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, C23.110(C22×D5), C22.159(C23×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
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
Generators and relations
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 >
►On 80 points
Generators in S80
(1 41)(2 47)(3 43)(4 49)(5 45)(6 67)(7 63)(8 69)(9 65)(10 61)(11 44)(12 50)(13 46)(14 42)(15 48)(16 68)(17 64)(18 70)(19 66)(20 62)(21 56)(22 74)(23 58)(24 76)(25 60)(26 78)(27 52)(28 80)(29 54)(30 72)(31 71)(32 55)(33 73)(34 57)(35 75)(36 59)(37 77)(38 51)(39 79)(40 53)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 28)(7 24)(8 30)(9 26)(10 22)(11 31)(12 37)(13 33)(14 39)(15 35)(16 36)(17 32)(18 38)(19 34)(20 40)(41 56)(42 79)(43 58)(44 71)(45 60)(46 73)(47 52)(48 75)(49 54)(50 77)(51 70)(53 62)(55 64)(57 66)(59 68)(61 74)(63 76)(65 78)(67 80)(69 72)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)(21 33)(22 34)(23 35)(24 36)(25 37)(26 38)(27 39)(28 40)(29 31)(30 32)(41 46)(42 47)(43 48)(44 49)(45 50)(51 78)(52 79)(53 80)(54 71)(55 72)(56 73)(57 74)(58 75)(59 76)(60 77)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 9)(2 10)(3 6)(4 7)(5 8)(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 65)(42 66)(43 67)(44 68)(45 69)(46 70)(47 61)(48 62)(49 63)(50 64)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 71)(60 72)
(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 8)(2 7)(3 6)(4 10)(5 9)(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 50)(42 49)(43 48)(44 47)(45 46)(51 55)(52 54)(56 60)(57 59)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,41)(2,47)(3,43)(4,49)(5,45)(6,67)(7,63)(8,69)(9,65)(10,61)(11,44)(12,50)(13,46)(14,42)(15,48)(16,68)(17,64)(18,70)(19,66)(20,62)(21,56)(22,74)(23,58)(24,76)(25,60)(26,78)(27,52)(28,80)(29,54)(30,72)(31,71)(32,55)(33,73)(34,57)(35,75)(36,59)(37,77)(38,51)(39,79)(40,53), (1,21)(2,27)(3,23)(4,29)(5,25)(6,28)(7,24)(8,30)(9,26)(10,22)(11,31)(12,37)(13,33)(14,39)(15,35)(16,36)(17,32)(18,38)(19,34)(20,40)(41,56)(42,79)(43,58)(44,71)(45,60)(46,73)(47,52)(48,75)(49,54)(50,77)(51,70)(53,62)(55,64)(57,66)(59,68)(61,74)(63,76)(65,78)(67,80)(69,72), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,46)(42,47)(43,48)(44,49)(45,50)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,66)(62,67)(63,68)(64,69)(65,70), (1,9)(2,10)(3,6)(4,7)(5,8)(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)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,71)(60,72), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(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)(41,50)(42,49)(43,48)(44,47)(45,46)(51,55)(52,54)(56,60)(57,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,41)(2,47)(3,43)(4,49)(5,45)(6,67)(7,63)(8,69)(9,65)(10,61)(11,44)(12,50)(13,46)(14,42)(15,48)(16,68)(17,64)(18,70)(19,66)(20,62)(21,56)(22,74)(23,58)(24,76)(25,60)(26,78)(27,52)(28,80)(29,54)(30,72)(31,71)(32,55)(33,73)(34,57)(35,75)(36,59)(37,77)(38,51)(39,79)(40,53), (1,21)(2,27)(3,23)(4,29)(5,25)(6,28)(7,24)(8,30)(9,26)(10,22)(11,31)(12,37)(13,33)(14,39)(15,35)(16,36)(17,32)(18,38)(19,34)(20,40)(41,56)(42,79)(43,58)(44,71)(45,60)(46,73)(47,52)(48,75)(49,54)(50,77)(51,70)(53,62)(55,64)(57,66)(59,68)(61,74)(63,76)(65,78)(67,80)(69,72), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,33)(22,34)(23,35)(24,36)(25,37)(26,38)(27,39)(28,40)(29,31)(30,32)(41,46)(42,47)(43,48)(44,49)(45,50)(51,78)(52,79)(53,80)(54,71)(55,72)(56,73)(57,74)(58,75)(59,76)(60,77)(61,66)(62,67)(63,68)(64,69)(65,70), (1,9)(2,10)(3,6)(4,7)(5,8)(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)(41,65)(42,66)(43,67)(44,68)(45,69)(46,70)(47,61)(48,62)(49,63)(50,64)(51,73)(52,74)(53,75)(54,76)(55,77)(56,78)(57,79)(58,80)(59,71)(60,72), (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), (1,8)(2,7)(3,6)(4,10)(5,9)(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)(41,50)(42,49)(43,48)(44,47)(45,46)(51,55)(52,54)(56,60)(57,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,41),(2,47),(3,43),(4,49),(5,45),(6,67),(7,63),(8,69),(9,65),(10,61),(11,44),(12,50),(13,46),(14,42),(15,48),(16,68),(17,64),(18,70),(19,66),(20,62),(21,56),(22,74),(23,58),(24,76),(25,60),(26,78),(27,52),(28,80),(29,54),(30,72),(31,71),(32,55),(33,73),(34,57),(35,75),(36,59),(37,77),(38,51),(39,79),(40,53)], [(1,21),(2,27),(3,23),(4,29),(5,25),(6,28),(7,24),(8,30),(9,26),(10,22),(11,31),(12,37),(13,33),(14,39),(15,35),(16,36),(17,32),(18,38),(19,34),(20,40),(41,56),(42,79),(43,58),(44,71),(45,60),(46,73),(47,52),(48,75),(49,54),(50,77),(51,70),(53,62),(55,64),(57,66),(59,68),(61,74),(63,76),(65,78),(67,80),(69,72)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19),(21,33),(22,34),(23,35),(24,36),(25,37),(26,38),(27,39),(28,40),(29,31),(30,32),(41,46),(42,47),(43,48),(44,49),(45,50),(51,78),(52,79),(53,80),(54,71),(55,72),(56,73),(57,74),(58,75),(59,76),(60,77),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,9),(2,10),(3,6),(4,7),(5,8),(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),(41,65),(42,66),(43,67),(44,68),(45,69),(46,70),(47,61),(48,62),(49,63),(50,64),(51,73),(52,74),(53,75),(54,76),(55,77),(56,78),(57,79),(58,80),(59,71),(60,72)], [(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)], [(1,8),(2,7),(3,6),(4,10),(5,9),(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),(41,50),(42,49),(43,48),(44,47),(45,46),(51,55),(52,54),(56,60),(57,59),(61,68),(62,67),(63,66),(64,65),(69,70),(71,79),(72,78),(73,77),(74,76)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀