direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×D5, C10.8C24, C20.22C23, Dic10⋊9C22, D10.17C23, Dic5.5C23, C10⋊2(C2×Q8), C5⋊2(C22×Q8), (Q8×C10)⋊5C2, (C2×C4).61D10, (C5×Q8)⋊5C22, C2.9(C23×D5), C4.22(C22×D5), (C2×Dic10)⋊13C2, (C2×C20).46C22, (C2×C10).66C23, (C4×D5).21C22, C22.31(C22×D5), (C2×Dic5).47C22, (C22×D5).45C22, (C2×C4×D5).6C2, SmallGroup(160,220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×D5
G = < a,b,c,d,e | a2=b4=d5=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 392 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, Q8, Q8, C23, D5, C10, C10, C22×C4, C2×Q8, C2×Q8, Dic5, C20, D10, C2×C10, C22×Q8, Dic10, C4×D5, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×Dic10, C2×C4×D5, Q8×D5, Q8×C10, C2×Q8×D5
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, C24, D10, C22×Q8, C22×D5, Q8×D5, C23×D5, C2×Q8×D5
►On 80 points
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 75)
(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(11 41 16 46)(12 42 17 47)(13 43 18 48)(14 44 19 49)(15 45 20 50)(21 71 26 76)(22 72 27 77)(23 73 28 78)(24 74 29 79)(25 75 30 80)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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 43)(2 42)(3 41)(4 45)(5 44)(6 46)(7 50)(8 49)(9 48)(10 47)(11 51)(12 55)(13 54)(14 53)(15 52)(16 56)(17 60)(18 59)(19 58)(20 57)(21 61)(22 65)(23 64)(24 63)(25 62)(26 66)(27 70)(28 69)(29 68)(30 67)(31 71)(32 75)(33 74)(34 73)(35 72)(36 76)(37 80)(38 79)(39 78)(40 77)
G:=sub<Sym(80)| (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,71,26,76)(22,72,27,77)(23,73,28,78)(24,74,29,79)(25,75,30,80)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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), (1,43)(2,42)(3,41)(4,45)(5,44)(6,46)(7,50)(8,49)(9,48)(10,47)(11,51)(12,55)(13,54)(14,53)(15,52)(16,56)(17,60)(18,59)(19,58)(20,57)(21,61)(22,65)(23,64)(24,63)(25,62)(26,66)(27,70)(28,69)(29,68)(30,67)(31,71)(32,75)(33,74)(34,73)(35,72)(36,76)(37,80)(38,79)(39,78)(40,77)>;
G:=Group( (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(11,41,16,46)(12,42,17,47)(13,43,18,48)(14,44,19,49)(15,45,20,50)(21,71,26,76)(22,72,27,77)(23,73,28,78)(24,74,29,79)(25,75,30,80)(31,66,36,61)(32,67,37,62)(33,68,38,63)(34,69,39,64)(35,70,40,65), (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), (1,43)(2,42)(3,41)(4,45)(5,44)(6,46)(7,50)(8,49)(9,48)(10,47)(11,51)(12,55)(13,54)(14,53)(15,52)(16,56)(17,60)(18,59)(19,58)(20,57)(21,61)(22,65)(23,64)(24,63)(25,62)(26,66)(27,70)(28,69)(29,68)(30,67)(31,71)(32,75)(33,74)(34,73)(35,72)(36,76)(37,80)(38,79)(39,78)(40,77) );
G=PermutationGroup([[(1,44),(2,45),(3,41),(4,42),(5,43),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(1,59,9,54),(2,60,10,55),(3,56,6,51),(4,57,7,52),(5,58,8,53),(11,41,16,46),(12,42,17,47),(13,43,18,48),(14,44,19,49),(15,45,20,50),(21,71,26,76),(22,72,27,77),(23,73,28,78),(24,74,29,79),(25,75,30,80),(31,66,36,61),(32,67,37,62),(33,68,38,63),(34,69,39,64),(35,70,40,65)], [(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)], [(1,43),(2,42),(3,41),(4,45),(5,44),(6,46),(7,50),(8,49),(9,48),(10,47),(11,51),(12,55),(13,54),(14,53),(15,52),(16,56),(17,60),(18,59),(19,58),(20,57),(21,61),(22,65),(23,64),(24,63),(25,62),(26,66),(27,70),(28,69),(29,68),(30,67),(31,71),(32,75),(33,74),(34,73),(35,72),(36,76),(37,80),(38,79),(39,78),(40,77)]])
C2×Q8×D5 is a maximal subgroup of
(Q8×D5)⋊C4 Q8⋊2D20 D10⋊4Q16 D10⋊8SD16 D10⋊5Q16 (C2×Q8)⋊4F5 (C2×Q8).7F5 (C2×F5)⋊Q8 C42.125D10 Q8⋊5D20 C10.162- 1+4 Dic10⋊21D4 Dic10⋊22D4 C42.141D10 Dic10⋊10D4 C42.171D10 D20⋊8Q8 C10.1072- 1+4 D5.2- 1+4
C2×Q8×D5 is a maximal quotient of
C10.102+ 1+4 Dic10⋊10Q8 C42.232D10 D20⋊10Q8 (Q8×Dic5)⋊C2 C10.502+ 1+4 Dic10⋊21D4 C10.512+ 1+4 C10.1182+ 1+4 C10.522+ 1+4 Dic10⋊7Q8 C42.236D10 C42.148D10 D20⋊7Q8 Dic10⋊8Q8 Dic10⋊9Q8 D20⋊8Q8 C42.241D10 C42.174D10 D20⋊9Q8
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D5 | D10 | D10 | Q8×D5 |
| kernel | C2×Q8×D5 | C2×Dic10 | C2×C4×D5 | Q8×D5 | Q8×C10 | D10 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 2 | 6 | 8 | 4 |
Matrix representation of C2×Q8×D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 23 |
| 0 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 9 | 32 |
| 0 | 1 | 0 | 0 |
| 40 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,23,32],[40,0,0,0,0,40,0,0,0,0,9,9,0,0,0,32],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,40,0,0,0,0,40] >;
C2×Q8×D5 in GAP, Magma, Sage, TeX
C_2\times Q_8\times D_5
% in TeX
G:=Group("C2xQ8xD5"); // GroupNames label
G:=SmallGroup(160,220);
// by ID
G=gap.SmallGroup(160,220);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,159,69,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations