metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊26D10, C10.762+ (1+4), (C4×D5)⋊5D4, C4⋊1D4⋊5D5, C4.34(D4×D5), (C2×D4)⋊12D10, C20.65(C2×D4), C20⋊D4⋊26C2, (C4×C20)⋊26C22, D10.81(C2×D4), C23⋊D10⋊26C2, C4.D20⋊25C2, (D4×C10)⋊32C22, C42⋊D5⋊23C2, Dic5.92(C2×D4), C10.93(C22×D4), Dic5⋊D4⋊36C2, C20.17D4⋊26C2, (C2×C20).635C23, (C2×C10).259C24, C5⋊5(C22.29C24), (C4×Dic5)⋊39C22, C23.D5⋊36C22, C2.80(D4⋊6D10), D10⋊C4⋊34C22, C23.65(C22×D5), (C2×Dic10)⋊34C22, (C2×D20).176C22, C10.D4⋊71C22, (C22×C10).73C23, (C23×D5).72C22, C22.280(C23×D5), (C2×Dic5).134C23, (C22×Dic5)⋊29C22, (C22×D5).237C23, (C2×D4×D5)⋊19C2, C2.66(C2×D4×D5), (C5×C4⋊1D4)⋊6C2, (C2×D4⋊2D5)⋊20C2, (C2×C5⋊D4)⋊26C22, (C2×C4×D5).147C22, (C2×C4).213(C22×D5), SmallGroup(320,1387)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1470 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×4], C23 [×11], D5 [×4], C10, C10 [×2], C10 [×4], C42, C42, C22⋊C4 [×10], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×D4 [×13], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×2], C20 [×2], D10 [×2], D10 [×16], C2×C10, C2×C10 [×12], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5, C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20, C2×C20 [×2], C5×D4 [×8], C22×D5, C22×D5 [×2], C22×D5 [×8], C22×C10 [×4], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C23.D5 [×4], C4×C20, C2×Dic10, C2×C4×D5, C2×D20, D4×D5 [×4], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×8], D4×C10 [×2], D4×C10 [×4], C23×D5 [×2], C42⋊D5, C4.D20, C20.17D4, C23⋊D10 [×4], Dic5⋊D4 [×4], C20⋊D4, C5×C4⋊1D4, C2×D4×D5, C2×D4⋊2D5, C42⋊26D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10 [×2], C42⋊26D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=b-1, dbd=a2b-1, dcd=c-1 >
►On 80 points
(1 42 16 47)(2 48 17 43)(3 44 18 49)(4 50 19 45)(5 46 20 41)(6 72 12 77)(7 78 13 73)(8 74 14 79)(9 80 15 75)(10 76 11 71)(21 66 32 60)(22 51 33 67)(23 68 34 52)(24 53 35 69)(25 70 36 54)(26 55 37 61)(27 62 38 56)(28 57 39 63)(29 64 40 58)(30 59 31 65)
(1 28 7 23)(2 24 8 29)(3 30 9 25)(4 26 10 21)(5 22 6 27)(11 32 19 37)(12 38 20 33)(13 34 16 39)(14 40 17 35)(15 36 18 31)(41 67 77 56)(42 57 78 68)(43 69 79 58)(44 59 80 70)(45 61 71 60)(46 51 72 62)(47 63 73 52)(48 53 74 64)(49 65 75 54)(50 55 76 66)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 19)(7 18)(8 17)(9 16)(10 20)(21 27)(22 26)(23 25)(28 30)(31 39)(32 38)(33 37)(34 36)(41 71)(42 80)(43 79)(44 78)(45 77)(46 76)(47 75)(48 74)(49 73)(50 72)(51 61)(52 70)(53 69)(54 68)(55 67)(56 66)(57 65)(58 64)(59 63)(60 62)
G:=sub<Sym(80)| (1,42,16,47)(2,48,17,43)(3,44,18,49)(4,50,19,45)(5,46,20,41)(6,72,12,77)(7,78,13,73)(8,74,14,79)(9,80,15,75)(10,76,11,71)(21,66,32,60)(22,51,33,67)(23,68,34,52)(24,53,35,69)(25,70,36,54)(26,55,37,61)(27,62,38,56)(28,57,39,63)(29,64,40,58)(30,59,31,65), (1,28,7,23)(2,24,8,29)(3,30,9,25)(4,26,10,21)(5,22,6,27)(11,32,19,37)(12,38,20,33)(13,34,16,39)(14,40,17,35)(15,36,18,31)(41,67,77,56)(42,57,78,68)(43,69,79,58)(44,59,80,70)(45,61,71,60)(46,51,72,62)(47,63,73,52)(48,53,74,64)(49,65,75,54)(50,55,76,66), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,25)(28,30)(31,39)(32,38)(33,37)(34,36)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62)>;
G:=Group( (1,42,16,47)(2,48,17,43)(3,44,18,49)(4,50,19,45)(5,46,20,41)(6,72,12,77)(7,78,13,73)(8,74,14,79)(9,80,15,75)(10,76,11,71)(21,66,32,60)(22,51,33,67)(23,68,34,52)(24,53,35,69)(25,70,36,54)(26,55,37,61)(27,62,38,56)(28,57,39,63)(29,64,40,58)(30,59,31,65), (1,28,7,23)(2,24,8,29)(3,30,9,25)(4,26,10,21)(5,22,6,27)(11,32,19,37)(12,38,20,33)(13,34,16,39)(14,40,17,35)(15,36,18,31)(41,67,77,56)(42,57,78,68)(43,69,79,58)(44,59,80,70)(45,61,71,60)(46,51,72,62)(47,63,73,52)(48,53,74,64)(49,65,75,54)(50,55,76,66), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,25)(28,30)(31,39)(32,38)(33,37)(34,36)(41,71)(42,80)(43,79)(44,78)(45,77)(46,76)(47,75)(48,74)(49,73)(50,72)(51,61)(52,70)(53,69)(54,68)(55,67)(56,66)(57,65)(58,64)(59,63)(60,62) );
G=PermutationGroup([(1,42,16,47),(2,48,17,43),(3,44,18,49),(4,50,19,45),(5,46,20,41),(6,72,12,77),(7,78,13,73),(8,74,14,79),(9,80,15,75),(10,76,11,71),(21,66,32,60),(22,51,33,67),(23,68,34,52),(24,53,35,69),(25,70,36,54),(26,55,37,61),(27,62,38,56),(28,57,39,63),(29,64,40,58),(30,59,31,65)], [(1,28,7,23),(2,24,8,29),(3,30,9,25),(4,26,10,21),(5,22,6,27),(11,32,19,37),(12,38,20,33),(13,34,16,39),(14,40,17,35),(15,36,18,31),(41,67,77,56),(42,57,78,68),(43,69,79,58),(44,59,80,70),(45,61,71,60),(46,51,72,62),(47,63,73,52),(48,53,74,64),(49,65,75,54),(50,55,76,66)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,19),(7,18),(8,17),(9,16),(10,20),(21,27),(22,26),(23,25),(28,30),(31,39),(32,38),(33,37),(34,36),(41,71),(42,80),(43,79),(44,78),(45,77),(46,76),(47,75),(48,74),(49,73),(50,72),(51,61),(52,70),(53,69),(54,68),(55,67),(56,66),(57,65),(58,64),(59,63),(60,62)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 39 | 39 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 40 |
| 0 | 0 | 1 | 40 | 0 | 0 | 0 | 0 |
| 35 | 1 | 2 | 35 | 0 | 0 | 0 | 0 |
| 38 | 21 | 40 | 0 | 0 | 0 | 0 | 0 |
| 39 | 21 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 40 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 11 | 6 | 35 | 35 | 0 | 0 | 0 | 0 |
| 16 | 12 | 6 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 40 |
| 7 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 8 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 30 | 35 | 6 | 6 | 0 | 0 | 0 | 0 |
| 25 | 29 | 1 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 2 | 1 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,2,40,1,0,0,0,0,0,1,0,1,0,0,0,0,2,39,1,0,0,0,0,0,0,39,0,40],[0,35,38,39,0,0,0,0,0,1,21,21,0,0,0,0,1,2,40,40,0,0,0,0,40,35,0,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,1],[0,34,11,16,0,0,0,0,6,7,6,12,0,0,0,0,0,0,35,6,0,0,0,0,0,0,35,40,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[7,8,30,25,0,0,0,0,35,34,35,29,0,0,0,0,0,0,6,1,0,0,0,0,0,0,6,35,0,0,0,0,0,0,0,0,1,0,1,40,0,0,0,0,0,40,0,40,0,0,0,0,0,0,40,2,0,0,0,0,0,0,0,1] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊6D10 |
| kernel | C42⋊26D10 | C42⋊D5 | C4.D20 | C20.17D4 | C23⋊D10 | Dic5⋊D4 | C20⋊D4 | C5×C4⋊1D4 | C2×D4×D5 | C2×D4⋊2D5 | C4×D5 | C4⋊1D4 | C42 | C2×D4 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 12 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{26}D_{10} % in TeX
G:=Group("C4^2:26D10"); // GroupNames label
G:=SmallGroup(320,1387);
// by ID
G=gap.SmallGroup(320,1387);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=b^-1,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations