metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.34D10, C10.282+ (1+4), (C2×D4)⋊6D10, C22≀C2⋊5D5, C22⋊C4⋊7D10, (C2×Dic5)⋊8D4, C23⋊D10⋊6C2, C20⋊D4⋊12C2, (D4×C10)⋊9C22, C24⋊2D5⋊7C2, D10⋊D4⋊14C2, C22⋊D20⋊10C2, C22.41(D4×D5), Dic5⋊D4⋊4C2, (C2×D20)⋊20C22, (C2×C20).30C23, Dic5.16(C2×D4), C10.58(C22×D4), (C2×C10).136C24, C5⋊2(C22.29C24), (C4×Dic5)⋊16C22, C2.30(D4⋊6D10), C23.D5⋊16C22, D10⋊C4⋊13C22, Dic5.5D4⋊13C2, (C2×Dic10)⋊21C22, C10.D4⋊11C22, C23.11D10⋊3C2, (C22×C10).10C23, (C23×C10).69C22, (C2×Dic5).61C23, (C23×D5).44C22, (C22×D5).55C23, C22.157(C23×D5), C23.109(C22×D5), (C22×Dic5)⋊15C22, C2.31(C2×D4×D5), (C2×C4×D5)⋊9C22, (C5×C22≀C2)⋊7C2, (C2×D4⋊2D5)⋊7C2, (C2×C10).55(C2×D4), (C2×C5⋊D4)⋊9C22, (C22×C5⋊D4)⋊10C2, (C5×C22⋊C4)⋊7C22, (C2×C4).30(C22×D5), SmallGroup(320,1264)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1406 in 334 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×10], C22, C22 [×2], C22 [×28], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×22], Q8 [×2], C23 [×2], C23 [×2], C23 [×11], D5 [×3], C10, C10 [×2], C10 [×5], C42 [×2], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×16], C2×Q8, C4○D4 [×4], C24, C24, Dic5 [×4], Dic5 [×3], C20 [×3], D10 [×13], C2×C10, C2×C10 [×2], C2×C10 [×15], C42⋊C2, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C4.4D4 [×2], C4⋊1D4 [×2], C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×2], D20 [×2], C2×Dic5 [×3], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×16], C2×C20, C2×C20 [×2], C5×D4 [×4], C22×D5, C22×D5 [×2], C22×D5 [×4], C22×C10 [×2], C22×C10 [×2], C22×C10 [×4], C22.29C24, C4×Dic5 [×2], C10.D4 [×2], D10⋊C4 [×4], C23.D5, C23.D5 [×2], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×D20 [×2], D4⋊2D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×2], C2×C5⋊D4 [×8], C2×C5⋊D4 [×4], D4×C10, D4×C10 [×2], C23×D5, C23×C10, C23.11D10, C22⋊D20, D10⋊D4 [×2], Dic5.5D4 [×2], C23⋊D10, Dic5⋊D4 [×2], C20⋊D4 [×2], C24⋊2D5, C5×C22≀C2, C2×D4⋊2D5, C22×C5⋊D4, C24.34D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊6D10 [×2], C24.34D10
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de-1 >
►On 80 points
(1 63)(2 31)(3 65)(4 33)(5 67)(6 35)(7 69)(8 37)(9 61)(10 39)(11 76)(12 50)(13 78)(14 42)(15 80)(16 44)(17 72)(18 46)(19 74)(20 48)(21 43)(22 71)(23 45)(24 73)(25 47)(26 75)(27 49)(28 77)(29 41)(30 79)(32 59)(34 51)(36 53)(38 55)(40 57)(52 68)(54 70)(56 62)(58 64)(60 66)
(1 24)(2 74)(3 26)(4 76)(5 28)(6 78)(7 30)(8 80)(9 22)(10 72)(11 33)(12 51)(13 35)(14 53)(15 37)(16 55)(17 39)(18 57)(19 31)(20 59)(21 70)(23 62)(25 64)(27 66)(29 68)(32 48)(34 50)(36 42)(38 44)(40 46)(41 52)(43 54)(45 56)(47 58)(49 60)(61 71)(63 73)(65 75)(67 77)(69 79)
(1 57)(2 58)(3 59)(4 60)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 27)(12 28)(13 29)(14 30)(15 21)(16 22)(17 23)(18 24)(19 25)(20 26)(31 64)(32 65)(33 66)(34 67)(35 68)(36 69)(37 70)(38 61)(39 62)(40 63)(41 78)(42 79)(43 80)(44 71)(45 72)(46 73)(47 74)(48 75)(49 76)(50 77)
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 61)(10 62)(11 49)(12 50)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 80)(22 71)(23 72)(24 73)(25 74)(26 75)(27 76)(28 77)(29 78)(30 79)(31 58)(32 59)(33 60)(34 51)(35 52)(36 53)(37 54)(38 55)(39 56)(40 57)
(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 62 63 10)(2 9 64 61)(3 70 65 8)(4 7 66 69)(5 68 67 6)(11 30 49 79)(12 78 50 29)(13 28 41 77)(14 76 42 27)(15 26 43 75)(16 74 44 25)(17 24 45 73)(18 72 46 23)(19 22 47 71)(20 80 48 21)(31 38 58 55)(32 54 59 37)(33 36 60 53)(34 52 51 35)(39 40 56 57)
G:=sub<Sym(80)| (1,63)(2,31)(3,65)(4,33)(5,67)(6,35)(7,69)(8,37)(9,61)(10,39)(11,76)(12,50)(13,78)(14,42)(15,80)(16,44)(17,72)(18,46)(19,74)(20,48)(21,43)(22,71)(23,45)(24,73)(25,47)(26,75)(27,49)(28,77)(29,41)(30,79)(32,59)(34,51)(36,53)(38,55)(40,57)(52,68)(54,70)(56,62)(58,64)(60,66), (1,24)(2,74)(3,26)(4,76)(5,28)(6,78)(7,30)(8,80)(9,22)(10,72)(11,33)(12,51)(13,35)(14,53)(15,37)(16,55)(17,39)(18,57)(19,31)(20,59)(21,70)(23,62)(25,64)(27,66)(29,68)(32,48)(34,50)(36,42)(38,44)(40,46)(41,52)(43,54)(45,56)(47,58)(49,60)(61,71)(63,73)(65,75)(67,77)(69,79), (1,57)(2,58)(3,59)(4,60)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,27)(12,28)(13,29)(14,30)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,61)(39,62)(40,63)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,80)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57), (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), (1,62,63,10)(2,9,64,61)(3,70,65,8)(4,7,66,69)(5,68,67,6)(11,30,49,79)(12,78,50,29)(13,28,41,77)(14,76,42,27)(15,26,43,75)(16,74,44,25)(17,24,45,73)(18,72,46,23)(19,22,47,71)(20,80,48,21)(31,38,58,55)(32,54,59,37)(33,36,60,53)(34,52,51,35)(39,40,56,57)>;
G:=Group( (1,63)(2,31)(3,65)(4,33)(5,67)(6,35)(7,69)(8,37)(9,61)(10,39)(11,76)(12,50)(13,78)(14,42)(15,80)(16,44)(17,72)(18,46)(19,74)(20,48)(21,43)(22,71)(23,45)(24,73)(25,47)(26,75)(27,49)(28,77)(29,41)(30,79)(32,59)(34,51)(36,53)(38,55)(40,57)(52,68)(54,70)(56,62)(58,64)(60,66), (1,24)(2,74)(3,26)(4,76)(5,28)(6,78)(7,30)(8,80)(9,22)(10,72)(11,33)(12,51)(13,35)(14,53)(15,37)(16,55)(17,39)(18,57)(19,31)(20,59)(21,70)(23,62)(25,64)(27,66)(29,68)(32,48)(34,50)(36,42)(38,44)(40,46)(41,52)(43,54)(45,56)(47,58)(49,60)(61,71)(63,73)(65,75)(67,77)(69,79), (1,57)(2,58)(3,59)(4,60)(5,51)(6,52)(7,53)(8,54)(9,55)(10,56)(11,27)(12,28)(13,29)(14,30)(15,21)(16,22)(17,23)(18,24)(19,25)(20,26)(31,64)(32,65)(33,66)(34,67)(35,68)(36,69)(37,70)(38,61)(39,62)(40,63)(41,78)(42,79)(43,80)(44,71)(45,72)(46,73)(47,74)(48,75)(49,76)(50,77), (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,61)(10,62)(11,49)(12,50)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,80)(22,71)(23,72)(24,73)(25,74)(26,75)(27,76)(28,77)(29,78)(30,79)(31,58)(32,59)(33,60)(34,51)(35,52)(36,53)(37,54)(38,55)(39,56)(40,57), (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), (1,62,63,10)(2,9,64,61)(3,70,65,8)(4,7,66,69)(5,68,67,6)(11,30,49,79)(12,78,50,29)(13,28,41,77)(14,76,42,27)(15,26,43,75)(16,74,44,25)(17,24,45,73)(18,72,46,23)(19,22,47,71)(20,80,48,21)(31,38,58,55)(32,54,59,37)(33,36,60,53)(34,52,51,35)(39,40,56,57) );
G=PermutationGroup([(1,63),(2,31),(3,65),(4,33),(5,67),(6,35),(7,69),(8,37),(9,61),(10,39),(11,76),(12,50),(13,78),(14,42),(15,80),(16,44),(17,72),(18,46),(19,74),(20,48),(21,43),(22,71),(23,45),(24,73),(25,47),(26,75),(27,49),(28,77),(29,41),(30,79),(32,59),(34,51),(36,53),(38,55),(40,57),(52,68),(54,70),(56,62),(58,64),(60,66)], [(1,24),(2,74),(3,26),(4,76),(5,28),(6,78),(7,30),(8,80),(9,22),(10,72),(11,33),(12,51),(13,35),(14,53),(15,37),(16,55),(17,39),(18,57),(19,31),(20,59),(21,70),(23,62),(25,64),(27,66),(29,68),(32,48),(34,50),(36,42),(38,44),(40,46),(41,52),(43,54),(45,56),(47,58),(49,60),(61,71),(63,73),(65,75),(67,77),(69,79)], [(1,57),(2,58),(3,59),(4,60),(5,51),(6,52),(7,53),(8,54),(9,55),(10,56),(11,27),(12,28),(13,29),(14,30),(15,21),(16,22),(17,23),(18,24),(19,25),(20,26),(31,64),(32,65),(33,66),(34,67),(35,68),(36,69),(37,70),(38,61),(39,62),(40,63),(41,78),(42,79),(43,80),(44,71),(45,72),(46,73),(47,74),(48,75),(49,76),(50,77)], [(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,61),(10,62),(11,49),(12,50),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,80),(22,71),(23,72),(24,73),(25,74),(26,75),(27,76),(28,77),(29,78),(30,79),(31,58),(32,59),(33,60),(34,51),(35,52),(36,53),(37,54),(38,55),(39,56),(40,57)], [(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)], [(1,62,63,10),(2,9,64,61),(3,70,65,8),(4,7,66,69),(5,68,67,6),(11,30,49,79),(12,78,50,29),(13,28,41,77),(14,76,42,27),(15,26,43,75),(16,74,44,25),(17,24,45,73),(18,72,46,23),(19,22,47,71),(20,80,48,21),(31,38,58,55),(32,54,59,37),(33,36,60,53),(34,52,51,35),(39,40,56,57)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 1 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 1 |
| 0 | 0 | 0 | 0 | 40 | 17 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 34 | 40 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 34 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 7 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,17,1,0,0,0,0,40,24,0,0,0,0,0,0,24,40,0,0,0,0,1,17],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,7,34,0,0,0,0,7,40,0,0],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,40,0,0,0,0,0,34,1,0,0] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | 4 | 4 | 4 | 20 | 20 | 20 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 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 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | 2+ (1+4) | D4×D5 | D4⋊6D10 |
| kernel | C24.34D10 | C23.11D10 | C22⋊D20 | D10⋊D4 | Dic5.5D4 | C23⋊D10 | Dic5⋊D4 | C20⋊D4 | C24⋊2D5 | C5×C22≀C2 | C2×D4⋊2D5 | C22×C5⋊D4 | C2×Dic5 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C10 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 6 | 6 | 2 | 2 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2^4._{34}D_{10} % in TeX
G:=Group("C2^4.34D10"); // GroupNames label
G:=SmallGroup(320,1264);
// by ID
G=gap.SmallGroup(320,1264);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,675,297,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^10=1,f^2=d,a*b=b*a,e*a*e^-1=f*a*f^-1=a*c=c*a,a*d=d*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations