metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D10, C10.802+ 1+4, C4⋊1D4⋊9D5, (C2×D4)⋊13D10, (C4×C20)⋊37C22, C23⋊D10⋊28C2, (D4×C10)⋊34C22, C42⋊2D5⋊19C2, Dic5⋊D4⋊39C2, (C2×C10).264C24, (C2×C20).638C23, C2.84(D4⋊6D10), C23.D5⋊38C22, C23.70(C22×D5), C5⋊5(C22.54C24), C10.D4⋊37C22, (C22×C10).78C23, (C23×D5).73C22, C22.285(C23×D5), D10⋊C4.75C22, C23.18D10⋊28C2, (C2×Dic5).138C23, (C22×Dic5)⋊30C22, (C22×D5).118C23, (C5×C4⋊1D4)⋊15C2, (C2×C4).216(C22×D5), (C2×C5⋊D4).80C22, SmallGroup(320,1392)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊28D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22.54C24, C10.D4, D10⋊C4, C23.D5, C4×C20, C22×Dic5, C2×C5⋊D4, D4×C10, C23×D5, C42⋊2D5, C23.18D10, C23⋊D10, Dic5⋊D4, C5×C4⋊1D4, C42⋊28D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D4⋊6D10, C42⋊28D10
►On 80 points
Generators in S80
(1 70 20 65)(2 66 16 61)(3 62 17 67)(4 68 18 63)(5 64 19 69)(6 77 15 72)(7 73 11 78)(8 79 12 74)(9 75 13 80)(10 71 14 76)(21 32 53 43)(22 44 54 33)(23 34 55 45)(24 46 56 35)(25 36 57 47)(26 48 58 37)(27 38 59 49)(28 50 60 39)(29 40 51 41)(30 42 52 31)
(1 28 9 23)(2 24 10 29)(3 30 6 25)(4 26 7 21)(5 22 8 27)(11 53 18 58)(12 59 19 54)(13 55 20 60)(14 51 16 56)(15 57 17 52)(31 72 47 67)(32 68 48 73)(33 74 49 69)(34 70 50 75)(35 76 41 61)(36 62 42 77)(37 78 43 63)(38 64 44 79)(39 80 45 65)(40 66 46 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 12)(2 11)(3 15)(4 14)(5 13)(6 17)(7 16)(8 20)(9 19)(10 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 58)(52 57)(53 56)(54 55)(59 60)(61 63)(64 70)(65 69)(66 68)(71 73)(74 80)(75 79)(76 78)
G:=sub<Sym(80)| (1,70,20,65)(2,66,16,61)(3,62,17,67)(4,68,18,63)(5,64,19,69)(6,77,15,72)(7,73,11,78)(8,79,12,74)(9,75,13,80)(10,71,14,76)(21,32,53,43)(22,44,54,33)(23,34,55,45)(24,46,56,35)(25,36,57,47)(26,48,58,37)(27,38,59,49)(28,50,60,39)(29,40,51,41)(30,42,52,31), (1,28,9,23)(2,24,10,29)(3,30,6,25)(4,26,7,21)(5,22,8,27)(11,53,18,58)(12,59,19,54)(13,55,20,60)(14,51,16,56)(15,57,17,52)(31,72,47,67)(32,68,48,73)(33,74,49,69)(34,70,50,75)(35,76,41,61)(36,62,42,77)(37,78,43,63)(38,64,44,79)(39,80,45,65)(40,66,46,71), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,58)(52,57)(53,56)(54,55)(59,60)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78)>;
G:=Group( (1,70,20,65)(2,66,16,61)(3,62,17,67)(4,68,18,63)(5,64,19,69)(6,77,15,72)(7,73,11,78)(8,79,12,74)(9,75,13,80)(10,71,14,76)(21,32,53,43)(22,44,54,33)(23,34,55,45)(24,46,56,35)(25,36,57,47)(26,48,58,37)(27,38,59,49)(28,50,60,39)(29,40,51,41)(30,42,52,31), (1,28,9,23)(2,24,10,29)(3,30,6,25)(4,26,7,21)(5,22,8,27)(11,53,18,58)(12,59,19,54)(13,55,20,60)(14,51,16,56)(15,57,17,52)(31,72,47,67)(32,68,48,73)(33,74,49,69)(34,70,50,75)(35,76,41,61)(36,62,42,77)(37,78,43,63)(38,64,44,79)(39,80,45,65)(40,66,46,71), (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,17)(7,16)(8,20)(9,19)(10,18)(21,24)(22,23)(25,30)(26,29)(27,28)(31,42)(32,41)(33,50)(34,49)(35,48)(36,47)(37,46)(38,45)(39,44)(40,43)(51,58)(52,57)(53,56)(54,55)(59,60)(61,63)(64,70)(65,69)(66,68)(71,73)(74,80)(75,79)(76,78) );
G=PermutationGroup([[(1,70,20,65),(2,66,16,61),(3,62,17,67),(4,68,18,63),(5,64,19,69),(6,77,15,72),(7,73,11,78),(8,79,12,74),(9,75,13,80),(10,71,14,76),(21,32,53,43),(22,44,54,33),(23,34,55,45),(24,46,56,35),(25,36,57,47),(26,48,58,37),(27,38,59,49),(28,50,60,39),(29,40,51,41),(30,42,52,31)], [(1,28,9,23),(2,24,10,29),(3,30,6,25),(4,26,7,21),(5,22,8,27),(11,53,18,58),(12,59,19,54),(13,55,20,60),(14,51,16,56),(15,57,17,52),(31,72,47,67),(32,68,48,73),(33,74,49,69),(34,70,50,75),(35,76,41,61),(36,62,42,77),(37,78,43,63),(38,64,44,79),(39,80,45,65),(40,66,46,71)], [(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,17),(7,16),(8,20),(9,19),(10,18),(21,24),(22,23),(25,30),(26,29),(27,28),(31,42),(32,41),(33,50),(34,49),(35,48),(36,47),(37,46),(38,45),(39,44),(40,43),(51,58),(52,57),(53,56),(54,55),(59,60),(61,63),(64,70),(65,69),(66,68),(71,73),(74,80),(75,79),(76,78)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2+ 1+4 | D4⋊6D10 |
| kernel | C42⋊28D10 | C42⋊2D5 | C23.18D10 | C23⋊D10 | Dic5⋊D4 | C5×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C10 | C2 |
| # reps | 1 | 2 | 3 | 3 | 6 | 1 | 2 | 2 | 12 | 3 | 12 |
Matrix representation of C42⋊28D10 ►in GL8(𝔽41)
| 18 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 39 | 30 | 36 | 12 |
| 0 | 0 | 0 | 0 | 13 | 12 | 13 | 2 |
| 0 | 0 | 0 | 0 | 25 | 29 | 2 | 13 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 34 | 38 | 3 |
| 0 | 0 | 0 | 0 | 6 | 17 | 38 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 18 | 6 |
| 0 | 0 | 0 | 0 | 13 | 28 | 35 | 23 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 3 | 16 |
| 0 | 0 | 0 | 0 | 7 | 35 | 16 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 7 |
| 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 35 | 1 | 40 |
| 0 | 0 | 0 | 0 | 40 | 35 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;
C42⋊28D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_{10} % in TeX
G:=Group("C4^2:28D10"); // GroupNames label
G:=SmallGroup(320,1392);
// by ID
G=gap.SmallGroup(320,1392);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations