metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C8×D5).6C4, (C2×C8).13F5, C8.29(C2×F5), (C2×C40).13C4, C40.26(C2×C4), D5⋊(C8.C4), (C4×D5).90D4, C4.31(C4⋊F5), D10.Q8⋊7C2, C40.C4⋊6C2, C20.31(C4⋊C4), (C4×D5).33Q8, D10.14(C2×Q8), C22.6(C4⋊F5), D10.32(C4⋊C4), C4.40(C22×F5), C4.F5.9C22, C20.80(C22×C4), Dic5.33(C2×D4), (C4×D5).80C23, (C8×D5).58C22, (C22×D5).20Q8, Dic5.33(C4⋊C4), (C2×Dic5).177D4, D5⋊M4(2).11C2, C5⋊3(C2×C8.C4), (D5×C2×C8).23C2, C2.19(C2×C4⋊F5), C10.16(C2×C4⋊C4), (C2×C5⋊2C8).27C4, C5⋊2C8.50(C2×C4), (C4×D5).89(C2×C4), (C2×C4).141(C2×F5), (C2×C10).24(C4⋊C4), (C2×C20).148(C2×C4), (C2×C4×D5).404C22, SmallGroup(320,1062)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 346 in 106 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], D5, C10, C10, C2×C8, C2×C8 [×7], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8.C4 [×4], C22×C8, C2×M4(2) [×2], C5⋊2C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C2×C5⋊2C8, C2×C40, D5⋊C8 [×2], C4.F5 [×4], C22.F5 [×2], C2×C4×D5, C40.C4 [×2], D10.Q8 [×2], D5×C2×C8, D5⋊M4(2) [×2], (C8×D5).C4
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, C8.C4 [×2], C2×C4⋊C4, C2×F5 [×3], C2×C8.C4, C4⋊F5 [×2], C22×F5, C2×C4⋊F5, (C8×D5).C4
Generators and relations
G = < a,b,c,d | a8=b5=c2=1, d4=a4, ab=ba, ac=ca, dad-1=a3, cbc=b-1, dbd-1=b3, dcd-1=b2c >
►On 80 points
(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 68 61 46 49)(2 69 62 47 50)(3 70 63 48 51)(4 71 64 41 52)(5 72 57 42 53)(6 65 58 43 54)(7 66 59 44 55)(8 67 60 45 56)(9 26 23 76 33)(10 27 24 77 34)(11 28 17 78 35)(12 29 18 79 36)(13 30 19 80 37)(14 31 20 73 38)(15 32 21 74 39)(16 25 22 75 40)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 76)(10 77)(11 78)(12 79)(13 80)(14 73)(15 74)(16 75)(17 28)(18 29)(19 30)(20 31)(21 32)(22 25)(23 26)(24 27)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)
(1 17 7 19 5 21 3 23)(2 20 8 22 6 24 4 18)(9 68 35 55 13 72 39 51)(10 71 36 50 14 67 40 54)(11 66 37 53 15 70 33 49)(12 69 38 56 16 65 34 52)(25 43 77 64 29 47 73 60)(26 46 78 59 30 42 74 63)(27 41 79 62 31 45 75 58)(28 44 80 57 32 48 76 61)
G:=sub<Sym(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), (1,68,61,46,49)(2,69,62,47,50)(3,70,63,48,51)(4,71,64,41,52)(5,72,57,42,53)(6,65,58,43,54)(7,66,59,44,55)(8,67,60,45,56)(9,26,23,76,33)(10,27,24,77,34)(11,28,17,78,35)(12,29,18,79,36)(13,30,19,80,37)(14,31,20,73,38)(15,32,21,74,39)(16,25,22,75,40), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,17,7,19,5,21,3,23)(2,20,8,22,6,24,4,18)(9,68,35,55,13,72,39,51)(10,71,36,50,14,67,40,54)(11,66,37,53,15,70,33,49)(12,69,38,56,16,65,34,52)(25,43,77,64,29,47,73,60)(26,46,78,59,30,42,74,63)(27,41,79,62,31,45,75,58)(28,44,80,57,32,48,76,61)>;
G:=Group( (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), (1,68,61,46,49)(2,69,62,47,50)(3,70,63,48,51)(4,71,64,41,52)(5,72,57,42,53)(6,65,58,43,54)(7,66,59,44,55)(8,67,60,45,56)(9,26,23,76,33)(10,27,24,77,34)(11,28,17,78,35)(12,29,18,79,36)(13,30,19,80,37)(14,31,20,73,38)(15,32,21,74,39)(16,25,22,75,40), (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,76)(10,77)(11,78)(12,79)(13,80)(14,73)(15,74)(16,75)(17,28)(18,29)(19,30)(20,31)(21,32)(22,25)(23,26)(24,27)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70), (1,17,7,19,5,21,3,23)(2,20,8,22,6,24,4,18)(9,68,35,55,13,72,39,51)(10,71,36,50,14,67,40,54)(11,66,37,53,15,70,33,49)(12,69,38,56,16,65,34,52)(25,43,77,64,29,47,73,60)(26,46,78,59,30,42,74,63)(27,41,79,62,31,45,75,58)(28,44,80,57,32,48,76,61) );
G=PermutationGroup([(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)], [(1,68,61,46,49),(2,69,62,47,50),(3,70,63,48,51),(4,71,64,41,52),(5,72,57,42,53),(6,65,58,43,54),(7,66,59,44,55),(8,67,60,45,56),(9,26,23,76,33),(10,27,24,77,34),(11,28,17,78,35),(12,29,18,79,36),(13,30,19,80,37),(14,31,20,73,38),(15,32,21,74,39),(16,25,22,75,40)], [(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,76),(10,77),(11,78),(12,79),(13,80),(14,73),(15,74),(16,75),(17,28),(18,29),(19,30),(20,31),(21,32),(22,25),(23,26),(24,27),(41,71),(42,72),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70)], [(1,17,7,19,5,21,3,23),(2,20,8,22,6,24,4,18),(9,68,35,55,13,72,39,51),(10,71,36,50,14,67,40,54),(11,66,37,53,15,70,33,49),(12,69,38,56,16,65,34,52),(25,43,77,64,29,47,73,60),(26,46,78,59,30,42,74,63),(27,41,79,62,31,45,75,58),(28,44,80,57,32,48,76,61)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 38 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 37 | 3 | 14 | 0 |
| 35 | 0 | 0 | 14 |
| 0 | 40 | 0 | 0 |
| 1 | 6 | 0 | 0 |
| 7 | 13 | 40 | 35 |
| 20 | 20 | 6 | 35 |
| 35 | 40 | 0 | 0 |
| 35 | 6 | 0 | 0 |
| 24 | 13 | 40 | 0 |
| 13 | 20 | 6 | 1 |
| 27 | 31 | 39 | 0 |
| 20 | 0 | 0 | 39 |
| 25 | 2 | 14 | 10 |
| 40 | 37 | 21 | 0 |
G:=sub<GL(4,GF(41))| [38,0,37,35,0,38,3,0,0,0,14,0,0,0,0,14],[0,1,7,20,40,6,13,20,0,0,40,6,0,0,35,35],[35,35,24,13,40,6,13,20,0,0,40,6,0,0,0,1],[27,20,25,40,31,0,2,37,39,0,14,21,0,39,10,0] >;
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 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 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 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 | D4 | Q8 | C8.C4 | F5 | C2×F5 | C2×F5 | C4⋊F5 | C4⋊F5 | (C8×D5).C4 |
| kernel | (C8×D5).C4 | C40.C4 | D10.Q8 | D5×C2×C8 | D5⋊M4(2) | 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_8\times D_5).C_4
% in TeX
G:=Group("(C8xD5).C4"); // GroupNames label
G:=SmallGroup(320,1062);
// by ID
G=gap.SmallGroup(320,1062);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,100,136,1684,102,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^5=c^2=1,d^4=a^4,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations