metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊7D10, C10.932+ 1+4, C4⋊C4⋊54D10, (C4×D20)⋊5C2, (C2×D20)⋊30C4, D20⋊36(C2×C4), (C4×C20)⋊4C22, C42⋊C2⋊6D5, D20⋊8C4⋊11C2, (C2×C10).65C24, C10.38(C23×C4), C4⋊Dic5⋊82C22, C2.1(D4⋊8D10), (C2×C20).583C23, C20.179(C22×C4), C22⋊C4.125D10, C5⋊3(C22.11C24), (C4×Dic5)⋊10C22, D10.13(C22×C4), (C22×D20).18C2, (C22×C4).188D10, D10⋊C4⋊60C22, C22.27(C23×D5), (C2×D20).263C22, (C23×D5).35C22, C23.153(C22×D5), C23.D5.94C22, C23.21D10⋊24C2, (C22×C10).135C23, (C22×C20).225C22, (C2×Dic5).205C23, (C22×D5).172C23, (C2×C4)⋊6(C4×D5), C4.58(C2×C4×D5), (C2×C20)⋊24(C2×C4), (C2×C4×D5)⋊43C22, (C5×C4⋊C4)⋊51C22, C2.19(D5×C22×C4), C22.27(C2×C4×D5), (D5×C22⋊C4)⋊25C2, (C22×D5)⋊9(C2×C4), (C5×C42⋊C2)⋊7C2, (C2×C4).271(C22×D5), (C2×C10).122(C22×C4), (C5×C22⋊C4).135C22, SmallGroup(320,1193)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊7D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, ac=ca, dad=a-1, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 1358 in 338 conjugacy classes, 151 normal (19 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.11C24, C4×Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C22×C20, C23×D5, C4×D20, D5×C22⋊C4, D20⋊8C4, C23.21D10, C5×C42⋊C2, C22×D20, C42⋊7D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, C4×D5, C22×D5, C22.11C24, C2×C4×D5, C23×D5, D5×C22×C4, D4⋊8D10, C42⋊7D10
►On 80 points
(1 65 25 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 70 30 51)(7 61 21 52)(8 62 22 53)(9 63 23 54)(10 64 24 55)(11 45 73 33)(12 46 74 34)(13 47 75 35)(14 48 76 36)(15 49 77 37)(16 50 78 38)(17 41 79 39)(18 42 80 40)(19 43 71 31)(20 44 72 32)
(1 33 6 50)(2 46 7 39)(3 35 8 42)(4 48 9 31)(5 37 10 44)(11 70 78 65)(12 52 79 57)(13 62 80 67)(14 54 71 59)(15 64 72 69)(16 56 73 51)(17 66 74 61)(18 58 75 53)(19 68 76 63)(20 60 77 55)(21 41 26 34)(22 40 27 47)(23 43 28 36)(24 32 29 49)(25 45 30 38)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 60)(7 59)(8 58)(9 57)(10 56)(11 44)(12 43)(13 42)(14 41)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 68)(22 67)(23 66)(24 65)(25 64)(26 63)(27 62)(28 61)(29 70)(30 69)(31 74)(32 73)(33 72)(34 71)(35 80)(36 79)(37 78)(38 77)(39 76)(40 75)
G:=sub<Sym(80)| (1,65,25,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,70,30,51)(7,61,21,52)(8,62,22,53)(9,63,23,54)(10,64,24,55)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32), (1,33,6,50)(2,46,7,39)(3,35,8,42)(4,48,9,31)(5,37,10,44)(11,70,78,65)(12,52,79,57)(13,62,80,67)(14,54,71,59)(15,64,72,69)(16,56,73,51)(17,66,74,61)(18,58,75,53)(19,68,76,63)(20,60,77,55)(21,41,26,34)(22,40,27,47)(23,43,28,36)(24,32,29,49)(25,45,30,38), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,60)(7,59)(8,58)(9,57)(10,56)(11,44)(12,43)(13,42)(14,41)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,70)(30,69)(31,74)(32,73)(33,72)(34,71)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)>;
G:=Group( (1,65,25,56)(2,66,26,57)(3,67,27,58)(4,68,28,59)(5,69,29,60)(6,70,30,51)(7,61,21,52)(8,62,22,53)(9,63,23,54)(10,64,24,55)(11,45,73,33)(12,46,74,34)(13,47,75,35)(14,48,76,36)(15,49,77,37)(16,50,78,38)(17,41,79,39)(18,42,80,40)(19,43,71,31)(20,44,72,32), (1,33,6,50)(2,46,7,39)(3,35,8,42)(4,48,9,31)(5,37,10,44)(11,70,78,65)(12,52,79,57)(13,62,80,67)(14,54,71,59)(15,64,72,69)(16,56,73,51)(17,66,74,61)(18,58,75,53)(19,68,76,63)(20,60,77,55)(21,41,26,34)(22,40,27,47)(23,43,28,36)(24,32,29,49)(25,45,30,38), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,60)(7,59)(8,58)(9,57)(10,56)(11,44)(12,43)(13,42)(14,41)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,70)(30,69)(31,74)(32,73)(33,72)(34,71)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75) );
G=PermutationGroup([[(1,65,25,56),(2,66,26,57),(3,67,27,58),(4,68,28,59),(5,69,29,60),(6,70,30,51),(7,61,21,52),(8,62,22,53),(9,63,23,54),(10,64,24,55),(11,45,73,33),(12,46,74,34),(13,47,75,35),(14,48,76,36),(15,49,77,37),(16,50,78,38),(17,41,79,39),(18,42,80,40),(19,43,71,31),(20,44,72,32)], [(1,33,6,50),(2,46,7,39),(3,35,8,42),(4,48,9,31),(5,37,10,44),(11,70,78,65),(12,52,79,57),(13,62,80,67),(14,54,71,59),(15,64,72,69),(16,56,73,51),(17,66,74,61),(18,58,75,53),(19,68,76,63),(20,60,77,55),(21,41,26,34),(22,40,27,47),(23,43,28,36),(24,32,29,49),(25,45,30,38)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,60),(7,59),(8,58),(9,57),(10,56),(11,44),(12,43),(13,42),(14,41),(15,50),(16,49),(17,48),(18,47),(19,46),(20,45),(21,68),(22,67),(23,66),(24,65),(25,64),(26,63),(27,62),(28,61),(29,70),(30,69),(31,74),(32,73),(33,72),(34,71),(35,80),(36,79),(37,78),(38,77),(39,76),(40,75)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2M | 4A | ··· | 4L | 4M | ··· | 4T | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | C4×D5 | 2+ 1+4 | D4⋊8D10 |
| kernel | C42⋊7D10 | C4×D20 | D5×C22⋊C4 | D20⋊8C4 | C23.21D10 | C5×C42⋊C2 | C22×D20 | C2×D20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C2 |
| # reps | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 16 | 2 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
Matrix representation of C42⋊7D10 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 32 | 0 | 0 |
| 0 | 0 | 9 | 11 | 0 | 0 |
| 0 | 0 | 5 | 36 | 30 | 32 |
| 0 | 0 | 1 | 36 | 9 | 11 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 39 | 0 |
| 0 | 0 | 33 | 40 | 0 | 39 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 40 | 1 | 8 | 1 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 9 | 35 | 7 | 7 |
| 0 | 0 | 30 | 27 | 34 | 40 |
| 35 | 7 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 27 | 0 | 0 |
| 0 | 0 | 11 | 27 | 0 | 0 |
| 0 | 0 | 12 | 29 | 27 | 14 |
| 0 | 0 | 27 | 29 | 30 | 14 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,30,9,5,1,0,0,32,11,36,36,0,0,0,0,30,9,0,0,0,0,32,11],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,32,33,0,40,0,0,0,40,0,1,0,0,39,0,9,8,0,0,0,39,0,1],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,34,7,9,30,0,0,34,1,35,27,0,0,0,0,7,34,0,0,0,0,7,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,14,11,12,27,0,0,27,27,29,29,0,0,0,0,27,30,0,0,0,0,14,14] >;
C42⋊7D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_7D_{10} % in TeX
G:=Group("C4^2:7D10"); // GroupNames label
G:=SmallGroup(320,1193);
// by ID
G=gap.SmallGroup(320,1193);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a^-1,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations