metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊8D10, C4⋊C4⋊42D10, (C4×D20)⋊7C2, (C2×C20)⋊10D4, (C2×C4)⋊11D20, (C4×C20)⋊5C22, C4.70(C2×D20), D10⋊1(C4○D4), C4⋊2D20⋊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×C10).138C23, (C22×C20).228C22, (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
Subgroups: 1310 in 330 conjugacy classes, 115 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×18], D4 [×14], Q8 [×2], C23, C23 [×10], D5 [×6], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×4], C22×C4, C22×C4 [×11], C2×D4 [×7], C2×Q8, C4○D4 [×4], C24, Dic5 [×4], C20 [×4], C20 [×4], D10 [×4], D10 [×18], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, Dic10 [×2], C4×D5 [×12], D20 [×10], C2×Dic5 [×4], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C22×D5 [×4], C22×D5 [×6], C22×C10, C22.19C24, C4⋊Dic5 [×4], D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×6], C2×C4×D5 [×4], C2×D20, C2×D20 [×4], C4○D20 [×4], C22×Dic5, C2×C5⋊D4 [×2], C22×C20, C23×D5, C4×D20 [×4], C22⋊D20 [×2], C22.D20 [×2], C4⋊2D20 [×2], D10⋊2Q8 [×2], C5×C42⋊C2, D5×C22×C4, C2×C4○D20, C42⋊8D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], D20 [×4], C22×D5 [×7], C22.19C24, C2×D20 [×6], C23×D5, C22×D20, D5×C4○D4 [×2], C42⋊8D10
Generators and relations
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 >
►On 80 points
(1 45 6 35)(2 31 7 41)(3 47 8 37)(4 33 9 43)(5 49 10 39)(11 65 75 70)(12 54 76 59)(13 67 77 62)(14 56 78 51)(15 69 79 64)(16 58 80 53)(17 61 71 66)(18 60 72 55)(19 63 73 68)(20 52 74 57)(21 50 26 40)(22 36 27 46)(23 42 28 32)(24 38 29 48)(25 44 30 34)
(1 58 26 70)(2 59 27 61)(3 60 28 62)(4 51 29 63)(5 52 30 64)(6 53 21 65)(7 54 22 66)(8 55 23 67)(9 56 24 68)(10 57 25 69)(11 45 80 40)(12 46 71 31)(13 47 72 32)(14 48 73 33)(15 49 74 34)(16 50 75 35)(17 41 76 36)(18 42 77 37)(19 43 78 38)(20 44 79 39)
(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 25)(2 24)(3 23)(4 22)(5 21)(6 30)(7 29)(8 28)(9 27)(10 26)(11 74)(12 73)(13 72)(14 71)(15 80)(16 79)(17 78)(18 77)(19 76)(20 75)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 50)(40 49)(51 66)(52 65)(53 64)(54 63)(55 62)(56 61)(57 70)(58 69)(59 68)(60 67)
G:=sub<Sym(80)| (1,45,6,35)(2,31,7,41)(3,47,8,37)(4,33,9,43)(5,49,10,39)(11,65,75,70)(12,54,76,59)(13,67,77,62)(14,56,78,51)(15,69,79,64)(16,58,80,53)(17,61,71,66)(18,60,72,55)(19,63,73,68)(20,52,74,57)(21,50,26,40)(22,36,27,46)(23,42,28,32)(24,38,29,48)(25,44,30,34), (1,58,26,70)(2,59,27,61)(3,60,28,62)(4,51,29,63)(5,52,30,64)(6,53,21,65)(7,54,22,66)(8,55,23,67)(9,56,24,68)(10,57,25,69)(11,45,80,40)(12,46,71,31)(13,47,72,32)(14,48,73,33)(15,49,74,34)(16,50,75,35)(17,41,76,36)(18,42,77,37)(19,43,78,38)(20,44,79,39), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,30)(7,29)(8,28)(9,27)(10,26)(11,74)(12,73)(13,72)(14,71)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,50)(40,49)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67)>;
G:=Group( (1,45,6,35)(2,31,7,41)(3,47,8,37)(4,33,9,43)(5,49,10,39)(11,65,75,70)(12,54,76,59)(13,67,77,62)(14,56,78,51)(15,69,79,64)(16,58,80,53)(17,61,71,66)(18,60,72,55)(19,63,73,68)(20,52,74,57)(21,50,26,40)(22,36,27,46)(23,42,28,32)(24,38,29,48)(25,44,30,34), (1,58,26,70)(2,59,27,61)(3,60,28,62)(4,51,29,63)(5,52,30,64)(6,53,21,65)(7,54,22,66)(8,55,23,67)(9,56,24,68)(10,57,25,69)(11,45,80,40)(12,46,71,31)(13,47,72,32)(14,48,73,33)(15,49,74,34)(16,50,75,35)(17,41,76,36)(18,42,77,37)(19,43,78,38)(20,44,79,39), (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), (1,25)(2,24)(3,23)(4,22)(5,21)(6,30)(7,29)(8,28)(9,27)(10,26)(11,74)(12,73)(13,72)(14,71)(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,50)(40,49)(51,66)(52,65)(53,64)(54,63)(55,62)(56,61)(57,70)(58,69)(59,68)(60,67) );
G=PermutationGroup([(1,45,6,35),(2,31,7,41),(3,47,8,37),(4,33,9,43),(5,49,10,39),(11,65,75,70),(12,54,76,59),(13,67,77,62),(14,56,78,51),(15,69,79,64),(16,58,80,53),(17,61,71,66),(18,60,72,55),(19,63,73,68),(20,52,74,57),(21,50,26,40),(22,36,27,46),(23,42,28,32),(24,38,29,48),(25,44,30,34)], [(1,58,26,70),(2,59,27,61),(3,60,28,62),(4,51,29,63),(5,52,30,64),(6,53,21,65),(7,54,22,66),(8,55,23,67),(9,56,24,68),(10,57,25,69),(11,45,80,40),(12,46,71,31),(13,47,72,32),(14,48,73,33),(15,49,74,34),(16,50,75,35),(17,41,76,36),(18,42,77,37),(19,43,78,38),(20,44,79,39)], [(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)], [(1,25),(2,24),(3,23),(4,22),(5,21),(6,30),(7,29),(8,28),(9,27),(10,26),(11,74),(12,73),(13,72),(14,71),(15,80),(16,79),(17,78),(18,77),(19,76),(20,75),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,50),(40,49),(51,66),(52,65),(53,64),(54,63),(55,62),(56,61),(57,70),(58,69),(59,68),(60,67)])
Matrix representation ►G ⊆ 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] >;
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⋊2D20 | 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 |
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