metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊9D10, C10.952+ (1+4), (C2×C4)⋊5D20, C4⋊C4⋊43D10, (C2×C20)⋊11D4, C4⋊D20⋊3C2, (C4×C20)⋊1C22, C4.71(C2×D20), C22⋊D20⋊4C2, C4⋊2D20⋊11C2, C20.287(C2×D4), C4.D20⋊3C2, (C2×D20)⋊5C22, C42⋊C2⋊9D5, (C22×D20)⋊14C2, (C2×C10).69C24, C22⋊C4.93D10, C10.13(C22×D4), C2.15(C22×D20), C22.20(C2×D20), C2.7(D4⋊8D10), D10⋊C4⋊3C22, (C2×C20).144C23, C5⋊1(C22.29C24), (C22×C4).190D10, C22.98(C23×D5), (C2×Dic10)⋊51C22, (C2×Dic5).23C23, (C23×D5).36C22, (C22×D5).19C23, C23.157(C22×D5), (C22×C20).229C22, (C22×C10).139C23, (C2×C4×D5)⋊1C22, (C2×C4○D20)⋊18C2, (C5×C4⋊C4)⋊53C22, (C2×C10).50(C2×D4), (C5×C42⋊C2)⋊11C2, (C2×C4).149(C22×D5), (C2×C5⋊D4).108C22, (C5×C22⋊C4).101C22, SmallGroup(320,1197)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1646 in 334 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×6], C22, C22 [×2], C22 [×28], C5, C2×C4 [×2], C2×C4 [×8], C2×C4 [×6], D4 [×22], Q8 [×2], C23, C23 [×14], D5 [×6], C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×19], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], C20 [×4], C20 [×4], D10 [×26], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×18], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×8], C22×D5 [×6], C22×D5 [×8], C22×C10, C22.29C24, D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×2], C2×D20, C2×D20 [×12], C2×D20 [×4], C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C23×D5 [×2], C4⋊D20 [×2], C4.D20 [×2], C22⋊D20 [×4], C4⋊2D20 [×4], C5×C42⋊C2, C22×D20, C2×C4○D20, C42⋊9D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], D20 [×4], C22×D5 [×7], C22.29C24, C2×D20 [×6], C23×D5, C22×D20, D4⋊8D10 [×2], C42⋊9D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, dad=a-1, bc=cb, dbd=b-1, dcd=c-1 >
►On 80 points
(1 45 6 33)(2 39 7 41)(3 47 8 35)(4 31 9 43)(5 49 10 37)(11 65 73 70)(12 54 74 59)(13 67 75 62)(14 56 76 51)(15 69 77 64)(16 58 78 53)(17 61 79 66)(18 60 80 55)(19 63 71 68)(20 52 72 57)(21 42 26 40)(22 36 27 48)(23 44 28 32)(24 38 29 50)(25 46 30 34)
(1 58 24 70)(2 59 25 61)(3 60 26 62)(4 51 27 63)(5 52 28 64)(6 53 29 65)(7 54 30 66)(8 55 21 67)(9 56 22 68)(10 57 23 69)(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)
(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 69)(2 68)(3 67)(4 66)(5 65)(6 64)(7 63)(8 62)(9 61)(10 70)(11 49)(12 48)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 50)(21 60)(22 59)(23 58)(24 57)(25 56)(26 55)(27 54)(28 53)(29 52)(30 51)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 72)(39 71)(40 80)
G:=sub<Sym(80)| (1,45,6,33)(2,39,7,41)(3,47,8,35)(4,31,9,43)(5,49,10,37)(11,65,73,70)(12,54,74,59)(13,67,75,62)(14,56,76,51)(15,69,77,64)(16,58,78,53)(17,61,79,66)(18,60,80,55)(19,63,71,68)(20,52,72,57)(21,42,26,40)(22,36,27,48)(23,44,28,32)(24,38,29,50)(25,46,30,34), (1,58,24,70)(2,59,25,61)(3,60,26,62)(4,51,27,63)(5,52,28,64)(6,53,29,65)(7,54,30,66)(8,55,21,67)(9,56,22,68)(10,57,23,69)(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), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,70)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80)>;
G:=Group( (1,45,6,33)(2,39,7,41)(3,47,8,35)(4,31,9,43)(5,49,10,37)(11,65,73,70)(12,54,74,59)(13,67,75,62)(14,56,76,51)(15,69,77,64)(16,58,78,53)(17,61,79,66)(18,60,80,55)(19,63,71,68)(20,52,72,57)(21,42,26,40)(22,36,27,48)(23,44,28,32)(24,38,29,50)(25,46,30,34), (1,58,24,70)(2,59,25,61)(3,60,26,62)(4,51,27,63)(5,52,28,64)(6,53,29,65)(7,54,30,66)(8,55,21,67)(9,56,22,68)(10,57,23,69)(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), (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), (1,69)(2,68)(3,67)(4,66)(5,65)(6,64)(7,63)(8,62)(9,61)(10,70)(11,49)(12,48)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,50)(21,60)(22,59)(23,58)(24,57)(25,56)(26,55)(27,54)(28,53)(29,52)(30,51)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,72)(39,71)(40,80) );
G=PermutationGroup([(1,45,6,33),(2,39,7,41),(3,47,8,35),(4,31,9,43),(5,49,10,37),(11,65,73,70),(12,54,74,59),(13,67,75,62),(14,56,76,51),(15,69,77,64),(16,58,78,53),(17,61,79,66),(18,60,80,55),(19,63,71,68),(20,52,72,57),(21,42,26,40),(22,36,27,48),(23,44,28,32),(24,38,29,50),(25,46,30,34)], [(1,58,24,70),(2,59,25,61),(3,60,26,62),(4,51,27,63),(5,52,28,64),(6,53,29,65),(7,54,30,66),(8,55,21,67),(9,56,22,68),(10,57,23,69),(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)], [(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)], [(1,69),(2,68),(3,67),(4,66),(5,65),(6,64),(7,63),(8,62),(9,61),(10,70),(11,49),(12,48),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,50),(21,60),(22,59),(23,58),(24,57),(25,56),(26,55),(27,54),(28,53),(29,52),(30,51),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,72),(39,71),(40,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 39 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 13 | 0 | 0 |
| 0 | 0 | 28 | 39 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 13 |
| 0 | 0 | 0 | 0 | 28 | 39 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 25 | 0 | 0 |
| 0 | 0 | 39 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 16 |
| 0 | 0 | 0 | 0 | 2 | 16 |
G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[1,1,0,0,0,0,0,40,0,0,0,0,0,0,16,39,0,0,0,0,25,25,0,0,0,0,0,0,25,2,0,0,0,0,16,16] >;
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 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 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | D20 | 2+ (1+4) | D4⋊8D10 |
| kernel | C42⋊9D10 | C4⋊D20 | C4.D20 | C22⋊D20 | C4⋊2D20 | C5×C42⋊C2 | C22×D20 | C2×C4○D20 | C2×C20 | C42⋊C2 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C10 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 16 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_9D_{10} % in TeX
G:=Group("C4^2:9D10"); // GroupNames label
G:=SmallGroup(320,1197);
// by ID
G=gap.SmallGroup(320,1197);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,184,675,570,80,12550]);
// 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,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations