metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4F5, (C2×C10)⋊8M4(2), (C23×C10).7C4, C23.51(C2×F5), C5⋊3(C24.4C4), (C2×Dic5).264D4, Dic5.117(C2×D4), C10.34(C2×M4(2)), C23.2F5⋊13C2, C22⋊2(C22.F5), (C22×Dic5).37C4, (C23×Dic5).13C2, C22.30(C22⋊F5), C22.100(C22×F5), Dic5.52(C22⋊C4), (C2×Dic5).360C23, (C22×Dic5).280C22, (C2×C5⋊C8)⋊3C22, (C2×C22.F5)⋊8C2, C2.39(C2×C22⋊F5), C10.39(C2×C22⋊C4), C2.12(C2×C22.F5), (C2×C10).92(C22×C4), (C22×C10).74(C2×C4), (C2×C10).63(C22⋊C4), (C2×Dic5).196(C2×C4), SmallGroup(320,1136)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 602 in 190 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×6], C22, C22 [×6], C22 [×14], C5, C8 [×4], C2×C4 [×18], C23, C23 [×2], C23 [×6], C10, C10 [×2], C10 [×6], C2×C8 [×4], M4(2) [×4], C22×C4 [×10], C24, Dic5 [×4], Dic5 [×2], C2×C10, C2×C10 [×6], C2×C10 [×14], C22⋊C8 [×4], C2×M4(2) [×2], C23×C4, C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×6], C2×Dic5 [×10], C22×C10, C22×C10 [×2], C22×C10 [×6], C24.4C4, C2×C5⋊C8 [×4], C22.F5 [×4], C22×Dic5 [×2], C22×Dic5 [×4], C22×Dic5 [×4], C23×C10, C23.2F5 [×4], C2×C22.F5 [×2], C23×Dic5, C24.4F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×M4(2) [×2], C2×F5 [×3], C24.4C4, C22.F5 [×4], C22⋊F5 [×2], C22×F5, C2×C22.F5 [×2], C2×C22⋊F5, C24.4F5
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >
►On 80 points
Generators in S80
(2 47)(4 41)(6 43)(8 45)(9 63)(11 57)(13 59)(15 61)(17 36)(19 38)(21 40)(23 34)(26 53)(28 55)(30 49)(32 51)(66 74)(68 76)(70 78)(72 80)
(1 5)(3 7)(10 14)(12 16)(18 22)(20 24)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(65 69)(67 71)(73 77)(75 79)
(1 46)(2 47)(3 48)(4 41)(5 42)(6 43)(7 44)(8 45)(9 63)(10 64)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(25 52)(26 53)(27 54)(28 55)(29 56)(30 49)(31 50)(32 51)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)(71 79)(72 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 35 58 29 73)(2 30 36 74 59)(3 75 31 60 37)(4 61 76 38 32)(5 39 62 25 77)(6 26 40 78 63)(7 79 27 64 33)(8 57 80 34 28)(9 43 53 21 70)(10 22 44 71 54)(11 72 23 55 45)(12 56 65 46 24)(13 47 49 17 66)(14 18 48 67 50)(15 68 19 51 41)(16 52 69 42 20)
(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)| (2,47)(4,41)(6,43)(8,45)(9,63)(11,57)(13,59)(15,61)(17,36)(19,38)(21,40)(23,34)(26,53)(28,55)(30,49)(32,51)(66,74)(68,76)(70,78)(72,80), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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), (1,35,58,29,73)(2,30,36,74,59)(3,75,31,60,37)(4,61,76,38,32)(5,39,62,25,77)(6,26,40,78,63)(7,79,27,64,33)(8,57,80,34,28)(9,43,53,21,70)(10,22,44,71,54)(11,72,23,55,45)(12,56,65,46,24)(13,47,49,17,66)(14,18,48,67,50)(15,68,19,51,41)(16,52,69,42,20), (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( (2,47)(4,41)(6,43)(8,45)(9,63)(11,57)(13,59)(15,61)(17,36)(19,38)(21,40)(23,34)(26,53)(28,55)(30,49)(32,51)(66,74)(68,76)(70,78)(72,80), (1,5)(3,7)(10,14)(12,16)(18,22)(20,24)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(65,69)(67,71)(73,77)(75,79), (1,46)(2,47)(3,48)(4,41)(5,42)(6,43)(7,44)(8,45)(9,63)(10,64)(11,57)(12,58)(13,59)(14,60)(15,61)(16,62)(17,36)(18,37)(19,38)(20,39)(21,40)(22,33)(23,34)(24,35)(25,52)(26,53)(27,54)(28,55)(29,56)(30,49)(31,50)(32,51)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78)(71,79)(72,80), (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), (1,35,58,29,73)(2,30,36,74,59)(3,75,31,60,37)(4,61,76,38,32)(5,39,62,25,77)(6,26,40,78,63)(7,79,27,64,33)(8,57,80,34,28)(9,43,53,21,70)(10,22,44,71,54)(11,72,23,55,45)(12,56,65,46,24)(13,47,49,17,66)(14,18,48,67,50)(15,68,19,51,41)(16,52,69,42,20), (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([(2,47),(4,41),(6,43),(8,45),(9,63),(11,57),(13,59),(15,61),(17,36),(19,38),(21,40),(23,34),(26,53),(28,55),(30,49),(32,51),(66,74),(68,76),(70,78),(72,80)], [(1,5),(3,7),(10,14),(12,16),(18,22),(20,24),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(65,69),(67,71),(73,77),(75,79)], [(1,46),(2,47),(3,48),(4,41),(5,42),(6,43),(7,44),(8,45),(9,63),(10,64),(11,57),(12,58),(13,59),(14,60),(15,61),(16,62),(17,36),(18,37),(19,38),(20,39),(21,40),(22,33),(23,34),(24,35),(25,52),(26,53),(27,54),(28,55),(29,56),(30,49),(31,50),(32,51),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78),(71,79),(72,80)], [(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)], [(1,35,58,29,73),(2,30,36,74,59),(3,75,31,60,37),(4,61,76,38,32),(5,39,62,25,77),(6,26,40,78,63),(7,79,27,64,33),(8,57,80,34,28),(9,43,53,21,70),(10,22,44,71,54),(11,72,23,55,45),(12,56,65,46,24),(13,47,49,17,66),(14,18,48,67,50),(15,68,19,51,41),(16,52,69,42,20)], [(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)])
Matrix representation ►G ⊆ 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 34 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 38 | 3 | 0 | 0 |
| 0 | 0 | 24 | 3 | 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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,38,24,0,0,0,0,3,3,0,0,1,0,0,0,0,0,0,1,0,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5 | 8A | ··· | 8H | 10A | ··· | 10O |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 20 | ··· | 20 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | M4(2) | F5 | C2×F5 | C22.F5 | C22⋊F5 |
| kernel | C24.4F5 | C23.2F5 | C2×C22.F5 | C23×Dic5 | C22×Dic5 | C23×C10 | C2×Dic5 | C2×C10 | C24 | C23 | C22 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 4 | 8 | 1 | 3 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2^4._4F_5
% in TeX
G:=Group("C2^4.4F5"); // GroupNames label
G:=SmallGroup(320,1136);
// by ID
G=gap.SmallGroup(320,1136);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,253,758,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀