metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊12D10, (C4×D4)⋊9D5, C4⋊C4⋊45D10, (C4×D5)⋊14D4, (C4×D20)⋊25C2, (D4×C20)⋊11C2, C4.219(D4×D5), D10⋊2(C4○D4), (C4×C20)⋊17C22, C22⋊C4⋊44D10, D10.72(C2×D4), C20.378(C2×D4), (C22×C4)⋊10D10, C22⋊D20⋊30C2, D10⋊D4⋊47C2, C23⋊D10⋊33C2, D10⋊Q8⋊51C2, (C2×D4).211D10, C22⋊2(C4○D20), C42⋊D5⋊13C2, (C2×C10).91C24, C4⋊Dic5⋊57C22, Dic5.83(C2×D4), C10.47(C22×D4), Dic5⋊D4⋊46C2, (C2×C20).157C23, (C22×C20)⋊15C22, C5⋊2(C22.19C24), (C4×Dic5)⋊51C22, D10.13D4⋊49C2, D10.12D4⋊53C2, C23.D5⋊49C22, D10⋊C4⋊65C22, (C2×Dic10)⋊52C22, (C2×D20).217C22, (D4×C10).304C22, C10.D4⋊70C22, (C2×Dic5).38C23, C22.116(C23×D5), C23.170(C22×D5), Dic5.14D4⋊50C2, (C22×C10).161C23, (C22×D5).180C23, (C23×D5).118C22, (C22×Dic5).243C22, C2.19(C2×D4×D5), (C2×C4○D20)⋊5C2, (C4×C5⋊D4)⋊42C2, C2.20(D5×C4○D4), (C2×C4×D5)⋊47C22, (D5×C22×C4)⋊22C2, (C2×C10)⋊1(C4○D4), (C5×C4⋊C4)⋊57C22, C10.39(C2×C4○D4), C2.43(C2×C4○D20), (C2×C5⋊D4)⋊37C22, (C5×C22⋊C4)⋊55C22, (C2×C4).156(C22×D5), SmallGroup(320,1219)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊12D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1246 in 330 conjugacy classes, 109 normal (91 characteristic)
C1, C2, C2, C4, 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×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22.19C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C42⋊D5, C4×D20, Dic5.14D4, C22⋊D20, D10.12D4, D10⋊D4, D10.13D4, D10⋊Q8, C4×C5⋊D4, C23⋊D10, Dic5⋊D4, D4×C20, D5×C22×C4, C2×C4○D20, C42⋊12D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.19C24, C4○D20, D4×D5, C23×D5, C2×C4○D20, C2×D4×D5, D5×C4○D4, C42⋊12D10
►On 80 points
(1 33 25 45)(2 46 26 34)(3 35 27 47)(4 48 28 36)(5 37 29 49)(6 50 30 38)(7 39 21 41)(8 42 22 40)(9 31 23 43)(10 44 24 32)(11 52 73 65)(12 66 74 53)(13 54 75 67)(14 68 76 55)(15 56 77 69)(16 70 78 57)(17 58 79 61)(18 62 80 59)(19 60 71 63)(20 64 72 51)
(1 57 6 52)(2 58 7 53)(3 59 8 54)(4 60 9 55)(5 51 10 56)(11 45 78 38)(12 46 79 39)(13 47 80 40)(14 48 71 31)(15 49 72 32)(16 50 73 33)(17 41 74 34)(18 42 75 35)(19 43 76 36)(20 44 77 37)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)
(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 10)(2 9)(3 8)(4 7)(5 6)(11 77)(12 76)(13 75)(14 74)(15 73)(16 72)(17 71)(18 80)(19 79)(20 78)(21 28)(22 27)(23 26)(24 25)(29 30)(31 41)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)
G:=sub<Sym(80)| (1,33,25,45)(2,46,26,34)(3,35,27,47)(4,48,28,36)(5,37,29,49)(6,50,30,38)(7,39,21,41)(8,42,22,40)(9,31,23,43)(10,44,24,32)(11,52,73,65)(12,66,74,53)(13,54,75,67)(14,68,76,55)(15,56,77,69)(16,70,78,57)(17,58,79,61)(18,62,80,59)(19,60,71,63)(20,64,72,51), (1,57,6,52)(2,58,7,53)(3,59,8,54)(4,60,9,55)(5,51,10,56)(11,45,78,38)(12,46,79,39)(13,47,80,40)(14,48,71,31)(15,49,72,32)(16,50,73,33)(17,41,74,34)(18,42,75,35)(19,43,76,36)(20,44,77,37)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(21,28)(22,27)(23,26)(24,25)(29,30)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)>;
G:=Group( (1,33,25,45)(2,46,26,34)(3,35,27,47)(4,48,28,36)(5,37,29,49)(6,50,30,38)(7,39,21,41)(8,42,22,40)(9,31,23,43)(10,44,24,32)(11,52,73,65)(12,66,74,53)(13,54,75,67)(14,68,76,55)(15,56,77,69)(16,70,78,57)(17,58,79,61)(18,62,80,59)(19,60,71,63)(20,64,72,51), (1,57,6,52)(2,58,7,53)(3,59,8,54)(4,60,9,55)(5,51,10,56)(11,45,78,38)(12,46,79,39)(13,47,80,40)(14,48,71,31)(15,49,72,32)(16,50,73,33)(17,41,74,34)(18,42,75,35)(19,43,76,36)(20,44,77,37)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,77)(12,76)(13,75)(14,74)(15,73)(16,72)(17,71)(18,80)(19,79)(20,78)(21,28)(22,27)(23,26)(24,25)(29,30)(31,41)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70) );
G=PermutationGroup([[(1,33,25,45),(2,46,26,34),(3,35,27,47),(4,48,28,36),(5,37,29,49),(6,50,30,38),(7,39,21,41),(8,42,22,40),(9,31,23,43),(10,44,24,32),(11,52,73,65),(12,66,74,53),(13,54,75,67),(14,68,76,55),(15,56,77,69),(16,70,78,57),(17,58,79,61),(18,62,80,59),(19,60,71,63),(20,64,72,51)], [(1,57,6,52),(2,58,7,53),(3,59,8,54),(4,60,9,55),(5,51,10,56),(11,45,78,38),(12,46,79,39),(13,47,80,40),(14,48,71,31),(15,49,72,32),(16,50,73,33),(17,41,74,34),(18,42,75,35),(19,43,76,36),(20,44,77,37),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,77),(12,76),(13,75),(14,74),(15,73),(16,72),(17,71),(18,80),(19,79),(20,78),(21,28),(22,27),(23,26),(24,25),(29,30),(31,41),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(51,52),(53,60),(54,59),(55,58),(56,57),(61,68),(62,67),(63,66),(64,65),(69,70)]])
68 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 | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 10 | 10 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | D4×D5 | D5×C4○D4 |
| kernel | C42⋊12D10 | C42⋊D5 | C4×D20 | Dic5.14D4 | C22⋊D20 | D10.12D4 | D10⋊D4 | D10.13D4 | D10⋊Q8 | C4×C5⋊D4 | C23⋊D10 | Dic5⋊D4 | D4×C20 | D5×C22×C4 | C2×C4○D20 | C4×D5 | C4×D4 | D10 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 4 | 4 |
Matrix representation of C42⋊12D10 ►in GL4(𝔽41) generated by
| 1 | 40 | 0 | 0 |
| 2 | 40 | 0 | 0 |
| 0 | 0 | 18 | 35 |
| 0 | 0 | 6 | 23 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 40 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 35 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 1 | 35 |
G:=sub<GL(4,GF(41))| [1,2,0,0,40,40,0,0,0,0,18,6,0,0,35,23],[9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[40,0,0,0,1,1,0,0,0,0,6,35,0,0,6,1],[40,0,0,0,0,40,0,0,0,0,6,1,0,0,6,35] >;
C42⋊12D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{12}D_{10} % in TeX
G:=Group("C4^2:12D10"); // GroupNames label
G:=SmallGroup(320,1219);
// by ID
G=gap.SmallGroup(320,1219);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,675,570,12550]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations