metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D10, C10.1262+ 1+4, (C2×Q8)⋊8D10, (C4×D20)⋊45C2, (C4×C20)⋊24C22, C22⋊C4⋊34D10, C4.4D4⋊12D5, C42⋊2D5⋊8C2, D10⋊D4⋊42C2, C23⋊D10⋊24C2, C22⋊D20⋊25C2, D10⋊3Q8⋊30C2, (C2×D4).110D10, C4⋊Dic5⋊41C22, (Q8×C10)⋊14C22, D10.14(C4○D4), Dic5⋊4D4⋊31C2, Dic5⋊D4⋊34C2, C20.23D4⋊22C2, (C2×C20).631C23, (C2×C10).222C24, C5⋊8(C22.32C24), (C4×Dic5)⋊36C22, D10.12D4⋊43C2, C2.50(D4⋊8D10), C2.75(D4⋊6D10), D10⋊C4⋊69C22, C23.44(C22×D5), (D4×C10).210C22, (C2×D20).232C22, C22.D20⋊25C2, C23.D10⋊39C2, C10.D4⋊36C22, (C22×C10).52C23, (C23×D5).65C22, (C22×D5).96C23, C22.243(C23×D5), C23.D5.56C22, (C2×Dic5).264C23, (C22×Dic5)⋊27C22, C2.78(D5×C4○D4), (C2×C4×D5)⋊52C22, (D5×C22⋊C4)⋊18C2, C10.189(C2×C4○D4), (C5×C4.4D4)⋊14C2, (C2×C5⋊D4)⋊24C22, (C5×C22⋊C4)⋊30C22, (C2×C4).197(C22×D5), SmallGroup(320,1350)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊20D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
Subgroups: 1070 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C22.32C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C4×D20, C42⋊2D5, C23.D10, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.12D4, D10⋊D4, C22.D20, C23⋊D10, Dic5⋊D4, D10⋊3Q8, C20.23D4, C5×C4.4D4, C42⋊20D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.32C24, C23×D5, D4⋊6D10, D5×C4○D4, D4⋊8D10, C42⋊20D10
►On 80 points
(1 61 6 53)(2 67 7 59)(3 63 8 55)(4 69 9 51)(5 65 10 57)(11 66 19 58)(12 62 20 54)(13 68 16 60)(14 64 17 56)(15 70 18 52)(21 48 26 37)(22 33 27 44)(23 50 28 39)(24 35 29 46)(25 42 30 31)(32 76 43 71)(34 78 45 73)(36 80 47 75)(38 72 49 77)(40 74 41 79)
(1 29 11 79)(2 25 12 75)(3 21 13 71)(4 27 14 77)(5 23 15 73)(6 24 19 74)(7 30 20 80)(8 26 16 76)(9 22 17 72)(10 28 18 78)(31 54 47 59)(32 63 48 68)(33 56 49 51)(34 65 50 70)(35 58 41 53)(36 67 42 62)(37 60 43 55)(38 69 44 64)(39 52 45 57)(40 61 46 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 13)(2 12)(3 11)(4 15)(5 14)(6 16)(7 20)(8 19)(9 18)(10 17)(21 24)(22 23)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 40)(38 39)(41 48)(42 47)(43 46)(44 45)(49 50)(51 57)(52 56)(53 55)(58 60)(61 63)(64 70)(65 69)(66 68)(71 74)(72 73)(75 80)(76 79)(77 78)
G:=sub<Sym(80)| (1,61,6,53)(2,67,7,59)(3,63,8,55)(4,69,9,51)(5,65,10,57)(11,66,19,58)(12,62,20,54)(13,68,16,60)(14,64,17,56)(15,70,18,52)(21,48,26,37)(22,33,27,44)(23,50,28,39)(24,35,29,46)(25,42,30,31)(32,76,43,71)(34,78,45,73)(36,80,47,75)(38,72,49,77)(40,74,41,79), (1,29,11,79)(2,25,12,75)(3,21,13,71)(4,27,14,77)(5,23,15,73)(6,24,19,74)(7,30,20,80)(8,26,16,76)(9,22,17,72)(10,28,18,78)(31,54,47,59)(32,63,48,68)(33,56,49,51)(34,65,50,70)(35,58,41,53)(36,67,42,62)(37,60,43,55)(38,69,44,64)(39,52,45,57)(40,61,46,66), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,24)(22,23)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,57)(52,56)(53,55)(58,60)(61,63)(64,70)(65,69)(66,68)(71,74)(72,73)(75,80)(76,79)(77,78)>;
G:=Group( (1,61,6,53)(2,67,7,59)(3,63,8,55)(4,69,9,51)(5,65,10,57)(11,66,19,58)(12,62,20,54)(13,68,16,60)(14,64,17,56)(15,70,18,52)(21,48,26,37)(22,33,27,44)(23,50,28,39)(24,35,29,46)(25,42,30,31)(32,76,43,71)(34,78,45,73)(36,80,47,75)(38,72,49,77)(40,74,41,79), (1,29,11,79)(2,25,12,75)(3,21,13,71)(4,27,14,77)(5,23,15,73)(6,24,19,74)(7,30,20,80)(8,26,16,76)(9,22,17,72)(10,28,18,78)(31,54,47,59)(32,63,48,68)(33,56,49,51)(34,65,50,70)(35,58,41,53)(36,67,42,62)(37,60,43,55)(38,69,44,64)(39,52,45,57)(40,61,46,66), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,16)(7,20)(8,19)(9,18)(10,17)(21,24)(22,23)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,48)(42,47)(43,46)(44,45)(49,50)(51,57)(52,56)(53,55)(58,60)(61,63)(64,70)(65,69)(66,68)(71,74)(72,73)(75,80)(76,79)(77,78) );
G=PermutationGroup([[(1,61,6,53),(2,67,7,59),(3,63,8,55),(4,69,9,51),(5,65,10,57),(11,66,19,58),(12,62,20,54),(13,68,16,60),(14,64,17,56),(15,70,18,52),(21,48,26,37),(22,33,27,44),(23,50,28,39),(24,35,29,46),(25,42,30,31),(32,76,43,71),(34,78,45,73),(36,80,47,75),(38,72,49,77),(40,74,41,79)], [(1,29,11,79),(2,25,12,75),(3,21,13,71),(4,27,14,77),(5,23,15,73),(6,24,19,74),(7,30,20,80),(8,26,16,76),(9,22,17,72),(10,28,18,78),(31,54,47,59),(32,63,48,68),(33,56,49,51),(34,65,50,70),(35,58,41,53),(36,67,42,62),(37,60,43,55),(38,69,44,64),(39,52,45,57),(40,61,46,66)], [(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)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,16),(7,20),(8,19),(9,18),(10,17),(21,24),(22,23),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,40),(38,39),(41,48),(42,47),(43,46),(44,45),(49,50),(51,57),(52,56),(53,55),(58,60),(61,63),(64,70),(65,69),(66,68),(71,74),(72,73),(75,80),(76,79),(77,78)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42⋊20D10 | C4×D20 | C42⋊2D5 | C23.D10 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.12D4 | D10⋊D4 | C22.D20 | C23⋊D10 | Dic5⋊D4 | D10⋊3Q8 | C20.23D4 | C5×C4.4D4 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C42⋊20D10 ►in GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 4 | 5 | 0 | 0 | 0 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,38,0,0,0,0,5,37,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,11,32,0,0,0,0,9,30],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;
C42⋊20D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_{10} % in TeX
G:=Group("C4^2:20D10"); // GroupNames label
G:=SmallGroup(320,1350);
// by ID
G=gap.SmallGroup(320,1350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,80,12550]);
// 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*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations