metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊21D10, C10.732+ 1+4, (C2×Q8)⋊9D10, (C4×C20)⋊33C22, C22⋊C4⋊35D10, C4.4D4⋊13D5, D10⋊3Q8⋊31C2, (C2×D4).111D10, C23⋊D10.6C2, C42⋊D5⋊36C2, C4⋊Dic5⋊42C22, (Q8×C10)⋊15C22, D10.37(C4○D4), Dic5⋊4D4⋊32C2, (C2×C10).223C24, (C2×C20).632C23, (C4×Dic5)⋊57C22, D10.12D4⋊44C2, C23.D5⋊34C22, C2.76(D4⋊6D10), C23.45(C22×D5), C5⋊8(C22.45C24), (D4×C10).211C22, C23.D10⋊40C2, C10.D4⋊67C22, (C22×C10).53C23, (C23×D5).66C22, C22.244(C23×D5), C23.18D10⋊25C2, (C2×Dic5).265C23, (C22×Dic5)⋊28C22, (C22×D5).227C23, D10⋊C4.136C22, C2.79(D5×C4○D4), (D5×C22⋊C4)⋊19C2, C10.190(C2×C4○D4), (C5×C4.4D4)⋊15C2, (C2×C4×D5).266C22, (C2×C4).74(C22×D5), (C5×C22⋊C4)⋊31C22, (C2×C5⋊D4).61C22, SmallGroup(320,1351)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊21D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, ad=da, cbc-1=dbd=a2b, dcd=c-1 >
Subgroups: 950 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C22⋊C4, C42⋊C2, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C22.45C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C4×C20, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C42⋊D5, C23.D10, D5×C22⋊C4, Dic5⋊4D4, D10.12D4, C23.18D10, C23⋊D10, D10⋊3Q8, C5×C4.4D4, C42⋊21D10
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, 2+ 1+4, C22×D5, C22.45C24, C23×D5, D4⋊6D10, D5×C4○D4, C42⋊21D10
►On 80 points
(1 34 11 51)(2 40 12 57)(3 36 13 53)(4 32 14 59)(5 38 15 55)(6 39 16 56)(7 35 17 52)(8 31 18 58)(9 37 19 54)(10 33 20 60)(21 64 26 77)(22 73 27 70)(23 66 28 79)(24 75 29 62)(25 68 30 71)(41 80 46 67)(42 63 47 76)(43 72 48 69)(44 65 49 78)(45 74 50 61)
(1 41 6 29)(2 47 7 25)(3 43 8 21)(4 49 9 27)(5 45 10 23)(11 46 16 24)(12 42 17 30)(13 48 18 26)(14 44 19 22)(15 50 20 28)(31 64 36 72)(32 78 37 70)(33 66 38 74)(34 80 39 62)(35 68 40 76)(51 67 56 75)(52 71 57 63)(53 69 58 77)(54 73 59 65)(55 61 60 79)
(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 14)(2 13)(3 12)(4 11)(5 15)(6 19)(7 18)(8 17)(9 16)(10 20)(21 25)(22 24)(26 30)(27 29)(31 52)(32 51)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 49)(42 48)(43 47)(44 46)(62 70)(63 69)(64 68)(65 67)(71 77)(72 76)(73 75)(78 80)
G:=sub<Sym(80)| (1,34,11,51)(2,40,12,57)(3,36,13,53)(4,32,14,59)(5,38,15,55)(6,39,16,56)(7,35,17,52)(8,31,18,58)(9,37,19,54)(10,33,20,60)(21,64,26,77)(22,73,27,70)(23,66,28,79)(24,75,29,62)(25,68,30,71)(41,80,46,67)(42,63,47,76)(43,72,48,69)(44,65,49,78)(45,74,50,61), (1,41,6,29)(2,47,7,25)(3,43,8,21)(4,49,9,27)(5,45,10,23)(11,46,16,24)(12,42,17,30)(13,48,18,26)(14,44,19,22)(15,50,20,28)(31,64,36,72)(32,78,37,70)(33,66,38,74)(34,80,39,62)(35,68,40,76)(51,67,56,75)(52,71,57,63)(53,69,58,77)(54,73,59,65)(55,61,60,79), (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), (1,14)(2,13)(3,12)(4,11)(5,15)(6,19)(7,18)(8,17)(9,16)(10,20)(21,25)(22,24)(26,30)(27,29)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,49)(42,48)(43,47)(44,46)(62,70)(63,69)(64,68)(65,67)(71,77)(72,76)(73,75)(78,80)>;
G:=Group( (1,34,11,51)(2,40,12,57)(3,36,13,53)(4,32,14,59)(5,38,15,55)(6,39,16,56)(7,35,17,52)(8,31,18,58)(9,37,19,54)(10,33,20,60)(21,64,26,77)(22,73,27,70)(23,66,28,79)(24,75,29,62)(25,68,30,71)(41,80,46,67)(42,63,47,76)(43,72,48,69)(44,65,49,78)(45,74,50,61), (1,41,6,29)(2,47,7,25)(3,43,8,21)(4,49,9,27)(5,45,10,23)(11,46,16,24)(12,42,17,30)(13,48,18,26)(14,44,19,22)(15,50,20,28)(31,64,36,72)(32,78,37,70)(33,66,38,74)(34,80,39,62)(35,68,40,76)(51,67,56,75)(52,71,57,63)(53,69,58,77)(54,73,59,65)(55,61,60,79), (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), (1,14)(2,13)(3,12)(4,11)(5,15)(6,19)(7,18)(8,17)(9,16)(10,20)(21,25)(22,24)(26,30)(27,29)(31,52)(32,51)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,49)(42,48)(43,47)(44,46)(62,70)(63,69)(64,68)(65,67)(71,77)(72,76)(73,75)(78,80) );
G=PermutationGroup([[(1,34,11,51),(2,40,12,57),(3,36,13,53),(4,32,14,59),(5,38,15,55),(6,39,16,56),(7,35,17,52),(8,31,18,58),(9,37,19,54),(10,33,20,60),(21,64,26,77),(22,73,27,70),(23,66,28,79),(24,75,29,62),(25,68,30,71),(41,80,46,67),(42,63,47,76),(43,72,48,69),(44,65,49,78),(45,74,50,61)], [(1,41,6,29),(2,47,7,25),(3,43,8,21),(4,49,9,27),(5,45,10,23),(11,46,16,24),(12,42,17,30),(13,48,18,26),(14,44,19,22),(15,50,20,28),(31,64,36,72),(32,78,37,70),(33,66,38,74),(34,80,39,62),(35,68,40,76),(51,67,56,75),(52,71,57,63),(53,69,58,77),(54,73,59,65),(55,61,60,79)], [(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)], [(1,14),(2,13),(3,12),(4,11),(5,15),(6,19),(7,18),(8,17),(9,16),(10,20),(21,25),(22,24),(26,30),(27,29),(31,52),(32,51),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,49),(42,48),(43,47),(44,46),(62,70),(63,69),(64,68),(65,67),(71,77),(72,76),(73,75),(78,80)]])
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 | D5×C4○D4 |
| kernel | C42⋊21D10 | C42⋊D5 | C23.D10 | D5×C22⋊C4 | Dic5⋊4D4 | D10.12D4 | C23.18D10 | C23⋊D10 | D10⋊3Q8 | C5×C4.4D4 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 8 | 2 | 8 | 2 | 2 | 1 | 4 | 8 |
Matrix representation of C42⋊21D10 ►in GL6(𝔽41)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 32 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 7 | 0 | 0 |
| 0 | 0 | 35 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 1 | 0 | 0 |
| 0 | 0 | 6 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,7,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
C42⋊21D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{21}D_{10} % in TeX
G:=Group("C4^2:21D10"); // GroupNames label
G:=SmallGroup(320,1351);
// by ID
G=gap.SmallGroup(320,1351);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,387,100,346,136,12550]);
// 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,a*d=d*a,c*b*c^-1=d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations