,18,50,10),(4,62,19,51,11),(5,63,20,52,12),(6,64,21,53,13),(7,57,22,54,14),(8,58,23,55,15),(25,45,73,33,65),(26,46,74,34,66),(27,47,75,35,67),(28,48,76,36,68),(29,41,77,37,69),(30,42,78,38,70),(31,43,79,39,71),(32,44,80,40,72)], [(2,32),(4,26),(6,28),(8,30),(9,72),(11,66),(13,68),(15,70),(17,80),(19,74),(21,76),(23,78),(34,51),(36,53),(38,55),(40,49),(42,58),(44,60),(46,62),(48,64)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(41,45),(43,47),(50,54),(52,56),(57,61),(59,63),(65,69),(67,71),(73,77),(75,79)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,72),(10,65),(11,66),(12,67),(13,68),(14,69),(15,70),(16,71),(17,80),(18,73),(19,74),(20,75),(21,76),(22,77),(23,78),(24,79),(33,50),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,49),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(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)], [(1,2,3,4,5,6,7,8),(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)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10L | 10M | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20AN | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | D4 | M4(2) | C5×D4 | C5×M4(2) |
| kernel | C5×C24.4C4 | C5×C22⋊C8 | C10×M4(2) | C23×C20 | C22×C20 | C23×C10 | C24.4C4 | C22⋊C8 | C2×M4(2) | C23×C4 | C22×C4 | C24 | C2×C20 | C2×C10 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 16 | 8 | 4 | 24 | 8 | 4 | 8 | 16 | 32 |
Matrix representation of C5×C24.4C4 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 9 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,2,40],[40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[0,9,0,0,1,0,0,0,0,0,1,40,0,0,0,40] >;
C5×C24.4C4 in GAP, Magma, Sage, TeX
C_5\times C_2^4._4C_4
% in TeX
G:=Group("C5xC2^4.4C4"); // GroupNames label
G:=SmallGroup(320,908);
// by ID
G=gap.SmallGroup(320,908);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,3446,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^5=b^2=c^2=d^2=e^2=1,f^4=e,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations