metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.48D10, C23.44D20, (C23×D5)⋊8C4, (C22×C4)⋊1D10, C5⋊2(C24⋊3C4), (D5×C24).1C2, C23.51(C4×D5), C10.37C22≀C2, D10⋊4(C22⋊C4), (C22×C20)⋊1C22, (C22×C10).67D4, C22.43(C2×D20), C22.100(D4×D5), C2.2(C23⋊D10), C2.4(C22⋊D20), (C22×D5).124D4, C23.52(C5⋊D4), C22⋊3(D10⋊C4), (C23×C10).38C22, (C22×Dic5)⋊2C22, (C23×D5).99C22, C23.282(C22×D5), (C22×C10).329C23, (C2×C22⋊C4)⋊2D5, (C10×C22⋊C4)⋊2C2, (C2×C23.D5)⋊2C2, C2.28(D5×C22⋊C4), (C2×D10⋊C4)⋊3C2, C22.126(C2×C4×D5), (C2×C10)⋊4(C22⋊C4), C2.9(C2×D10⋊C4), (C2×C10).321(C2×D4), C10.77(C2×C22⋊C4), C22.50(C2×C5⋊D4), (C22×C10).121(C2×C4), (C2×C10).209(C22×C4), (C22×D5).104(C2×C4), SmallGroup(320,582)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.48D10
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 2318 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, C23, D5, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C2×C22⋊C4, C25, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C22×C10, C22×C10, C24⋊3C4, D10⋊C4, C23.D5, C5×C22⋊C4, C22×Dic5, C22×C20, C23×D5, C23×D5, C23×C10, C2×D10⋊C4, C2×C23.D5, C10×C22⋊C4, D5×C24, C24.48D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C22≀C2, C4×D5, D20, C5⋊D4, C22×D5, C24⋊3C4, D10⋊C4, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D5×C22⋊C4, C22⋊D20, C2×D10⋊C4, C23⋊D10, C24.48D10
►On 80 points
(1 57)(2 78)(3 59)(4 80)(5 41)(6 62)(7 43)(8 64)(9 45)(10 66)(11 47)(12 68)(13 49)(14 70)(15 51)(16 72)(17 53)(18 74)(19 55)(20 76)(21 58)(22 79)(23 60)(24 61)(25 42)(26 63)(27 44)(28 65)(29 46)(30 67)(31 48)(32 69)(33 50)(34 71)(35 52)(36 73)(37 54)(38 75)(39 56)(40 77)
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(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 46 67 39)(2 38 68 45)(3 44 69 37)(4 36 70 43)(5 42 71 35)(6 34 72 41)(7 60 73 33)(8 32 74 59)(9 58 75 31)(10 30 76 57)(11 56 77 29)(12 28 78 55)(13 54 79 27)(14 26 80 53)(15 52 61 25)(16 24 62 51)(17 50 63 23)(18 22 64 49)(19 48 65 21)(20 40 66 47)
G:=sub<Sym(80)| (1,57)(2,78)(3,59)(4,80)(5,41)(6,62)(7,43)(8,64)(9,45)(10,66)(11,47)(12,68)(13,49)(14,70)(15,51)(16,72)(17,53)(18,74)(19,55)(20,76)(21,58)(22,79)(23,60)(24,61)(25,42)(26,63)(27,44)(28,65)(29,46)(30,67)(31,48)(32,69)(33,50)(34,71)(35,52)(36,73)(37,54)(38,75)(39,56)(40,77), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(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), (1,46,67,39)(2,38,68,45)(3,44,69,37)(4,36,70,43)(5,42,71,35)(6,34,72,41)(7,60,73,33)(8,32,74,59)(9,58,75,31)(10,30,76,57)(11,56,77,29)(12,28,78,55)(13,54,79,27)(14,26,80,53)(15,52,61,25)(16,24,62,51)(17,50,63,23)(18,22,64,49)(19,48,65,21)(20,40,66,47)>;
G:=Group( (1,57)(2,78)(3,59)(4,80)(5,41)(6,62)(7,43)(8,64)(9,45)(10,66)(11,47)(12,68)(13,49)(14,70)(15,51)(16,72)(17,53)(18,74)(19,55)(20,76)(21,58)(22,79)(23,60)(24,61)(25,42)(26,63)(27,44)(28,65)(29,46)(30,67)(31,48)(32,69)(33,50)(34,71)(35,52)(36,73)(37,54)(38,75)(39,56)(40,77), (1,67)(2,68)(3,69)(4,70)(5,71)(6,72)(7,73)(8,74)(9,75)(10,76)(11,77)(12,78)(13,79)(14,80)(15,61)(16,62)(17,63)(18,64)(19,65)(20,66)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (1,40)(2,21)(3,22)(4,23)(5,24)(6,25)(7,26)(8,27)(9,28)(10,29)(11,30)(12,31)(13,32)(14,33)(15,34)(16,35)(17,36)(18,37)(19,38)(20,39)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(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), (1,46,67,39)(2,38,68,45)(3,44,69,37)(4,36,70,43)(5,42,71,35)(6,34,72,41)(7,60,73,33)(8,32,74,59)(9,58,75,31)(10,30,76,57)(11,56,77,29)(12,28,78,55)(13,54,79,27)(14,26,80,53)(15,52,61,25)(16,24,62,51)(17,50,63,23)(18,22,64,49)(19,48,65,21)(20,40,66,47) );
G=PermutationGroup([[(1,57),(2,78),(3,59),(4,80),(5,41),(6,62),(7,43),(8,64),(9,45),(10,66),(11,47),(12,68),(13,49),(14,70),(15,51),(16,72),(17,53),(18,74),(19,55),(20,76),(21,58),(22,79),(23,60),(24,61),(25,42),(26,63),(27,44),(28,65),(29,46),(30,67),(31,48),(32,69),(33,50),(34,71),(35,52),(36,73),(37,54),(38,75),(39,56),(40,77)], [(1,67),(2,68),(3,69),(4,70),(5,71),(6,72),(7,73),(8,74),(9,75),(10,76),(11,77),(12,78),(13,79),(14,80),(15,61),(16,62),(17,63),(18,64),(19,65),(20,66),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(1,40),(2,21),(3,22),(4,23),(5,24),(6,25),(7,26),(8,27),(9,28),(10,29),(11,30),(12,31),(13,32),(14,33),(15,34),(16,35),(17,36),(18,37),(19,38),(20,39),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(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)], [(1,46,67,39),(2,38,68,45),(3,44,69,37),(4,36,70,43),(5,42,71,35),(6,34,72,41),(7,60,73,33),(8,32,74,59),(9,58,75,31),(10,30,76,57),(11,56,77,29),(12,28,78,55),(13,54,79,27),(14,26,80,53),(15,52,61,25),(16,24,62,51),(17,50,63,23),(18,22,64,49),(19,48,65,21),(20,40,66,47)]])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 |
| kernel | C24.48D10 | C2×D10⋊C4 | C2×C23.D5 | C10×C22⋊C4 | D5×C24 | C23×D5 | C22×D5 | C22×C10 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C24.48D10 ►in GL6(𝔽41)
| 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 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 21 | 24 | 0 | 0 | 0 | 0 |
| 38 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 37 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 36 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,38,0,0,0,0,24,18,0,0,0,0,0,0,1,37,0,0,0,0,21,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,36,0,0,0,0,24,18,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
C24.48D10 in GAP, Magma, Sage, TeX
C_2^4._{48}D_{10} % in TeX
G:=Group("C2^4.48D10"); // GroupNames label
G:=SmallGroup(320,582);
// by ID
G=gap.SmallGroup(320,582);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations