metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊10D10, C10.962+ 1+4, C4⋊C4⋊44D10, (C4×D20)⋊8C2, (C4×C20)⋊6C22, D10⋊Q8⋊4C2, C4.D20⋊6C2, C42⋊2D5⋊3C2, C42⋊D5⋊1C2, C22⋊D20.2C2, C42⋊C2⋊11D5, (C2×C10).71C24, C4⋊Dic5⋊56C22, C22⋊C4.95D10, Dic5⋊4D4⋊43C2, D10.29(C4○D4), D10.13D4⋊4C2, C2.8(D4⋊8D10), (C2×C20).146C23, (C2×Dic10)⋊5C22, (C4×Dic5)⋊50C22, (C2×D20).25C22, (C22×C4).192D10, D10⋊C4⋊40C22, Dic5.14D4⋊4C2, C5⋊2(C22.45C24), C23.D5.4C22, C22.18(C4○D20), C10.D4⋊32C22, (C23×D5).37C22, (C22×D5).21C23, C23.159(C22×D5), C22.100(C23×D5), C23.23D10⋊27C2, (C22×C20).435C22, (C22×C10).141C23, (C2×Dic5).208C23, (C22×Dic5).88C22, C4⋊C4⋊D5⋊4C2, C2.10(D5×C4○D4), (C2×C4×D5)⋊45C22, (C5×C4⋊C4)⋊54C22, (D5×C22⋊C4)⋊26C2, C10.28(C2×C4○D4), C2.30(C2×C4○D20), (C2×D10⋊C4)⋊40C2, (C2×C5⋊D4).9C22, (C5×C42⋊C2)⋊13C2, (C2×C10).41(C4○D4), (C2×C4).274(C22×D5), (C5×C22⋊C4).138C22, SmallGroup(320,1199)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊10D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 998 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.45C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C42⋊D5, C4×D20, C4.D20, C42⋊2D5, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.13D4, D10⋊Q8, C4⋊C4⋊D5, C2×D10⋊C4, C23.23D10, C5×C42⋊C2, C42⋊10D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.45C24, C4○D20, C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42⋊10D10
►On 80 points
(1 53 13 41)(2 59 14 47)(3 55 15 43)(4 51 11 49)(5 57 12 45)(6 56 16 44)(7 52 17 50)(8 58 18 46)(9 54 19 42)(10 60 20 48)(21 71 31 65)(22 77 32 61)(23 73 33 67)(24 79 34 63)(25 75 35 69)(26 76 36 70)(27 72 37 66)(28 78 38 62)(29 74 39 68)(30 80 40 64)
(1 28 8 23)(2 29 9 24)(3 30 10 25)(4 26 6 21)(5 27 7 22)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 62 46 67)(42 63 47 68)(43 64 48 69)(44 65 49 70)(45 66 50 61)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(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 7)(2 6)(3 10)(4 9)(5 8)(11 19)(12 18)(13 17)(14 16)(15 20)(21 39)(22 38)(23 37)(24 36)(25 40)(26 34)(27 33)(28 32)(29 31)(30 35)(41 45)(42 44)(46 50)(47 49)(51 59)(52 58)(53 57)(54 56)(61 73)(62 72)(63 71)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)
G:=sub<Sym(80)| (1,53,13,41)(2,59,14,47)(3,55,15,43)(4,51,11,49)(5,57,12,45)(6,56,16,44)(7,52,17,50)(8,58,18,46)(9,54,19,42)(10,60,20,48)(21,71,31,65)(22,77,32,61)(23,73,33,67)(24,79,34,63)(25,75,35,69)(26,76,36,70)(27,72,37,66)(28,78,38,62)(29,74,39,68)(30,80,40,64), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,62,46,67)(42,63,47,68)(43,64,48,69)(44,65,49,70)(45,66,50,61)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,39)(22,38)(23,37)(24,36)(25,40)(26,34)(27,33)(28,32)(29,31)(30,35)(41,45)(42,44)(46,50)(47,49)(51,59)(52,58)(53,57)(54,56)(61,73)(62,72)(63,71)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)>;
G:=Group( (1,53,13,41)(2,59,14,47)(3,55,15,43)(4,51,11,49)(5,57,12,45)(6,56,16,44)(7,52,17,50)(8,58,18,46)(9,54,19,42)(10,60,20,48)(21,71,31,65)(22,77,32,61)(23,73,33,67)(24,79,34,63)(25,75,35,69)(26,76,36,70)(27,72,37,66)(28,78,38,62)(29,74,39,68)(30,80,40,64), (1,28,8,23)(2,29,9,24)(3,30,10,25)(4,26,6,21)(5,27,7,22)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,62,46,67)(42,63,47,68)(43,64,48,69)(44,65,49,70)(45,66,50,61)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,19)(12,18)(13,17)(14,16)(15,20)(21,39)(22,38)(23,37)(24,36)(25,40)(26,34)(27,33)(28,32)(29,31)(30,35)(41,45)(42,44)(46,50)(47,49)(51,59)(52,58)(53,57)(54,56)(61,73)(62,72)(63,71)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74) );
G=PermutationGroup([[(1,53,13,41),(2,59,14,47),(3,55,15,43),(4,51,11,49),(5,57,12,45),(6,56,16,44),(7,52,17,50),(8,58,18,46),(9,54,19,42),(10,60,20,48),(21,71,31,65),(22,77,32,61),(23,73,33,67),(24,79,34,63),(25,75,35,69),(26,76,36,70),(27,72,37,66),(28,78,38,62),(29,74,39,68),(30,80,40,64)], [(1,28,8,23),(2,29,9,24),(3,30,10,25),(4,26,6,21),(5,27,7,22),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,62,46,67),(42,63,47,68),(43,64,48,69),(44,65,49,70),(45,66,50,61),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(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)], [(1,7),(2,6),(3,10),(4,9),(5,8),(11,19),(12,18),(13,17),(14,16),(15,20),(21,39),(22,38),(23,37),(24,36),(25,40),(26,34),(27,33),(28,32),(29,31),(30,35),(41,45),(42,44),(46,50),(47,49),(51,59),(52,58),(53,57),(54,56),(61,73),(62,72),(63,71),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74)]])
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ 1+4 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42⋊10D10 | C42⋊D5 | C4×D20 | C4.D20 | C42⋊2D5 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C2×D10⋊C4 | C23.23D10 | C5×C42⋊C2 | C42⋊C2 | D10 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
Matrix representation of C42⋊10D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 0 |
| 0 | 0 | 29 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 34 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 37 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,29,0,0,0,0,4,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,40,4,0,0,0,0,0,1] >;
C42⋊10D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}D_{10} % in TeX
G:=Group("C4^2:10D10"); // GroupNames label
G:=SmallGroup(320,1199);
// by ID
G=gap.SmallGroup(320,1199);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,136,12550]);
// 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*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations