metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊5D10, C10.302+ 1+4, C22≀C2⋊7D5, C22⋊C4⋊8D10, C23⋊D10⋊7C2, (C2×D4).87D10, C24⋊2D5⋊9C2, (C2×C20).32C23, (C23×D5)⋊8C22, C20.17D4⋊13C2, (C2×C10).138C24, (C23×C10)⋊11C22, C5⋊1(C24⋊C22), (C4×Dic5)⋊18C22, C23.D5⋊18C22, C2.32(D4⋊6D10), D10⋊C4⋊15C22, Dic5.5D4⋊15C2, (C2×Dic10)⋊23C22, (D4×C10).112C22, (C2×Dic5).63C23, (C22×D5).57C23, C22.159(C23×D5), C23.110(C22×D5), (C22×C10).183C23, (C5×C22≀C2)⋊9C2, (C5×C22⋊C4)⋊8C22, (C2×C4).32(C22×D5), (C2×C5⋊D4).22C22, SmallGroup(320,1266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊5D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=ac=ca, ad=da, faf=acd, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 1094 in 260 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×26], C5, C2×C4 [×3], C2×C4 [×6], D4 [×9], Q8 [×3], C23, C23 [×3], C23 [×8], D5 [×2], C10 [×3], C10 [×4], C42 [×3], C22⋊C4 [×3], C22⋊C4 [×15], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×3], C24, C24, Dic5 [×6], C20 [×3], D10 [×10], C2×C10, C2×C10 [×16], C22≀C2, C22≀C2 [×5], C4.4D4 [×9], Dic10 [×3], C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C22×D5 [×2], C22×D5 [×3], C22×C10, C22×C10 [×3], C22×C10 [×3], C24⋊C22, C4×Dic5 [×3], D10⋊C4 [×6], C23.D5 [×9], C5×C22⋊C4 [×3], C2×Dic10 [×3], C2×C5⋊D4 [×6], D4×C10 [×3], C23×D5, C23×C10, Dic5.5D4 [×6], C20.17D4 [×3], C23⋊D10 [×3], C24⋊2D5 [×2], C5×C22≀C2, C24⋊5D10
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 [×3], C24⋊5D10
►On 80 points
Generators in S80
(1 58)(2 54)(3 60)(4 56)(5 52)(6 69)(7 65)(8 61)(9 67)(10 63)(11 51)(12 57)(13 53)(14 59)(15 55)(16 68)(17 64)(18 70)(19 66)(20 62)(21 45)(22 74)(23 47)(24 76)(25 49)(26 78)(27 41)(28 80)(29 43)(30 72)(31 71)(32 44)(33 73)(34 46)(35 75)(36 48)(37 77)(38 50)(39 79)(40 42)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 30)(7 26)(8 22)(9 28)(10 24)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 54)(42 62)(43 56)(44 64)(45 58)(46 66)(47 60)(48 68)(49 52)(50 70)(51 71)(53 73)(55 75)(57 77)(59 79)(61 74)(63 76)(65 78)(67 80)(69 72)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 17)(7 18)(8 19)(9 20)(10 16)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 79)(42 80)(43 71)(44 72)(45 73)(46 74)(47 75)(48 76)(49 77)(50 78)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 74)(42 75)(43 76)(44 77)(45 78)(46 79)(47 80)(48 71)(49 72)(50 73)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(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 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 43)(44 50)(45 49)(46 48)(51 54)(52 53)(55 60)(56 59)(57 58)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,58)(2,54)(3,60)(4,56)(5,52)(6,69)(7,65)(8,61)(9,67)(10,63)(11,51)(12,57)(13,53)(14,59)(15,55)(16,68)(17,64)(18,70)(19,66)(20,62)(21,45)(22,74)(23,47)(24,76)(25,49)(26,78)(27,41)(28,80)(29,43)(30,72)(31,71)(32,44)(33,73)(34,46)(35,75)(36,48)(37,77)(38,50)(39,79)(40,42), (1,21)(2,27)(3,23)(4,29)(5,25)(6,30)(7,26)(8,22)(9,28)(10,24)(11,31)(12,37)(13,33)(14,39)(15,35)(16,36)(17,32)(18,38)(19,34)(20,40)(41,54)(42,62)(43,56)(44,64)(45,58)(46,66)(47,60)(48,68)(49,52)(50,70)(51,71)(53,73)(55,75)(57,77)(59,79)(61,74)(63,76)(65,78)(67,80)(69,72), (1,13)(2,14)(3,15)(4,11)(5,12)(6,17)(7,18)(8,19)(9,20)(10,16)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70), (1,7)(2,8)(3,9)(4,10)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,71)(49,72)(50,73)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,19)(12,18)(13,17)(14,16)(15,20)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,43)(44,50)(45,49)(46,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,58)(2,54)(3,60)(4,56)(5,52)(6,69)(7,65)(8,61)(9,67)(10,63)(11,51)(12,57)(13,53)(14,59)(15,55)(16,68)(17,64)(18,70)(19,66)(20,62)(21,45)(22,74)(23,47)(24,76)(25,49)(26,78)(27,41)(28,80)(29,43)(30,72)(31,71)(32,44)(33,73)(34,46)(35,75)(36,48)(37,77)(38,50)(39,79)(40,42), (1,21)(2,27)(3,23)(4,29)(5,25)(6,30)(7,26)(8,22)(9,28)(10,24)(11,31)(12,37)(13,33)(14,39)(15,35)(16,36)(17,32)(18,38)(19,34)(20,40)(41,54)(42,62)(43,56)(44,64)(45,58)(46,66)(47,60)(48,68)(49,52)(50,70)(51,71)(53,73)(55,75)(57,77)(59,79)(61,74)(63,76)(65,78)(67,80)(69,72), (1,13)(2,14)(3,15)(4,11)(5,12)(6,17)(7,18)(8,19)(9,20)(10,16)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,79)(42,80)(43,71)(44,72)(45,73)(46,74)(47,75)(48,76)(49,77)(50,78)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70), (1,7)(2,8)(3,9)(4,10)(5,6)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,74)(42,75)(43,76)(44,77)(45,78)(46,79)(47,80)(48,71)(49,72)(50,73)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,19)(12,18)(13,17)(14,16)(15,20)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,43)(44,50)(45,49)(46,48)(51,54)(52,53)(55,60)(56,59)(57,58)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,58),(2,54),(3,60),(4,56),(5,52),(6,69),(7,65),(8,61),(9,67),(10,63),(11,51),(12,57),(13,53),(14,59),(15,55),(16,68),(17,64),(18,70),(19,66),(20,62),(21,45),(22,74),(23,47),(24,76),(25,49),(26,78),(27,41),(28,80),(29,43),(30,72),(31,71),(32,44),(33,73),(34,46),(35,75),(36,48),(37,77),(38,50),(39,79),(40,42)], [(1,21),(2,27),(3,23),(4,29),(5,25),(6,30),(7,26),(8,22),(9,28),(10,24),(11,31),(12,37),(13,33),(14,39),(15,35),(16,36),(17,32),(18,38),(19,34),(20,40),(41,54),(42,62),(43,56),(44,64),(45,58),(46,66),(47,60),(48,68),(49,52),(50,70),(51,71),(53,73),(55,75),(57,77),(59,79),(61,74),(63,76),(65,78),(67,80),(69,72)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,17),(7,18),(8,19),(9,20),(10,16),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,31),(30,32),(41,79),(42,80),(43,71),(44,72),(45,73),(46,74),(47,75),(48,76),(49,77),(50,78),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,7),(2,8),(3,9),(4,10),(5,6),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,74),(42,75),(43,76),(44,77),(45,78),(46,79),(47,80),(48,71),(49,72),(50,73),(51,68),(52,69),(53,70),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,19),(12,18),(13,17),(14,16),(15,20),(21,32),(22,31),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(41,43),(44,50),(45,49),(46,48),(51,54),(52,53),(55,60),(56,59),(57,58),(61,68),(62,67),(63,66),(64,65),(69,70),(71,79),(72,78),(73,77),(74,76)])
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | D10 | 2+ 1+4 | D4⋊6D10 |
| kernel | C24⋊5D10 | Dic5.5D4 | C20.17D4 | C23⋊D10 | C24⋊2D5 | C5×C22≀C2 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C10 | C2 |
| # reps | 1 | 6 | 3 | 3 | 2 | 1 | 2 | 6 | 6 | 2 | 3 | 12 |
Matrix representation of C24⋊5D10 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 1 | 0 | 0 | 0 | 0 | 0 |
| 13 | 28 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 28 | 40 | 0 |
| 0 | 0 | 0 | 0 | 31 | 38 | 0 | 40 |
| 17 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 24 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 17 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 38 | 23 | 36 |
| 0 | 0 | 0 | 0 | 3 | 0 | 40 | 18 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 34 | 20 | 0 | 0 | 0 | 0 | 0 |
| 6 | 34 | 38 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 7 | 39 | 27 |
| 0 | 0 | 0 | 0 | 34 | 7 | 2 | 37 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 35 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 35 |
| 7 | 34 | 0 | 20 | 0 | 0 | 0 | 0 |
| 1 | 34 | 23 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 7 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 35 | 6 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6 |
G:=sub<GL(8,GF(41))| [40,0,0,13,0,0,0,0,0,40,28,28,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,28,31,0,0,0,0,0,1,28,38,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[17,35,0,0,0,0,0,0,7,24,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,24,1,3,3,0,0,0,0,40,17,38,0,0,0,0,0,0,0,23,40,0,0,0,0,0,0,36,18],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[0,6,0,0,0,0,0,0,34,34,0,0,0,0,0,0,20,38,1,35,0,0,0,0,0,23,6,6,0,0,0,0,0,0,0,0,40,34,0,0,0,0,0,0,7,7,0,0,0,0,0,0,39,2,0,7,0,0,0,0,27,37,35,35],[7,1,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,23,6,6,0,0,0,0,20,38,1,35,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,35,1,0,0,0,0,0,0,6,6] >;
C24⋊5D10 in GAP, Magma, Sage, TeX
C_2^4\rtimes_5D_{10} % in TeX
G:=Group("C2^4:5D10"); // GroupNames label
G:=SmallGroup(320,1266);
// by ID
G=gap.SmallGroup(320,1266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=f^2=1,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations