metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊8D10, C4⋊C4⋊42D10, (C4×D20)⋊7C2, (C2×C4)⋊11D20, (C2×C20)⋊10D4, (C4×C20)⋊5C22, C4.70(C2×D20), D10⋊1(C4○D4), C4⋊D20⋊42C2, C20.223(C2×D4), C42⋊C2⋊8D5, C22⋊D20⋊29C2, D10⋊2Q8⋊47C2, (C2×D20)⋊52C22, (C2×C10).68C24, C4⋊Dic5⋊55C22, C22⋊C4.92D10, C2.14(C22×D20), C22.19(C2×D20), C10.12(C22×D4), (C2×C20).143C23, C5⋊1(C22.19C24), (C22×C4).365D10, D10⋊C4⋊51C22, C22.97(C23×D5), (C2×Dic10)⋊61C22, C22.D20⋊32C2, (C22×D5).18C23, C23.156(C22×D5), (C22×C20).228C22, (C22×C10).138C23, (C2×Dic5).207C23, (C23×D5).116C22, (C22×Dic5).240C22, C2.9(D5×C4○D4), (D5×C22×C4)⋊2C2, (C2×C4×D5)⋊44C22, (C2×C4○D20)⋊17C2, (C5×C4⋊C4)⋊52C22, (C2×C10).49(C2×D4), C10.133(C2×C4○D4), (C5×C42⋊C2)⋊10C2, (C2×C4).575(C22×D5), (C2×C5⋊D4).107C22, (C5×C22⋊C4).100C22, SmallGroup(320,1196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊8D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, bd=db, dcd=c-1 >
Subgroups: 1310 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C22.19C24, C4⋊Dic5, D10⋊C4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C4×D20, C22⋊D20, C22.D20, C4⋊D20, D10⋊2Q8, C5×C42⋊C2, D5×C22×C4, C2×C4○D20, C42⋊8D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, D20, C22×D5, C22.19C24, C2×D20, C23×D5, C22×D20, D5×C4○D4, C42⋊8D10
►On 80 points
(1 45 6 37)(2 33 7 41)(3 47 8 39)(4 35 9 43)(5 49 10 31)(11 65 77 70)(12 54 78 59)(13 67 79 62)(14 56 80 51)(15 69 71 64)(16 58 72 53)(17 61 73 66)(18 60 74 55)(19 63 75 68)(20 52 76 57)(21 46 26 38)(22 34 27 42)(23 48 28 40)(24 36 29 44)(25 50 30 32)
(1 58 30 70)(2 59 21 61)(3 60 22 62)(4 51 23 63)(5 52 24 64)(6 53 25 65)(7 54 26 66)(8 55 27 67)(9 56 28 68)(10 57 29 69)(11 45 72 32)(12 46 73 33)(13 47 74 34)(14 48 75 35)(15 49 76 36)(16 50 77 37)(17 41 78 38)(18 42 79 39)(19 43 80 40)(20 44 71 31)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 30)(11 76)(12 75)(13 74)(14 73)(15 72)(16 71)(17 80)(18 79)(19 78)(20 77)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 42)(40 41)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
G:=sub<Sym(80)| (1,45,6,37)(2,33,7,41)(3,47,8,39)(4,35,9,43)(5,49,10,31)(11,65,77,70)(12,54,78,59)(13,67,79,62)(14,56,80,51)(15,69,71,64)(16,58,72,53)(17,61,73,66)(18,60,74,55)(19,63,75,68)(20,52,76,57)(21,46,26,38)(22,34,27,42)(23,48,28,40)(24,36,29,44)(25,50,30,32), (1,58,30,70)(2,59,21,61)(3,60,22,62)(4,51,23,63)(5,52,24,64)(6,53,25,65)(7,54,26,66)(8,55,27,67)(9,56,28,68)(10,57,29,69)(11,45,72,32)(12,46,73,33)(13,47,74,34)(14,48,75,35)(15,49,76,36)(16,50,77,37)(17,41,78,38)(18,42,79,39)(19,43,80,40)(20,44,71,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,80)(18,79)(19,78)(20,77)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67)>;
G:=Group( (1,45,6,37)(2,33,7,41)(3,47,8,39)(4,35,9,43)(5,49,10,31)(11,65,77,70)(12,54,78,59)(13,67,79,62)(14,56,80,51)(15,69,71,64)(16,58,72,53)(17,61,73,66)(18,60,74,55)(19,63,75,68)(20,52,76,57)(21,46,26,38)(22,34,27,42)(23,48,28,40)(24,36,29,44)(25,50,30,32), (1,58,30,70)(2,59,21,61)(3,60,22,62)(4,51,23,63)(5,52,24,64)(6,53,25,65)(7,54,26,66)(8,55,27,67)(9,56,28,68)(10,57,29,69)(11,45,72,32)(12,46,73,33)(13,47,74,34)(14,48,75,35)(15,49,76,36)(16,50,77,37)(17,41,78,38)(18,42,79,39)(19,43,80,40)(20,44,71,31), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,30)(11,76)(12,75)(13,74)(14,73)(15,72)(16,71)(17,80)(18,79)(19,78)(20,77)(31,50)(32,49)(33,48)(34,47)(35,46)(36,45)(37,44)(38,43)(39,42)(40,41)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67) );
G=PermutationGroup([[(1,45,6,37),(2,33,7,41),(3,47,8,39),(4,35,9,43),(5,49,10,31),(11,65,77,70),(12,54,78,59),(13,67,79,62),(14,56,80,51),(15,69,71,64),(16,58,72,53),(17,61,73,66),(18,60,74,55),(19,63,75,68),(20,52,76,57),(21,46,26,38),(22,34,27,42),(23,48,28,40),(24,36,29,44),(25,50,30,32)], [(1,58,30,70),(2,59,21,61),(3,60,22,62),(4,51,23,63),(5,52,24,64),(6,53,25,65),(7,54,26,66),(8,55,27,67),(9,56,28,68),(10,57,29,69),(11,45,72,32),(12,46,73,33),(13,47,74,34),(14,48,75,35),(15,49,76,36),(16,50,77,37),(17,41,78,38),(18,42,79,39),(19,43,80,40),(20,44,71,31)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,30),(11,76),(12,75),(13,74),(14,73),(15,72),(16,71),(17,80),(18,79),(19,78),(20,77),(31,50),(32,49),(33,48),(34,47),(35,46),(36,45),(37,44),(38,43),(39,42),(40,41),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67)]])
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 | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| 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 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D10 | D20 | D5×C4○D4 |
| kernel | C42⋊8D10 | C4×D20 | C22⋊D20 | C22.D20 | C4⋊D20 | D10⋊2Q8 | C5×C42⋊C2 | D5×C22×C4 | C2×C4○D20 | C2×C20 | C42⋊C2 | D10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C2 |
| # reps | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 8 | 4 | 4 | 4 | 2 | 16 | 8 |
Matrix representation of C42⋊8D10 ►in GL4(𝔽41) generated by
| 30 | 32 | 0 | 0 |
| 9 | 11 | 0 | 0 |
| 0 | 0 | 40 | 32 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 34 | 34 | 0 | 0 |
| 7 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 18 | 40 |
| 34 | 34 | 0 | 0 |
| 1 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [30,9,0,0,32,11,0,0,0,0,40,0,0,0,32,1],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,9],[34,7,0,0,34,1,0,0,0,0,1,18,0,0,0,40],[34,1,0,0,34,7,0,0,0,0,40,0,0,0,0,40] >;
C42⋊8D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_8D_{10} % in TeX
G:=Group("C4^2:8D10"); // GroupNames label
G:=SmallGroup(320,1196);
// by ID
G=gap.SmallGroup(320,1196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations