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
Generators and relations for C42⋊26D10
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 >
Subgroups: 1470 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22.29C24, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C4×C20, C2×Dic10, C2×C4×D5, C2×D20, D4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, D4×C10, C23×D5, C42⋊D5, C4.D20, C20.17D4, C23⋊D10, Dic5⋊D4, C20⋊D4, C5×C4⋊1D4, C2×D4×D5, C2×D4⋊2D5, C42⋊26D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C22.29C24, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, C42⋊26D10
►On 80 points
(1 64 20 69)(2 70 16 65)(3 66 17 61)(4 62 18 67)(5 68 19 63)(6 75 12 80)(7 71 13 76)(8 77 14 72)(9 73 15 78)(10 79 11 74)(21 38 54 44)(22 45 55 39)(23 40 56 46)(24 47 57 31)(25 32 58 48)(26 49 59 33)(27 34 60 50)(28 41 51 35)(29 36 52 42)(30 43 53 37)
(1 22 6 27)(2 28 7 23)(3 24 8 29)(4 30 9 25)(5 26 10 21)(11 54 19 59)(12 60 20 55)(13 56 16 51)(14 52 17 57)(15 58 18 53)(31 72 42 61)(32 62 43 73)(33 74 44 63)(34 64 45 75)(35 76 46 65)(36 66 47 77)(37 78 48 67)(38 68 49 79)(39 80 50 69)(40 70 41 71)
(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 11)(2 15)(3 14)(4 13)(5 12)(6 19)(7 18)(8 17)(9 16)(10 20)(21 27)(22 26)(23 25)(28 30)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 50)(39 49)(40 48)(51 53)(54 60)(55 59)(56 58)(61 72)(62 71)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)
G:=sub<Sym(80)| (1,64,20,69)(2,70,16,65)(3,66,17,61)(4,62,18,67)(5,68,19,63)(6,75,12,80)(7,71,13,76)(8,77,14,72)(9,73,15,78)(10,79,11,74)(21,38,54,44)(22,45,55,39)(23,40,56,46)(24,47,57,31)(25,32,58,48)(26,49,59,33)(27,34,60,50)(28,41,51,35)(29,36,52,42)(30,43,53,37), (1,22,6,27)(2,28,7,23)(3,24,8,29)(4,30,9,25)(5,26,10,21)(11,54,19,59)(12,60,20,55)(13,56,16,51)(14,52,17,57)(15,58,18,53)(31,72,42,61)(32,62,43,73)(33,74,44,63)(34,64,45,75)(35,76,46,65)(36,66,47,77)(37,78,48,67)(38,68,49,79)(39,80,50,69)(40,70,41,71), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,25)(28,30)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,53)(54,60)(55,59)(56,58)(61,72)(62,71)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)>;
G:=Group( (1,64,20,69)(2,70,16,65)(3,66,17,61)(4,62,18,67)(5,68,19,63)(6,75,12,80)(7,71,13,76)(8,77,14,72)(9,73,15,78)(10,79,11,74)(21,38,54,44)(22,45,55,39)(23,40,56,46)(24,47,57,31)(25,32,58,48)(26,49,59,33)(27,34,60,50)(28,41,51,35)(29,36,52,42)(30,43,53,37), (1,22,6,27)(2,28,7,23)(3,24,8,29)(4,30,9,25)(5,26,10,21)(11,54,19,59)(12,60,20,55)(13,56,16,51)(14,52,17,57)(15,58,18,53)(31,72,42,61)(32,62,43,73)(33,74,44,63)(34,64,45,75)(35,76,46,65)(36,66,47,77)(37,78,48,67)(38,68,49,79)(39,80,50,69)(40,70,41,71), (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), (1,11)(2,15)(3,14)(4,13)(5,12)(6,19)(7,18)(8,17)(9,16)(10,20)(21,27)(22,26)(23,25)(28,30)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,50)(39,49)(40,48)(51,53)(54,60)(55,59)(56,58)(61,72)(62,71)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73) );
G=PermutationGroup([[(1,64,20,69),(2,70,16,65),(3,66,17,61),(4,62,18,67),(5,68,19,63),(6,75,12,80),(7,71,13,76),(8,77,14,72),(9,73,15,78),(10,79,11,74),(21,38,54,44),(22,45,55,39),(23,40,56,46),(24,47,57,31),(25,32,58,48),(26,49,59,33),(27,34,60,50),(28,41,51,35),(29,36,52,42),(30,43,53,37)], [(1,22,6,27),(2,28,7,23),(3,24,8,29),(4,30,9,25),(5,26,10,21),(11,54,19,59),(12,60,20,55),(13,56,16,51),(14,52,17,57),(15,58,18,53),(31,72,42,61),(32,62,43,73),(33,74,44,63),(34,64,45,75),(35,76,46,65),(36,66,47,77),(37,78,48,67),(38,68,49,79),(39,80,50,69),(40,70,41,71)], [(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)], [(1,11),(2,15),(3,14),(4,13),(5,12),(6,19),(7,18),(8,17),(9,16),(10,20),(21,27),(22,26),(23,25),(28,30),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,50),(39,49),(40,48),(51,53),(54,60),(55,59),(56,58),(61,72),(62,71),(63,80),(64,79),(65,78),(66,77),(67,76),(68,75),(69,74),(70,73)]])
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 |
Matrix representation of C42⋊26D10 ►in 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] >;
C42⋊26D10 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