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, C22⋊D20⋊10C2, D10⋊D4⋊14C2, 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, C23.D5⋊16C22, C2.30(D4⋊6D10), D10⋊C4⋊13C22, Dic5.5D4⋊13C2, (C2×Dic10)⋊21C22, C23.11D10⋊3C2, C10.D4⋊11C22, (C22×C10).10C23, (C23×C10).69C22, (C2×Dic5).61C23, (C22×D5).55C23, (C23×D5).44C22, C23.109(C22×D5), C22.157(C23×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
Generators and relations for C24.34D10
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 >
Subgroups: 1406 in 334 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C22.29C24, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C2×Dic10, C2×C4×D5, C2×D20, D4⋊2D5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D4×C10, C23×D5, C23×C10, 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, C24.34D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C22.29C24, D4×D5, C23×D5, C2×D4×D5, D4⋊6D10, C24.34D10
►On 80 points
(1 63)(2 31)(3 65)(4 33)(5 67)(6 35)(7 69)(8 37)(9 61)(10 39)(11 46)(12 74)(13 48)(14 76)(15 50)(16 78)(17 42)(18 80)(19 44)(20 72)(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 57)(12 31)(13 59)(14 33)(15 51)(16 35)(17 53)(18 37)(19 55)(20 39)(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 24)(12 25)(13 26)(14 27)(15 28)(16 29)(17 30)(18 21)(19 22)(20 23)(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 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(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 72 46 23)(12 22 47 71)(13 80 48 21)(14 30 49 79)(15 78 50 29)(16 28 41 77)(17 76 42 27)(18 26 43 75)(19 74 44 25)(20 24 45 73)(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)(11,46)(12,74)(13,48)(14,76)(15,50)(16,78)(17,42)(18,80)(19,44)(20,72)(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)(11,57)(12,31)(13,59)(14,33)(15,51)(16,35)(17,53)(18,37)(19,55)(20,39)(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)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,21)(19,22)(20,23)(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)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(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)(11,72,46,23)(12,22,47,71)(13,80,48,21)(14,30,49,79)(15,78,50,29)(16,28,41,77)(17,76,42,27)(18,26,43,75)(19,74,44,25)(20,24,45,73)(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)(11,46)(12,74)(13,48)(14,76)(15,50)(16,78)(17,42)(18,80)(19,44)(20,72)(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)(11,57)(12,31)(13,59)(14,33)(15,51)(16,35)(17,53)(18,37)(19,55)(20,39)(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)(11,24)(12,25)(13,26)(14,27)(15,28)(16,29)(17,30)(18,21)(19,22)(20,23)(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)(11,46)(12,47)(13,48)(14,49)(15,50)(16,41)(17,42)(18,43)(19,44)(20,45)(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)(11,72,46,23)(12,22,47,71)(13,80,48,21)(14,30,49,79)(15,78,50,29)(16,28,41,77)(17,76,42,27)(18,26,43,75)(19,74,44,25)(20,24,45,73)(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),(11,46),(12,74),(13,48),(14,76),(15,50),(16,78),(17,42),(18,80),(19,44),(20,72),(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),(11,57),(12,31),(13,59),(14,33),(15,51),(16,35),(17,53),(18,37),(19,55),(20,39),(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),(11,24),(12,25),(13,26),(14,27),(15,28),(16,29),(17,30),(18,21),(19,22),(20,23),(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),(11,46),(12,47),(13,48),(14,49),(15,50),(16,41),(17,42),(18,43),(19,44),(20,45),(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),(11,72,46,23),(12,22,47,71),(13,80,48,21),(14,30,49,79),(15,78,50,29),(16,28,41,77),(17,76,42,27),(18,26,43,75),(19,74,44,25),(20,24,45,73),(31,38,58,55),(32,54,59,37),(33,36,60,53),(34,52,51,35),(39,40,56,57)]])
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 |
Matrix representation of C24.34D10 ►in 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] >;
C24.34D10 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