metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.56D10, C22≀C2⋊9D5, (C2×Dic5)⋊20D4, (D4×Dic5)⋊10C2, C24⋊2D5⋊4C2, C22.39(D4×D5), Dic5⋊D4⋊1C2, Dic5⋊4D4⋊1C2, (C2×D4).148D10, (C2×C20).25C23, C4⋊Dic5⋊23C22, C22⋊C4.44D10, Dic5.84(C2×D4), (C23×Dic5)⋊5C2, C10.53(C22×D4), D10⋊C4⋊8C22, C22⋊3(D4⋊2D5), (C2×C10).130C24, C5⋊3(C22.19C24), (C4×Dic5)⋊12C22, C10.D4⋊6C22, C22.D20⋊8C2, (C22×C10).7C23, C23.D5⋊11C22, (C2×Dic10)⋊18C22, (D4×C10).109C22, C23.18D10⋊2C2, C23.11D10⋊1C2, (C23×C10).66C22, (C22×D5).52C23, C22.151(C23×D5), C23.175(C22×D5), Dic5.14D4⋊11C2, (C2×Dic5).229C23, (C22×Dic5)⋊10C22, C2.26(C2×D4×D5), (C2×C4×D5)⋊4C22, (C5×C22≀C2)⋊2C2, (C2×D4⋊2D5)⋊5C2, (C2×C10)⋊9(C4○D4), C10.75(C2×C4○D4), (C2×C10).52(C2×D4), (C2×C5⋊D4)⋊6C22, C2.26(C2×D4⋊2D5), (C2×C4).25(C22×D5), (C5×C22⋊C4).1C22, SmallGroup(320,1258)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.56D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, ac=ca, eae-1=faf-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 1070 in 330 conjugacy classes, 111 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, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C22×C10, C22.19C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C23.D5, C5×C22⋊C4, C5×C22⋊C4, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C2×C5⋊D4, D4×C10, D4×C10, C23×C10, C23.11D10, Dic5.14D4, Dic5⋊4D4, C22.D20, D4×Dic5, C23.18D10, Dic5⋊D4, C24⋊2D5, C5×C22≀C2, C2×D4⋊2D5, C23×Dic5, C24.56D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.19C24, D4×D5, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, C24.56D10
►On 80 points
(1 17)(2 42)(3 19)(4 44)(5 11)(6 46)(7 13)(8 48)(9 15)(10 50)(12 38)(14 40)(16 32)(18 34)(20 36)(21 52)(22 79)(23 54)(24 71)(25 56)(26 73)(27 58)(28 75)(29 60)(30 77)(31 49)(33 41)(35 43)(37 45)(39 47)(51 68)(53 70)(55 62)(57 64)(59 66)(61 80)(63 72)(65 74)(67 76)(69 78)
(1 33)(2 42)(3 35)(4 44)(5 37)(6 46)(7 39)(8 48)(9 31)(10 50)(11 45)(12 38)(13 47)(14 40)(15 49)(16 32)(17 41)(18 34)(19 43)(20 36)(21 52)(22 70)(23 54)(24 62)(25 56)(26 64)(27 58)(28 66)(29 60)(30 68)(51 77)(53 79)(55 71)(57 73)(59 75)(61 80)(63 72)(65 74)(67 76)(69 78)
(1 17)(2 18)(3 19)(4 20)(5 11)(6 12)(7 13)(8 14)(9 15)(10 16)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 69)(22 70)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(51 77)(52 78)(53 79)(54 80)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)
(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 76 33 60)(2 75 34 59)(3 74 35 58)(4 73 36 57)(5 72 37 56)(6 71 38 55)(7 80 39 54)(8 79 40 53)(9 78 31 52)(10 77 32 51)(11 25 45 63)(12 24 46 62)(13 23 47 61)(14 22 48 70)(15 21 49 69)(16 30 50 68)(17 29 41 67)(18 28 42 66)(19 27 43 65)(20 26 44 64)
G:=sub<Sym(80)| (1,17)(2,42)(3,19)(4,44)(5,11)(6,46)(7,13)(8,48)(9,15)(10,50)(12,38)(14,40)(16,32)(18,34)(20,36)(21,52)(22,79)(23,54)(24,71)(25,56)(26,73)(27,58)(28,75)(29,60)(30,77)(31,49)(33,41)(35,43)(37,45)(39,47)(51,68)(53,70)(55,62)(57,64)(59,66)(61,80)(63,72)(65,74)(67,76)(69,78), (1,33)(2,42)(3,35)(4,44)(5,37)(6,46)(7,39)(8,48)(9,31)(10,50)(11,45)(12,38)(13,47)(14,40)(15,49)(16,32)(17,41)(18,34)(19,43)(20,36)(21,52)(22,70)(23,54)(24,62)(25,56)(26,64)(27,58)(28,66)(29,60)(30,68)(51,77)(53,79)(55,71)(57,73)(59,75)(61,80)(63,72)(65,74)(67,76)(69,78), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,69)(22,70)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (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), (1,76,33,60)(2,75,34,59)(3,74,35,58)(4,73,36,57)(5,72,37,56)(6,71,38,55)(7,80,39,54)(8,79,40,53)(9,78,31,52)(10,77,32,51)(11,25,45,63)(12,24,46,62)(13,23,47,61)(14,22,48,70)(15,21,49,69)(16,30,50,68)(17,29,41,67)(18,28,42,66)(19,27,43,65)(20,26,44,64)>;
G:=Group( (1,17)(2,42)(3,19)(4,44)(5,11)(6,46)(7,13)(8,48)(9,15)(10,50)(12,38)(14,40)(16,32)(18,34)(20,36)(21,52)(22,79)(23,54)(24,71)(25,56)(26,73)(27,58)(28,75)(29,60)(30,77)(31,49)(33,41)(35,43)(37,45)(39,47)(51,68)(53,70)(55,62)(57,64)(59,66)(61,80)(63,72)(65,74)(67,76)(69,78), (1,33)(2,42)(3,35)(4,44)(5,37)(6,46)(7,39)(8,48)(9,31)(10,50)(11,45)(12,38)(13,47)(14,40)(15,49)(16,32)(17,41)(18,34)(19,43)(20,36)(21,52)(22,70)(23,54)(24,62)(25,56)(26,64)(27,58)(28,66)(29,60)(30,68)(51,77)(53,79)(55,71)(57,73)(59,75)(61,80)(63,72)(65,74)(67,76)(69,78), (1,17)(2,18)(3,19)(4,20)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(21,78)(22,79)(23,80)(24,71)(25,72)(26,73)(27,74)(28,75)(29,76)(30,77)(31,49)(32,50)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48)(51,68)(52,69)(53,70)(54,61)(55,62)(56,63)(57,64)(58,65)(59,66)(60,67), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,31)(10,32)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(21,69)(22,70)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(51,77)(52,78)(53,79)(54,80)(55,71)(56,72)(57,73)(58,74)(59,75)(60,76), (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), (1,76,33,60)(2,75,34,59)(3,74,35,58)(4,73,36,57)(5,72,37,56)(6,71,38,55)(7,80,39,54)(8,79,40,53)(9,78,31,52)(10,77,32,51)(11,25,45,63)(12,24,46,62)(13,23,47,61)(14,22,48,70)(15,21,49,69)(16,30,50,68)(17,29,41,67)(18,28,42,66)(19,27,43,65)(20,26,44,64) );
G=PermutationGroup([[(1,17),(2,42),(3,19),(4,44),(5,11),(6,46),(7,13),(8,48),(9,15),(10,50),(12,38),(14,40),(16,32),(18,34),(20,36),(21,52),(22,79),(23,54),(24,71),(25,56),(26,73),(27,58),(28,75),(29,60),(30,77),(31,49),(33,41),(35,43),(37,45),(39,47),(51,68),(53,70),(55,62),(57,64),(59,66),(61,80),(63,72),(65,74),(67,76),(69,78)], [(1,33),(2,42),(3,35),(4,44),(5,37),(6,46),(7,39),(8,48),(9,31),(10,50),(11,45),(12,38),(13,47),(14,40),(15,49),(16,32),(17,41),(18,34),(19,43),(20,36),(21,52),(22,70),(23,54),(24,62),(25,56),(26,64),(27,58),(28,66),(29,60),(30,68),(51,77),(53,79),(55,71),(57,73),(59,75),(61,80),(63,72),(65,74),(67,76),(69,78)], [(1,17),(2,18),(3,19),(4,20),(5,11),(6,12),(7,13),(8,14),(9,15),(10,16),(21,78),(22,79),(23,80),(24,71),(25,72),(26,73),(27,74),(28,75),(29,76),(30,77),(31,49),(32,50),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48),(51,68),(52,69),(53,70),(54,61),(55,62),(56,63),(57,64),(58,65),(59,66),(60,67)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,31),(10,32),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,41),(18,42),(19,43),(20,44),(21,69),(22,70),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(51,77),(52,78),(53,79),(54,80),(55,71),(56,72),(57,73),(58,74),(59,75),(60,76)], [(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)], [(1,76,33,60),(2,75,34,59),(3,74,35,58),(4,73,36,57),(5,72,37,56),(6,71,38,55),(7,80,39,54),(8,79,40,53),(9,78,31,52),(10,77,32,51),(11,25,45,63),(12,24,46,62),(13,23,47,61),(14,22,48,70),(15,21,49,69),(16,30,50,68),(17,29,41,67),(18,28,42,66),(19,27,43,65),(20,26,44,64)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 10S | 10T | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 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 | 20 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | D4×D5 | D4⋊2D5 |
| kernel | C24.56D10 | C23.11D10 | Dic5.14D4 | Dic5⋊4D4 | C22.D20 | D4×Dic5 | C23.18D10 | Dic5⋊D4 | C24⋊2D5 | C5×C22≀C2 | C2×D4⋊2D5 | C23×Dic5 | C2×Dic5 | C22≀C2 | C2×C10 | C22⋊C4 | C2×D4 | C24 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 8 | 6 | 6 | 2 | 4 | 8 |
Matrix representation of C24.56D10 ►in GL6(𝔽41)
| 40 | 23 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 23 | 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 |
| 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 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 9 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 40 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 13 | 14 | 0 | 0 |
| 0 | 0 | 29 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 9 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,23,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,9,0,0,0,0,0,40,0,0,0,0,0,0,0,6,0,0,0,0,34,35,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[9,40,0,0,0,0,0,32,0,0,0,0,0,0,13,29,0,0,0,0,14,28,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;
C24.56D10 in GAP, Magma, Sage, TeX
C_2^4._{56}D_{10} % in TeX
G:=Group("C2^4.56D10"); // GroupNames label
G:=SmallGroup(320,1258);
// by ID
G=gap.SmallGroup(320,1258);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,570,185,12550]);
// 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,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,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^-1=e^-1>;
// generators/relations