metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C8)⋊6F5, (C2×C40)⋊6C4, (C8×D5)⋊7C4, C40⋊C4⋊6C2, D5.D8⋊5C2, C8.27(C2×F5), C40.24(C2×C4), (C4×D5).89D4, C4.30(C4⋊F5), C20.30(C4⋊C4), (C4×D5).32Q8, D5.1(C4○D8), D10.31(C2×D4), C5⋊(C23.25D4), C4⋊F5.17C22, C22.5(C4⋊F5), D10.29(C4⋊C4), C4.37(C22×F5), C20.77(C22×C4), (C2×Dic5).35Q8, Dic5.15(C2×Q8), (C22×D5).99D4, (C8×D5).54C22, (C4×D5).77C23, Dic5.30(C4⋊C4), D10.C23.11C2, (D5×C2×C8).22C2, (C2×C5⋊2C8)⋊20C4, C2.16(C2×C4⋊F5), C10.13(C2×C4⋊C4), C5⋊2C8.47(C2×C4), (C4×D5).86(C2×C4), (C2×C4).138(C2×F5), (C2×C10).21(C4⋊C4), (C2×C20).147(C2×C4), (C2×C4×D5).403C22, SmallGroup(320,1059)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 442 in 114 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], D5, C10, C10, C42 [×2], C22⋊C4 [×2], C4⋊C4 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C22×C8, C5⋊2C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C2×F5 [×4], C22×D5, C23.25D4, C8×D5 [×4], C2×C5⋊2C8, C2×C40, C4×F5 [×2], C4⋊F5 [×4], C22⋊F5 [×2], C2×C4×D5, C40⋊C4 [×2], D5.D8 [×2], D5×C2×C8, D10.C23 [×2], (C2×C8)⋊6F5
Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C2×C4⋊C4, C4○D8 [×2], C2×F5 [×3], C23.25D4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, (C2×C8)⋊6F5
Generators and relations
G = < a,b,c,d | a2=b8=c5=d4=1, ab=ba, ac=ca, dad-1=ab4, bc=cb, dbd-1=b3, dcd-1=c3 >
►On 80 points
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 69)(10 70)(11 71)(12 72)(13 65)(14 66)(15 67)(16 68)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 73)(48 74)
(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 53 44 71 59)(2 54 45 72 60)(3 55 46 65 61)(4 56 47 66 62)(5 49 48 67 63)(6 50 41 68 64)(7 51 42 69 57)(8 52 43 70 58)(9 35 19 29 76)(10 36 20 30 77)(11 37 21 31 78)(12 38 22 32 79)(13 39 23 25 80)(14 40 24 26 73)(15 33 17 27 74)(16 34 18 28 75)
(1 19 5 23)(2 22 6 18)(3 17 7 21)(4 20 8 24)(9 49 80 59)(10 52 73 62)(11 55 74 57)(12 50 75 60)(13 53 76 63)(14 56 77 58)(15 51 78 61)(16 54 79 64)(25 44 35 67)(26 47 36 70)(27 42 37 65)(28 45 38 68)(29 48 39 71)(30 43 40 66)(31 46 33 69)(32 41 34 72)
G:=sub<Sym(80)| (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74), (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), (1,53,44,71,59)(2,54,45,72,60)(3,55,46,65,61)(4,56,47,66,62)(5,49,48,67,63)(6,50,41,68,64)(7,51,42,69,57)(8,52,43,70,58)(9,35,19,29,76)(10,36,20,30,77)(11,37,21,31,78)(12,38,22,32,79)(13,39,23,25,80)(14,40,24,26,73)(15,33,17,27,74)(16,34,18,28,75), (1,19,5,23)(2,22,6,18)(3,17,7,21)(4,20,8,24)(9,49,80,59)(10,52,73,62)(11,55,74,57)(12,50,75,60)(13,53,76,63)(14,56,77,58)(15,51,78,61)(16,54,79,64)(25,44,35,67)(26,47,36,70)(27,42,37,65)(28,45,38,68)(29,48,39,71)(30,43,40,66)(31,46,33,69)(32,41,34,72)>;
G:=Group( (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,69)(10,70)(11,71)(12,72)(13,65)(14,66)(15,67)(16,68)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,75)(42,76)(43,77)(44,78)(45,79)(46,80)(47,73)(48,74), (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), (1,53,44,71,59)(2,54,45,72,60)(3,55,46,65,61)(4,56,47,66,62)(5,49,48,67,63)(6,50,41,68,64)(7,51,42,69,57)(8,52,43,70,58)(9,35,19,29,76)(10,36,20,30,77)(11,37,21,31,78)(12,38,22,32,79)(13,39,23,25,80)(14,40,24,26,73)(15,33,17,27,74)(16,34,18,28,75), (1,19,5,23)(2,22,6,18)(3,17,7,21)(4,20,8,24)(9,49,80,59)(10,52,73,62)(11,55,74,57)(12,50,75,60)(13,53,76,63)(14,56,77,58)(15,51,78,61)(16,54,79,64)(25,44,35,67)(26,47,36,70)(27,42,37,65)(28,45,38,68)(29,48,39,71)(30,43,40,66)(31,46,33,69)(32,41,34,72) );
G=PermutationGroup([(1,21),(2,22),(3,23),(4,24),(5,17),(6,18),(7,19),(8,20),(9,69),(10,70),(11,71),(12,72),(13,65),(14,66),(15,67),(16,68),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,75),(42,76),(43,77),(44,78),(45,79),(46,80),(47,73),(48,74)], [(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)], [(1,53,44,71,59),(2,54,45,72,60),(3,55,46,65,61),(4,56,47,66,62),(5,49,48,67,63),(6,50,41,68,64),(7,51,42,69,57),(8,52,43,70,58),(9,35,19,29,76),(10,36,20,30,77),(11,37,21,31,78),(12,38,22,32,79),(13,39,23,25,80),(14,40,24,26,73),(15,33,17,27,74),(16,34,18,28,75)], [(1,19,5,23),(2,22,6,18),(3,17,7,21),(4,20,8,24),(9,49,80,59),(10,52,73,62),(11,55,74,57),(12,50,75,60),(13,53,76,63),(14,56,77,58),(15,51,78,61),(16,54,79,64),(25,44,35,67),(26,47,36,70),(27,42,37,65),(28,45,38,68),(29,48,39,71),(30,43,40,66),(31,46,33,69),(32,41,34,72)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 23 | 0 | 0 | 0 | 0 |
| 9 | 9 | 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 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 15 | 0 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 |
G:=sub<GL(6,GF(41))| [32,9,0,0,0,0,23,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,15,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,1,0,0,0,0,40,0,1,0,0,0,40,0,0,1,0,0,40,0,0,0],[40,1,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,1,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,40,0,1] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 1 | 1 | 2 | 5 | 5 | 10 | 20 | ··· | 20 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | Q8 | Q8 | D4 | C4○D8 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | (C2×C8)⋊6F5 |
| kernel | (C2×C8)⋊6F5 | C40⋊C4 | D5.D8 | D5×C2×C8 | D10.C23 | C8×D5 | C2×C5⋊2C8 | C2×C40 | C4×D5 | C4×D5 | C2×Dic5 | C22×D5 | D5 | C2×C8 | C8 | C2×C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 2 | 1 | 2 | 4 | 2 | 2 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 2 | 2 | 8 |
In GAP, Magma, Sage, TeX
(C_2\times C_8)\rtimes_6F_5
% in TeX
G:=Group("(C2xC8):6F5"); // GroupNames label
G:=SmallGroup(320,1059);
// by ID
G=gap.SmallGroup(320,1059);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^8=c^5=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^4,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations