metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊22D10, C10.1292+ 1+4, (C2×Q8)⋊10D10, (C4×C20)⋊29C22, C22⋊C4⋊21D10, C4.4D4⋊16D5, C23⋊D10⋊25C2, C22⋊D20⋊26C2, (C2×D4).112D10, C4.D20⋊30C2, (C2×C20).83C23, (Q8×C10)⋊16C22, C20.23D4⋊24C2, D10⋊C4⋊6C22, (C2×C10).227C24, C5⋊2(C24⋊C22), (C4×Dic5)⋊37C22, (C2×D20).36C22, (C23×D5)⋊12C22, C2.77(D4⋊6D10), C2.53(D4⋊8D10), C23.D5⋊35C22, C23.49(C22×D5), Dic5.5D4⋊43C2, (C2×Dic10)⋊10C22, (D4×C10).212C22, (C22×C10).57C23, (C22×D5).99C23, C22.248(C23×D5), (C2×Dic5).117C23, (C5×C4.4D4)⋊19C2, (C5×C22⋊C4)⋊32C22, (C2×C4).200(C22×D5), (C2×C5⋊D4).65C22, SmallGroup(320,1355)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊22D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1286 in 260 conjugacy classes, 91 normal (17 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, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4.4D4, C4.4D4, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C24⋊C22, C4×Dic5, D10⋊C4, C23.D5, C4×C20, C5×C22⋊C4, C2×Dic10, C2×D20, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C4.D20, C22⋊D20, Dic5.5D4, C23⋊D10, C20.23D4, C5×C4.4D4, C42⋊22D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C24⋊C22, C23×D5, D4⋊6D10, D4⋊8D10, C42⋊22D10
►On 80 points
(1 42 13 47)(2 48 14 43)(3 44 15 49)(4 50 11 45)(5 46 12 41)(6 56 16 51)(7 52 17 57)(8 58 18 53)(9 54 19 59)(10 60 20 55)(21 67 33 79)(22 80 34 68)(23 69 35 71)(24 72 36 70)(25 61 37 73)(26 74 38 62)(27 63 39 75)(28 76 40 64)(29 65 31 77)(30 78 32 66)
(1 38 18 21)(2 34 19 27)(3 40 20 23)(4 36 16 29)(5 32 17 25)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(41 78 52 73)(42 62 53 67)(43 80 54 75)(44 64 55 69)(45 72 56 77)(46 66 57 61)(47 74 58 79)(48 68 59 63)(49 76 60 71)(50 70 51 65)
(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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 60)(50 59)(61 74)(62 73)(63 72)(64 71)(65 80)(66 79)(67 78)(68 77)(69 76)(70 75)
G:=sub<Sym(80)| (1,42,13,47)(2,48,14,43)(3,44,15,49)(4,50,11,45)(5,46,12,41)(6,56,16,51)(7,52,17,57)(8,58,18,53)(9,54,19,59)(10,60,20,55)(21,67,33,79)(22,80,34,68)(23,69,35,71)(24,72,36,70)(25,61,37,73)(26,74,38,62)(27,63,39,75)(28,76,40,64)(29,65,31,77)(30,78,32,66), (1,38,18,21)(2,34,19,27)(3,40,20,23)(4,36,16,29)(5,32,17,25)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,78,52,73)(42,62,53,67)(43,80,54,75)(44,64,55,69)(45,72,56,77)(46,66,57,61)(47,74,58,79)(48,68,59,63)(49,76,60,71)(50,70,51,65), (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,60)(50,59)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)>;
G:=Group( (1,42,13,47)(2,48,14,43)(3,44,15,49)(4,50,11,45)(5,46,12,41)(6,56,16,51)(7,52,17,57)(8,58,18,53)(9,54,19,59)(10,60,20,55)(21,67,33,79)(22,80,34,68)(23,69,35,71)(24,72,36,70)(25,61,37,73)(26,74,38,62)(27,63,39,75)(28,76,40,64)(29,65,31,77)(30,78,32,66), (1,38,18,21)(2,34,19,27)(3,40,20,23)(4,36,16,29)(5,32,17,25)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,78,52,73)(42,62,53,67)(43,80,54,75)(44,64,55,69)(45,72,56,77)(46,66,57,61)(47,74,58,79)(48,68,59,63)(49,76,60,71)(50,70,51,65), (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,60)(50,59)(61,74)(62,73)(63,72)(64,71)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75) );
G=PermutationGroup([[(1,42,13,47),(2,48,14,43),(3,44,15,49),(4,50,11,45),(5,46,12,41),(6,56,16,51),(7,52,17,57),(8,58,18,53),(9,54,19,59),(10,60,20,55),(21,67,33,79),(22,80,34,68),(23,69,35,71),(24,72,36,70),(25,61,37,73),(26,74,38,62),(27,63,39,75),(28,76,40,64),(29,65,31,77),(30,78,32,66)], [(1,38,18,21),(2,34,19,27),(3,40,20,23),(4,36,16,29),(5,32,17,25),(6,31,11,24),(7,37,12,30),(8,33,13,26),(9,39,14,22),(10,35,15,28),(41,78,52,73),(42,62,53,67),(43,80,54,75),(44,64,55,69),(45,72,56,77),(46,66,57,61),(47,74,58,79),(48,68,59,63),(49,76,60,71),(50,70,51,65)], [(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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,40),(29,39),(30,38),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,60),(50,59),(61,74),(62,73),(63,72),(64,71),(65,80),(66,79),(67,78),(68,77),(69,76),(70,75)]])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 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 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 | D4⋊8D10 |
| kernel | C42⋊22D10 | C4.D20 | C22⋊D20 | Dic5.5D4 | C23⋊D10 | C20.23D4 | C5×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 |
| # reps | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 3 | 4 | 8 |
Matrix representation of C42⋊22D10 ►in GL8(𝔽41)
| 17 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 24 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 34 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 7 | 34 |
| 0 | 0 | 0 | 0 | 14 | 2 | 1 | 0 |
| 0 | 0 | 0 | 0 | 14 | 14 | 0 | 1 |
| 1 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 40 | 0 | 0 | 0 | 0 |
| 2 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 30 | 32 | 27 | 30 |
| 0 | 0 | 0 | 0 | 9 | 11 | 27 | 27 |
| 0 | 0 | 0 | 0 | 13 | 22 | 30 | 9 |
| 0 | 0 | 0 | 0 | 28 | 13 | 32 | 11 |
| 40 | 7 | 1 | 34 | 0 | 0 | 0 | 0 |
| 34 | 7 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 14 | 40 | 7 |
| 0 | 0 | 0 | 0 | 0 | 2 | 34 | 7 |
| 7 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 7 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 34 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 40 |
G:=sub<GL(8,GF(41))| [17,40,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,17,40,0,0,0,0,0,0,1,24,0,0,0,0,0,0,0,0,40,0,14,14,0,0,0,0,0,40,2,14,0,0,0,0,34,7,1,0,0,0,0,0,1,34,0,1],[1,0,2,0,0,0,0,0,0,1,0,2,0,0,0,0,40,0,40,0,0,0,0,0,0,40,0,40,0,0,0,0,0,0,0,0,30,9,13,28,0,0,0,0,32,11,22,13,0,0,0,0,27,27,30,32,0,0,0,0,30,27,9,11],[40,34,0,0,0,0,0,0,7,7,0,0,0,0,0,0,1,7,1,7,0,0,0,0,34,34,34,34,0,0,0,0,0,0,0,0,34,7,2,0,0,0,0,0,34,1,14,2,0,0,0,0,0,0,40,34,0,0,0,0,0,0,7,7],[7,7,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,0,7,7,0,0,0,0,0,0,40,34,0,0,0,0,0,0,0,0,7,40,0,0,0,0,0,0,7,34,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,40] >;
C42⋊22D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{22}D_{10} % in TeX
G:=Group("C4^2:22D10"); // GroupNames label
G:=SmallGroup(320,1355);
// by ID
G=gap.SmallGroup(320,1355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,192,12550]);
// 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^-1,d*a*d=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations