direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×M4(2)⋊4C4, M4(2)⋊4C20, (C2×C8)⋊2C20, (C2×C40)⋊14C4, C20.85(C4⋊C4), (C2×C20).38Q8, (C2×C20).510D4, C22⋊C4.1C20, C23.7(C2×C20), C22.3(C4×C20), (C5×M4(2))⋊16C4, (C2×C10).32C42, C42⋊C2.3C10, (C2×M4(2)).9C10, C20.153(C22⋊C4), (C10×M4(2)).21C2, (C22×C20).389C22, C10.47(C2.C42), C4.5(C5×C4⋊C4), (C2×C4).3(C5×Q8), C22.6(C5×C4⋊C4), (C2×C4).15(C2×C20), (C2×C4).115(C5×D4), C4.27(C5×C22⋊C4), (C2×C10).51(C4⋊C4), (C2×C20).499(C2×C4), (C5×C22⋊C4).10C4, C22.9(C5×C22⋊C4), (C22×C4).19(C2×C10), (C22×C10).86(C2×C4), C2.9(C5×C2.C42), (C5×C42⋊C2).17C2, (C2×C10).136(C22⋊C4), SmallGroup(320,149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M4(2)⋊4C4
G = < a,b,c,d | a5=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=bc, dcd-1=b4c >
Subgroups: 138 in 90 conjugacy classes, 54 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C42⋊C2, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, M4(2)⋊4C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C5×C42⋊C2, C10×M4(2), C5×M4(2)⋊4C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C2×C20, C5×D4, C5×Q8, M4(2)⋊4C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×M4(2)⋊4C4
►On 80 points
(1 62 19 54 11)(2 63 20 55 12)(3 64 21 56 13)(4 57 22 49 14)(5 58 23 50 15)(6 59 24 51 16)(7 60 17 52 9)(8 61 18 53 10)(25 42 76 33 68)(26 43 77 34 69)(27 44 78 35 70)(28 45 79 36 71)(29 46 80 37 72)(30 47 73 38 65)(31 48 74 39 66)(32 41 75 40 67)
(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)
(1 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 71)(10 68)(11 65)(12 70)(13 67)(14 72)(15 69)(16 66)(17 79)(18 76)(19 73)(20 78)(21 75)(22 80)(23 77)(24 74)(33 53)(34 50)(35 55)(36 52)(37 49)(38 54)(39 51)(40 56)(41 64)(42 61)(43 58)(44 63)(45 60)(46 57)(47 62)(48 59)
(2 31 6 27)(3 7)(4 29 8 25)(9 13)(10 68 14 72)(12 66 16 70)(17 21)(18 76 22 80)(20 74 24 78)(26 30)(33 49 37 53)(34 38)(35 55 39 51)(42 57 46 61)(43 47)(44 63 48 59)(52 56)(60 64)(65 69)(73 77)
G:=sub<Sym(80)| (1,62,19,54,11)(2,63,20,55,12)(3,64,21,56,13)(4,57,22,49,14)(5,58,23,50,15)(6,59,24,51,16)(7,60,17,52,9)(8,61,18,53,10)(25,42,76,33,68)(26,43,77,34,69)(27,44,78,35,70)(28,45,79,36,71)(29,46,80,37,72)(30,47,73,38,65)(31,48,74,39,66)(32,41,75,40,67), (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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(17,79)(18,76)(19,73)(20,78)(21,75)(22,80)(23,77)(24,74)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59), (2,31,6,27)(3,7)(4,29,8,25)(9,13)(10,68,14,72)(12,66,16,70)(17,21)(18,76,22,80)(20,74,24,78)(26,30)(33,49,37,53)(34,38)(35,55,39,51)(42,57,46,61)(43,47)(44,63,48,59)(52,56)(60,64)(65,69)(73,77)>;
G:=Group( (1,62,19,54,11)(2,63,20,55,12)(3,64,21,56,13)(4,57,22,49,14)(5,58,23,50,15)(6,59,24,51,16)(7,60,17,52,9)(8,61,18,53,10)(25,42,76,33,68)(26,43,77,34,69)(27,44,78,35,70)(28,45,79,36,71)(29,46,80,37,72)(30,47,73,38,65)(31,48,74,39,66)(32,41,75,40,67), (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), (1,30)(2,27)(3,32)(4,29)(5,26)(6,31)(7,28)(8,25)(9,71)(10,68)(11,65)(12,70)(13,67)(14,72)(15,69)(16,66)(17,79)(18,76)(19,73)(20,78)(21,75)(22,80)(23,77)(24,74)(33,53)(34,50)(35,55)(36,52)(37,49)(38,54)(39,51)(40,56)(41,64)(42,61)(43,58)(44,63)(45,60)(46,57)(47,62)(48,59), (2,31,6,27)(3,7)(4,29,8,25)(9,13)(10,68,14,72)(12,66,16,70)(17,21)(18,76,22,80)(20,74,24,78)(26,30)(33,49,37,53)(34,38)(35,55,39,51)(42,57,46,61)(43,47)(44,63,48,59)(52,56)(60,64)(65,69)(73,77) );
G=PermutationGroup([[(1,62,19,54,11),(2,63,20,55,12),(3,64,21,56,13),(4,57,22,49,14),(5,58,23,50,15),(6,59,24,51,16),(7,60,17,52,9),(8,61,18,53,10),(25,42,76,33,68),(26,43,77,34,69),(27,44,78,35,70),(28,45,79,36,71),(29,46,80,37,72),(30,47,73,38,65),(31,48,74,39,66),(32,41,75,40,67)], [(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)], [(1,30),(2,27),(3,32),(4,29),(5,26),(6,31),(7,28),(8,25),(9,71),(10,68),(11,65),(12,70),(13,67),(14,72),(15,69),(16,66),(17,79),(18,76),(19,73),(20,78),(21,75),(22,80),(23,77),(24,74),(33,53),(34,50),(35,55),(36,52),(37,49),(38,54),(39,51),(40,56),(41,64),(42,61),(43,58),(44,63),(45,60),(46,57),(47,62),(48,59)], [(2,31,6,27),(3,7),(4,29,8,25),(9,13),(10,68,14,72),(12,66,16,70),(17,21),(18,76,22,80),(20,74,24,78),(26,30),(33,49,37,53),(34,38),(35,55,39,51),(42,57,46,61),(43,47),(44,63,48,59),(52,56),(60,64),(65,69),(73,77)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20T | 20U | ··· | 20AJ | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | C20 | D4 | Q8 | C5×D4 | C5×Q8 | M4(2)⋊4C4 | C5×M4(2)⋊4C4 |
| kernel | C5×M4(2)⋊4C4 | C5×C42⋊C2 | C10×M4(2) | C5×C22⋊C4 | C2×C40 | C5×M4(2) | M4(2)⋊4C4 | C42⋊C2 | C2×M4(2) | C22⋊C4 | C2×C8 | M4(2) | C2×C20 | C2×C20 | C2×C4 | C2×C4 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 16 | 16 | 16 | 3 | 1 | 12 | 4 | 2 | 8 |
Matrix representation of C5×M4(2)⋊4C4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,0,0,40,0,0,1,0,32,0,0,0,0,9,0,0],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,40,0,0,0,0,0,1,0,0,40,0] >;
C5×M4(2)⋊4C4 in GAP, Magma, Sage, TeX
C_5\times M_4(2)\rtimes_4C_4
% in TeX
G:=Group("C5xM4(2):4C4"); // GroupNames label
G:=SmallGroup(320,149);
// by ID
G=gap.SmallGroup(320,149);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,3511,124]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b*c,d*c*d^-1=b^4*c>;
// generators/relations