metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊11D10, C10.152+ 1+4, (D4×D5)⋊5C4, D4⋊9(C4×D5), (C4×D4)⋊7D5, C4⋊C4⋊56D10, (D4×C20)⋊9C2, D20⋊23(C2×C4), (C4×D20)⋊24C2, (D4×Dic5)⋊8C2, (C22×C4)⋊4D10, (C4×C20)⋊16C22, C22⋊C4⋊53D10, D20⋊8C4⋊14C2, (C2×D4).245D10, C42⋊D5⋊11C2, C20.67(C22×C4), C10.43(C23×C4), (C2×C10).89C24, C4⋊Dic5⋊73C22, Dic5⋊4D4⋊45C2, C2.3(D4⋊6D10), C2.3(D4⋊8D10), (C2×C20).587C23, (C22×C20)⋊36C22, C5⋊4(C22.11C24), (C4×Dic5)⋊11C22, D10.17(C22×C4), C23.D5⋊48C22, D10⋊C4⋊62C22, C22.32(C23×D5), (C2×D20).266C22, (D4×C10).253C22, C10.D4⋊64C22, Dic5.17(C22×C4), (C22×Dic5)⋊8C22, (C23×D5).38C22, C23.168(C22×D5), (C22×C10).159C23, (C2×Dic5).211C23, (C22×D5).178C23, (C2×D4×D5).9C2, C4.32(C2×C4×D5), (C4×D5)⋊3(C2×C4), C5⋊D4⋊7(C2×C4), (C5×D4)⋊22(C2×C4), (C4×C5⋊D4)⋊40C2, C22.2(C2×C4×D5), (C2×C4×D5)⋊46C22, C4⋊C4⋊7D5⋊14C2, (C5×C4⋊C4)⋊56C22, C2.24(D5×C22×C4), (D5×C22⋊C4)⋊27C2, (C22×D5)⋊10(C2×C4), (C2×C10).9(C22×C4), (C2×D10⋊C4)⋊34C2, (C5×C22⋊C4)⋊63C22, (C2×C4).282(C22×D5), (C2×C5⋊D4).118C22, SmallGroup(320,1217)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊11D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 1294 in 338 conjugacy classes, 151 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, D4, C23, C23, D5, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C4×D4, C22×D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22.11C24, C4×Dic5, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, D4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, C23×D5, C42⋊D5, C4×D20, D5×C22⋊C4, Dic5⋊4D4, C4⋊C4⋊7D5, D20⋊8C4, C2×D10⋊C4, C4×C5⋊D4, D4×Dic5, D4×C20, C2×D4×D5, C42⋊11D10
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C24, D10, C23×C4, 2+ 1+4, C4×D5, C22×D5, C22.11C24, C2×C4×D5, C23×D5, D5×C22×C4, D4⋊6D10, D4⋊8D10, C42⋊11D10
►On 80 points
(1 70 30 53)(2 54 21 61)(3 62 22 55)(4 56 23 63)(5 64 24 57)(6 58 25 65)(7 66 26 59)(8 60 27 67)(9 68 28 51)(10 52 29 69)(11 45 73 40)(12 31 74 46)(13 47 75 32)(14 33 76 48)(15 49 77 34)(16 35 78 50)(17 41 79 36)(18 37 80 42)(19 43 71 38)(20 39 72 44)
(1 35 6 40)(2 36 7 31)(3 37 8 32)(4 38 9 33)(5 39 10 34)(11 70 78 58)(12 61 79 59)(13 62 80 60)(14 63 71 51)(15 64 72 52)(16 65 73 53)(17 66 74 54)(18 67 75 55)(19 68 76 56)(20 69 77 57)(21 41 26 46)(22 42 27 47)(23 43 28 48)(24 44 29 49)(25 45 30 50)
(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 72)(12 71)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 28)(22 27)(23 26)(24 25)(29 30)(31 43)(32 42)(33 41)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)(40 44)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)
G:=sub<Sym(80)| (1,70,30,53)(2,54,21,61)(3,62,22,55)(4,56,23,63)(5,64,24,57)(6,58,25,65)(7,66,26,59)(8,60,27,67)(9,68,28,51)(10,52,29,69)(11,45,73,40)(12,31,74,46)(13,47,75,32)(14,33,76,48)(15,49,77,34)(16,35,78,50)(17,41,79,36)(18,37,80,42)(19,43,71,38)(20,39,72,44), (1,35,6,40)(2,36,7,31)(3,37,8,32)(4,38,9,33)(5,39,10,34)(11,70,78,58)(12,61,79,59)(13,62,80,60)(14,63,71,51)(15,64,72,52)(16,65,73,53)(17,66,74,54)(18,67,75,55)(19,68,76,56)(20,69,77,57)(21,41,26,46)(22,42,27,47)(23,43,28,48)(24,44,29,49)(25,45,30,50), (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)(11,72)(12,71)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,28)(22,27)(23,26)(24,25)(29,30)(31,43)(32,42)(33,41)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)>;
G:=Group( (1,70,30,53)(2,54,21,61)(3,62,22,55)(4,56,23,63)(5,64,24,57)(6,58,25,65)(7,66,26,59)(8,60,27,67)(9,68,28,51)(10,52,29,69)(11,45,73,40)(12,31,74,46)(13,47,75,32)(14,33,76,48)(15,49,77,34)(16,35,78,50)(17,41,79,36)(18,37,80,42)(19,43,71,38)(20,39,72,44), (1,35,6,40)(2,36,7,31)(3,37,8,32)(4,38,9,33)(5,39,10,34)(11,70,78,58)(12,61,79,59)(13,62,80,60)(14,63,71,51)(15,64,72,52)(16,65,73,53)(17,66,74,54)(18,67,75,55)(19,68,76,56)(20,69,77,57)(21,41,26,46)(22,42,27,47)(23,43,28,48)(24,44,29,49)(25,45,30,50), (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)(11,72)(12,71)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,28)(22,27)(23,26)(24,25)(29,30)(31,43)(32,42)(33,41)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)(40,44)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70) );
G=PermutationGroup([[(1,70,30,53),(2,54,21,61),(3,62,22,55),(4,56,23,63),(5,64,24,57),(6,58,25,65),(7,66,26,59),(8,60,27,67),(9,68,28,51),(10,52,29,69),(11,45,73,40),(12,31,74,46),(13,47,75,32),(14,33,76,48),(15,49,77,34),(16,35,78,50),(17,41,79,36),(18,37,80,42),(19,43,71,38),(20,39,72,44)], [(1,35,6,40),(2,36,7,31),(3,37,8,32),(4,38,9,33),(5,39,10,34),(11,70,78,58),(12,61,79,59),(13,62,80,60),(14,63,71,51),(15,64,72,52),(16,65,73,53),(17,66,74,54),(18,67,75,55),(19,68,76,56),(20,69,77,57),(21,41,26,46),(22,42,27,47),(23,43,28,48),(24,44,29,49),(25,45,30,50)], [(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),(11,72),(12,71),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,28),(22,27),(23,26),(24,25),(29,30),(31,43),(32,42),(33,41),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45),(40,44),(51,54),(52,53),(55,60),(56,59),(57,58),(61,68),(62,67),(63,66),(64,65),(69,70)]])
74 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | ··· | 2M | 4A | ··· | 4J | 4K | ··· | 4T | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
74 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | D10 | D10 | D10 | D10 | D10 | C4×D5 | 2+ 1+4 | D4⋊6D10 | D4⋊8D10 |
| kernel | C42⋊11D10 | C42⋊D5 | C4×D20 | D5×C22⋊C4 | Dic5⋊4D4 | C4⋊C4⋊7D5 | D20⋊8C4 | C2×D10⋊C4 | C4×C5⋊D4 | D4×Dic5 | D4×C20 | C2×D4×D5 | D4×D5 | C4×D4 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | D4 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 16 | 2 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 4 | 4 |
Matrix representation of C42⋊11D10 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 34 | 7 | 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 |
| 7 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 35 |
| 0 | 0 | 0 | 0 | 40 | 6 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,40,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,2,28,0,0,0,0,13,39,0,0,0,0,0,0,2,28,0,0,0,0,13,39],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,6,35,0,0,0,0,6,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40],[1,7,0,0,0,0,0,40,0,0,0,0,0,0,6,1,0,0,0,0,6,35,0,0,0,0,0,0,35,40,0,0,0,0,35,6] >;
C42⋊11D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{11}D_{10} % in TeX
G:=Group("C4^2:11D10"); // GroupNames label
G:=SmallGroup(320,1217);
// by ID
G=gap.SmallGroup(320,1217);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,1123,80,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations