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, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C2×Q8, C24, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C22≀C2, C4.4D4, Dic10, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C24⋊C22, C4×Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C5⋊D4, D4×C10, C23×D5, C23×C10, Dic5.5D4, C20.17D4, C23⋊D10, C24⋊2D5, C5×C22≀C2, C24⋊5D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C24⋊C22, C23×D5, D4⋊6D10, C24⋊5D10
►On 80 points
Generators in S80
(1 47)(2 43)(3 49)(4 45)(5 41)(6 63)(7 69)(8 65)(9 61)(10 67)(11 44)(12 50)(13 46)(14 42)(15 48)(16 68)(17 64)(18 70)(19 66)(20 62)(21 58)(22 74)(23 60)(24 76)(25 52)(26 78)(27 54)(28 80)(29 56)(30 72)(31 71)(32 57)(33 73)(34 59)(35 75)(36 51)(37 77)(38 53)(39 79)(40 55)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 24)(7 30)(8 26)(9 22)(10 28)(11 35)(12 31)(13 37)(14 33)(15 39)(16 36)(17 32)(18 38)(19 34)(20 40)(41 52)(42 73)(43 54)(44 75)(45 56)(46 77)(47 58)(48 79)(49 60)(50 71)(51 68)(53 70)(55 62)(57 64)(59 66)(61 74)(63 76)(65 78)(67 80)(69 72)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(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 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 75)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 20)(12 16)(13 17)(14 18)(15 19)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 69)(42 70)(43 61)(44 62)(45 63)(46 64)(47 65)(48 66)(49 67)(50 68)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(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 7)(2 6)(3 10)(4 9)(5 8)(11 20)(12 19)(13 18)(14 17)(15 16)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(41 42)(43 50)(44 49)(45 48)(46 47)(51 59)(52 58)(53 57)(54 56)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,47)(2,43)(3,49)(4,45)(5,41)(6,63)(7,69)(8,65)(9,61)(10,67)(11,44)(12,50)(13,46)(14,42)(15,48)(16,68)(17,64)(18,70)(19,66)(20,62)(21,58)(22,74)(23,60)(24,76)(25,52)(26,78)(27,54)(28,80)(29,56)(30,72)(31,71)(32,57)(33,73)(34,59)(35,75)(36,51)(37,77)(38,53)(39,79)(40,55), (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,35)(12,31)(13,37)(14,33)(15,39)(16,36)(17,32)(18,38)(19,34)(20,40)(41,52)(42,73)(43,54)(44,75)(45,56)(46,77)(47,58)(48,79)(49,60)(50,71)(51,68)(53,70)(55,62)(57,64)(59,66)(61,74)(63,76)(65,78)(67,80)(69,72), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(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)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70), (1,8)(2,9)(3,10)(4,6)(5,7)(11,20)(12,16)(13,17)(14,18)(15,19)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,20)(12,19)(13,18)(14,17)(15,16)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,42)(43,50)(44,49)(45,48)(46,47)(51,59)(52,58)(53,57)(54,56)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,47)(2,43)(3,49)(4,45)(5,41)(6,63)(7,69)(8,65)(9,61)(10,67)(11,44)(12,50)(13,46)(14,42)(15,48)(16,68)(17,64)(18,70)(19,66)(20,62)(21,58)(22,74)(23,60)(24,76)(25,52)(26,78)(27,54)(28,80)(29,56)(30,72)(31,71)(32,57)(33,73)(34,59)(35,75)(36,51)(37,77)(38,53)(39,79)(40,55), (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,35)(12,31)(13,37)(14,33)(15,39)(16,36)(17,32)(18,38)(19,34)(20,40)(41,52)(42,73)(43,54)(44,75)(45,56)(46,77)(47,58)(48,79)(49,60)(50,71)(51,68)(53,70)(55,62)(57,64)(59,66)(61,74)(63,76)(65,78)(67,80)(69,72), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(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)(51,76)(52,77)(53,78)(54,79)(55,80)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70), (1,8)(2,9)(3,10)(4,6)(5,7)(11,20)(12,16)(13,17)(14,18)(15,19)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,69)(42,70)(43,61)(44,62)(45,63)(46,64)(47,65)(48,66)(49,67)(50,68)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,20)(12,19)(13,18)(14,17)(15,16)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(41,42)(43,50)(44,49)(45,48)(46,47)(51,59)(52,58)(53,57)(54,56)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,47),(2,43),(3,49),(4,45),(5,41),(6,63),(7,69),(8,65),(9,61),(10,67),(11,44),(12,50),(13,46),(14,42),(15,48),(16,68),(17,64),(18,70),(19,66),(20,62),(21,58),(22,74),(23,60),(24,76),(25,52),(26,78),(27,54),(28,80),(29,56),(30,72),(31,71),(32,57),(33,73),(34,59),(35,75),(36,51),(37,77),(38,53),(39,79),(40,55)], [(1,21),(2,27),(3,23),(4,29),(5,25),(6,24),(7,30),(8,26),(9,22),(10,28),(11,35),(12,31),(13,37),(14,33),(15,39),(16,36),(17,32),(18,38),(19,34),(20,40),(41,52),(42,73),(43,54),(44,75),(45,56),(46,77),(47,58),(48,79),(49,60),(50,71),(51,68),(53,70),(55,62),(57,64),(59,66),(61,74),(63,76),(65,78),(67,80),(69,72)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(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),(51,76),(52,77),(53,78),(54,79),(55,80),(56,71),(57,72),(58,73),(59,74),(60,75),(61,66),(62,67),(63,68),(64,69),(65,70)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,20),(12,16),(13,17),(14,18),(15,19),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,69),(42,70),(43,61),(44,62),(45,63),(46,64),(47,65),(48,66),(49,67),(50,68),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,7),(2,6),(3,10),(4,9),(5,8),(11,20),(12,19),(13,18),(14,17),(15,16),(21,32),(22,31),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(41,42),(43,50),(44,49),(45,48),(46,47),(51,59),(52,58),(53,57),(54,56),(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