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
Subgroups: 1286 in 260 conjugacy classes, 91 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×9], C22, C22 [×26], C5, C2×C4, C2×C4 [×4], C2×C4 [×4], D4 [×9], Q8 [×3], C23 [×2], C23 [×10], D5 [×4], C10, C10 [×2], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×14], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×2], C24 [×2], Dic5 [×4], C20 [×5], D10 [×20], C2×C10, C2×C10 [×6], C22≀C2 [×6], C4.4D4, C4.4D4 [×8], Dic10 [×2], D20 [×4], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C5×D4, C5×Q8, C22×D5 [×4], C22×D5 [×6], C22×C10 [×2], C24⋊C22, C4×Dic5 [×2], D10⋊C4 [×12], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×D20 [×4], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C23×D5 [×2], C4.D20 [×2], C22⋊D20 [×4], Dic5.5D4 [×4], C23⋊D10 [×2], C20.23D4 [×2], C5×C4.4D4, C42⋊22D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×3], C22×D5 [×7], C24⋊C22, C23×D5, D4⋊6D10, D4⋊8D10 [×2], C42⋊22D10
Generators and relations
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 >
►On 80 points
(1 46 13 41)(2 42 14 47)(3 48 15 43)(4 44 11 49)(5 50 12 45)(6 68 17 63)(7 64 18 69)(8 70 19 65)(9 66 20 61)(10 62 16 67)(21 51 33 78)(22 79 34 52)(23 53 35 80)(24 71 36 54)(25 55 37 72)(26 73 38 56)(27 57 39 74)(28 75 40 58)(29 59 31 76)(30 77 32 60)
(1 38 19 21)(2 34 20 27)(3 40 16 23)(4 36 17 29)(5 32 18 25)(6 31 11 24)(7 37 12 30)(8 33 13 26)(9 39 14 22)(10 35 15 28)(41 73 70 78)(42 52 61 57)(43 75 62 80)(44 54 63 59)(45 77 64 72)(46 56 65 51)(47 79 66 74)(48 58 67 53)(49 71 68 76)(50 60 69 55)
(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)(17 20)(18 19)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 40)(29 39)(30 38)(41 64)(42 63)(43 62)(44 61)(45 70)(46 69)(47 68)(48 67)(49 66)(50 65)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)(57 71)(58 80)(59 79)(60 78)
G:=sub<Sym(80)| (1,46,13,41)(2,42,14,47)(3,48,15,43)(4,44,11,49)(5,50,12,45)(6,68,17,63)(7,64,18,69)(8,70,19,65)(9,66,20,61)(10,62,16,67)(21,51,33,78)(22,79,34,52)(23,53,35,80)(24,71,36,54)(25,55,37,72)(26,73,38,56)(27,57,39,74)(28,75,40,58)(29,59,31,76)(30,77,32,60), (1,38,19,21)(2,34,20,27)(3,40,16,23)(4,36,17,29)(5,32,18,25)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,73,70,78)(42,52,61,57)(43,75,62,80)(44,54,63,59)(45,77,64,72)(46,56,65,51)(47,79,66,74)(48,58,67,53)(49,71,68,76)(50,60,69,55), (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)(17,20)(18,19)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,64)(42,63)(43,62)(44,61)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,80)(59,79)(60,78)>;
G:=Group( (1,46,13,41)(2,42,14,47)(3,48,15,43)(4,44,11,49)(5,50,12,45)(6,68,17,63)(7,64,18,69)(8,70,19,65)(9,66,20,61)(10,62,16,67)(21,51,33,78)(22,79,34,52)(23,53,35,80)(24,71,36,54)(25,55,37,72)(26,73,38,56)(27,57,39,74)(28,75,40,58)(29,59,31,76)(30,77,32,60), (1,38,19,21)(2,34,20,27)(3,40,16,23)(4,36,17,29)(5,32,18,25)(6,31,11,24)(7,37,12,30)(8,33,13,26)(9,39,14,22)(10,35,15,28)(41,73,70,78)(42,52,61,57)(43,75,62,80)(44,54,63,59)(45,77,64,72)(46,56,65,51)(47,79,66,74)(48,58,67,53)(49,71,68,76)(50,60,69,55), (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)(17,20)(18,19)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,40)(29,39)(30,38)(41,64)(42,63)(43,62)(44,61)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,77)(52,76)(53,75)(54,74)(55,73)(56,72)(57,71)(58,80)(59,79)(60,78) );
G=PermutationGroup([(1,46,13,41),(2,42,14,47),(3,48,15,43),(4,44,11,49),(5,50,12,45),(6,68,17,63),(7,64,18,69),(8,70,19,65),(9,66,20,61),(10,62,16,67),(21,51,33,78),(22,79,34,52),(23,53,35,80),(24,71,36,54),(25,55,37,72),(26,73,38,56),(27,57,39,74),(28,75,40,58),(29,59,31,76),(30,77,32,60)], [(1,38,19,21),(2,34,20,27),(3,40,16,23),(4,36,17,29),(5,32,18,25),(6,31,11,24),(7,37,12,30),(8,33,13,26),(9,39,14,22),(10,35,15,28),(41,73,70,78),(42,52,61,57),(43,75,62,80),(44,54,63,59),(45,77,64,72),(46,56,65,51),(47,79,66,74),(48,58,67,53),(49,71,68,76),(50,60,69,55)], [(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),(17,20),(18,19),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,40),(29,39),(30,38),(41,64),(42,63),(43,62),(44,61),(45,70),(46,69),(47,68),(48,67),(49,66),(50,65),(51,77),(52,76),(53,75),(54,74),(55,73),(56,72),(57,71),(58,80),(59,79),(60,78)])
Matrix representation ►G ⊆ 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] >;
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 |
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