metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊4D10, C10.262+ (1+4), C5⋊2(D42), C5⋊D4⋊4D4, D10⋊6(C2×D4), C22⋊3(D4×D5), (C2×D4)⋊18D10, C22≀C2⋊3D5, Dic5⋊3(C2×D4), C22⋊D20⋊9C2, C23⋊D10⋊4C2, C20⋊D4⋊11C2, C22⋊C4⋊24D10, (D4×C10)⋊7C22, D10⋊D4⋊13C2, Dic5⋊4D4⋊2C2, Dic5⋊D4⋊2C2, (C2×D20)⋊19C22, (C2×C20).28C23, C10.56(C22×D4), (C23×D5)⋊7C22, (C2×C10).134C24, (C23×C10)⋊10C22, (C4×Dic5)⋊14C22, C10.D4⋊9C22, C2.28(D4⋊6D10), C23.D5⋊15C22, D10⋊C4⋊11C22, (C22×D5).53C23, C23.108(C22×D5), C22.155(C23×D5), (C22×C10).181C23, (C2×Dic5).231C23, (C22×Dic5)⋊13C22, (C2×D4×D5)⋊7C2, C2.29(C2×D4×D5), (C2×C10)⋊6(C2×D4), (C2×C4×D5)⋊7C22, (C5×C22≀C2)⋊5C2, (C22×C5⋊D4)⋊8C2, (C2×C5⋊D4)⋊39C22, (C5×C22⋊C4)⋊5C22, (C2×C4).28(C22×D5), SmallGroup(320,1262)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1910 in 428 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×9], C22, C22 [×4], C22 [×40], C5, C2×C4, C2×C4 [×2], C2×C4 [×12], D4 [×34], C23 [×2], C23 [×2], C23 [×24], D5 [×6], C10, C10 [×2], C10 [×6], C42, C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×29], C24, C24 [×3], Dic5 [×4], Dic5 [×2], C20 [×3], D10 [×4], D10 [×22], C2×C10, C2×C10 [×4], C2×C10 [×14], C4×D4 [×2], C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C4⋊1D4, C22×D4 [×4], C4×D5 [×4], D20 [×5], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×8], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×5], C22×D5 [×4], C22×D5 [×15], C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], D42, C4×Dic5, C10.D4 [×2], D10⋊C4 [×4], C23.D5, C5×C22⋊C4, C5×C22⋊C4 [×2], C2×C4×D5 [×2], C2×D20, C2×D20 [×2], D4×D5 [×8], C22×Dic5 [×2], C2×C5⋊D4 [×10], C2×C5⋊D4 [×8], D4×C10, D4×C10 [×2], C23×D5, C23×D5 [×2], C23×C10, Dic5⋊4D4 [×2], C22⋊D20 [×2], D10⋊D4 [×2], C23⋊D10, Dic5⋊D4 [×2], C20⋊D4, C5×C22≀C2, C2×D4×D5 [×2], C22×C5⋊D4 [×2], C24⋊4D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], D5, C2×D4 [×12], C24, D10 [×7], C22×D4 [×2], 2+ (1+4), C22×D5 [×7], D42, D4×D5 [×4], C23×D5, C2×D4×D5 [×2], D4⋊6D10, C24⋊4D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >
►On 80 points
(1 47)(2 43)(3 49)(4 45)(5 41)(6 51)(7 57)(8 53)(9 59)(10 55)(11 50)(12 46)(13 42)(14 48)(15 44)(16 58)(17 54)(18 60)(19 56)(20 52)(21 71)(22 66)(23 73)(24 68)(25 75)(26 70)(27 77)(28 62)(29 79)(30 64)(31 63)(32 80)(33 65)(34 72)(35 67)(36 74)(37 69)(38 76)(39 61)(40 78)
(1 21)(2 27)(3 23)(4 29)(5 25)(6 24)(7 30)(8 26)(9 22)(10 28)(11 31)(12 37)(13 33)(14 39)(15 35)(16 38)(17 34)(18 40)(19 36)(20 32)(41 75)(42 65)(43 77)(44 67)(45 79)(46 69)(47 71)(48 61)(49 73)(50 63)(51 68)(52 80)(53 70)(54 72)(55 62)(56 74)(57 64)(58 76)(59 66)(60 78)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 19)(7 20)(8 16)(9 17)(10 18)(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 56)(52 57)(53 58)(54 59)(55 60)(61 77)(62 78)(63 79)(64 80)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)
(1 8)(2 9)(3 10)(4 6)(5 7)(11 19)(12 20)(13 16)(14 17)(15 18)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 57)(42 58)(43 59)(44 60)(45 51)(46 52)(47 53)(48 54)(49 55)(50 56)(61 72)(62 73)(63 74)(64 75)(65 76)(66 77)(67 78)(68 79)(69 80)(70 71)
(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 17)(12 16)(13 20)(14 19)(15 18)(21 32)(22 31)(23 40)(24 39)(25 38)(26 37)(27 36)(28 35)(29 34)(30 33)(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)| (1,47)(2,43)(3,49)(4,45)(5,41)(6,51)(7,57)(8,53)(9,59)(10,55)(11,50)(12,46)(13,42)(14,48)(15,44)(16,58)(17,54)(18,60)(19,56)(20,52)(21,71)(22,66)(23,73)(24,68)(25,75)(26,70)(27,77)(28,62)(29,79)(30,64)(31,63)(32,80)(33,65)(34,72)(35,67)(36,74)(37,69)(38,76)(39,61)(40,78), (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,31)(12,37)(13,33)(14,39)(15,35)(16,38)(17,34)(18,40)(19,36)(20,32)(41,75)(42,65)(43,77)(44,67)(45,79)(46,69)(47,71)(48,61)(49,73)(50,63)(51,68)(52,80)(53,70)(54,72)(55,62)(56,74)(57,64)(58,76)(59,66)(60,78), (1,13)(2,14)(3,15)(4,11)(5,12)(6,19)(7,20)(8,16)(9,17)(10,18)(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)(51,56)(52,57)(53,58)(54,59)(55,60)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76), (1,8)(2,9)(3,10)(4,6)(5,7)(11,19)(12,20)(13,16)(14,17)(15,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,57)(42,58)(43,59)(44,60)(45,51)(46,52)(47,53)(48,54)(49,55)(50,56)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,71), (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)(11,17)(12,16)(13,20)(14,19)(15,18)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(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( (1,47)(2,43)(3,49)(4,45)(5,41)(6,51)(7,57)(8,53)(9,59)(10,55)(11,50)(12,46)(13,42)(14,48)(15,44)(16,58)(17,54)(18,60)(19,56)(20,52)(21,71)(22,66)(23,73)(24,68)(25,75)(26,70)(27,77)(28,62)(29,79)(30,64)(31,63)(32,80)(33,65)(34,72)(35,67)(36,74)(37,69)(38,76)(39,61)(40,78), (1,21)(2,27)(3,23)(4,29)(5,25)(6,24)(7,30)(8,26)(9,22)(10,28)(11,31)(12,37)(13,33)(14,39)(15,35)(16,38)(17,34)(18,40)(19,36)(20,32)(41,75)(42,65)(43,77)(44,67)(45,79)(46,69)(47,71)(48,61)(49,73)(50,63)(51,68)(52,80)(53,70)(54,72)(55,62)(56,74)(57,64)(58,76)(59,66)(60,78), (1,13)(2,14)(3,15)(4,11)(5,12)(6,19)(7,20)(8,16)(9,17)(10,18)(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)(51,56)(52,57)(53,58)(54,59)(55,60)(61,77)(62,78)(63,79)(64,80)(65,71)(66,72)(67,73)(68,74)(69,75)(70,76), (1,8)(2,9)(3,10)(4,6)(5,7)(11,19)(12,20)(13,16)(14,17)(15,18)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,57)(42,58)(43,59)(44,60)(45,51)(46,52)(47,53)(48,54)(49,55)(50,56)(61,72)(62,73)(63,74)(64,75)(65,76)(66,77)(67,78)(68,79)(69,80)(70,71), (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)(11,17)(12,16)(13,20)(14,19)(15,18)(21,32)(22,31)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(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([(1,47),(2,43),(3,49),(4,45),(5,41),(6,51),(7,57),(8,53),(9,59),(10,55),(11,50),(12,46),(13,42),(14,48),(15,44),(16,58),(17,54),(18,60),(19,56),(20,52),(21,71),(22,66),(23,73),(24,68),(25,75),(26,70),(27,77),(28,62),(29,79),(30,64),(31,63),(32,80),(33,65),(34,72),(35,67),(36,74),(37,69),(38,76),(39,61),(40,78)], [(1,21),(2,27),(3,23),(4,29),(5,25),(6,24),(7,30),(8,26),(9,22),(10,28),(11,31),(12,37),(13,33),(14,39),(15,35),(16,38),(17,34),(18,40),(19,36),(20,32),(41,75),(42,65),(43,77),(44,67),(45,79),(46,69),(47,71),(48,61),(49,73),(50,63),(51,68),(52,80),(53,70),(54,72),(55,62),(56,74),(57,64),(58,76),(59,66),(60,78)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,19),(7,20),(8,16),(9,17),(10,18),(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),(51,56),(52,57),(53,58),(54,59),(55,60),(61,77),(62,78),(63,79),(64,80),(65,71),(66,72),(67,73),(68,74),(69,75),(70,76)], [(1,8),(2,9),(3,10),(4,6),(5,7),(11,19),(12,20),(13,16),(14,17),(15,18),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,57),(42,58),(43,59),(44,60),(45,51),(46,52),(47,53),(48,54),(49,55),(50,56),(61,72),(62,73),(63,74),(64,75),(65,76),(66,77),(67,78),(68,79),(69,80),(70,71)], [(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),(11,17),(12,16),(13,20),(14,19),(15,18),(21,32),(22,31),(23,40),(24,39),(25,38),(26,37),(27,36),(28,35),(29,34),(30,33),(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)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 23 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 18 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 18 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 32 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,23,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,18,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,18,0,0,0,0,0,40],[1,32,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,40,0,0,0,0,0,0,40] >;
53 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
53 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊6D10 |
| kernel | C24⋊4D10 | Dic5⋊4D4 | C22⋊D20 | D10⋊D4 | C23⋊D10 | Dic5⋊D4 | C20⋊D4 | C5×C22≀C2 | C2×D4×D5 | C22×C5⋊D4 | C5⋊D4 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C2 |
| # reps | 1 | 2 | 2 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 8 | 2 | 6 | 6 | 2 | 1 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2^4\rtimes_4D_{10} % in TeX
G:=Group("C2^4:4D10"); // GroupNames label
G:=SmallGroup(320,1262);
// by ID
G=gap.SmallGroup(320,1262);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,1571,297,12550]);
// 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=f*a*f=a*c=c*a,a*d=d*a,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