direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊D5, Q8⋊3D10, C10⋊3SD16, C20.18D4, C20.14C23, D20.9C22, (C2×Q8)⋊1D5, C5⋊4(C2×SD16), (Q8×C10)⋊1C2, C5⋊2C8⋊9C22, (C2×D20).9C2, (C2×C10).41D4, C10.53(C2×D4), (C2×C4).53D10, C4.8(C5⋊D4), (C5×Q8)⋊3C22, C4.14(C22×D5), (C2×C20).36C22, C22.23(C5⋊D4), (C2×C5⋊2C8)⋊6C2, C2.17(C2×C5⋊D4), SmallGroup(160,162)
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=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >
Subgroups: 248 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C10, C2×C8, SD16, C2×D4, C2×Q8, C20, C20, D10, C2×C10, C2×SD16, C5⋊2C8, D20, D20, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, C2×C5⋊2C8, Q8⋊D5, C2×D20, Q8×C10, C2×Q8⋊D5
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C2×SD16, C5⋊D4, C22×D5, Q8⋊D5, C2×C5⋊D4, C2×Q8⋊D5
►On 80 points
(1 41)(2 42)(3 43)(4 44)(5 45)(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 16 6 11)(2 17 7 12)(3 18 8 13)(4 19 9 14)(5 20 10 15)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 56 46 51)(42 57 47 52)(43 58 48 53)(44 59 49 54)(45 60 50 55)(61 71 66 76)(62 72 67 77)(63 73 68 78)(64 74 69 79)(65 75 70 80)
(1 66 6 61)(2 67 7 62)(3 68 8 63)(4 69 9 64)(5 70 10 65)(11 76 16 71)(12 77 17 72)(13 78 18 73)(14 79 19 74)(15 80 20 75)(21 41 26 46)(22 42 27 47)(23 43 28 48)(24 44 29 49)(25 45 30 50)(31 51 36 56)(32 52 37 57)(33 53 38 58)(34 54 39 59)(35 55 40 60)
(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 5)(2 4)(6 10)(7 9)(11 20)(12 19)(13 18)(14 17)(15 16)(21 35)(22 34)(23 33)(24 32)(25 31)(26 40)(27 39)(28 38)(29 37)(30 36)(41 45)(42 44)(46 50)(47 49)(51 60)(52 59)(53 58)(54 57)(55 56)(61 75)(62 74)(63 73)(64 72)(65 71)(66 80)(67 79)(68 78)(69 77)(70 76)
G:=sub<Sym(80)| (1,41)(2,42)(3,43)(4,44)(5,45)(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), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,66,6,61)(2,67,7,62)(3,68,8,63)(4,69,9,64)(5,70,10,65)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,41,26,46)(22,42,27,47)(23,43,28,48)(24,44,29,49)(25,45,30,50)(31,51,36,56)(32,52,37,57)(33,53,38,58)(34,54,39,59)(35,55,40,60), (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), (1,5)(2,4)(6,10)(7,9)(11,20)(12,19)(13,18)(14,17)(15,16)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76)>;
G:=Group( (1,41)(2,42)(3,43)(4,44)(5,45)(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), (1,16,6,11)(2,17,7,12)(3,18,8,13)(4,19,9,14)(5,20,10,15)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,56,46,51)(42,57,47,52)(43,58,48,53)(44,59,49,54)(45,60,50,55)(61,71,66,76)(62,72,67,77)(63,73,68,78)(64,74,69,79)(65,75,70,80), (1,66,6,61)(2,67,7,62)(3,68,8,63)(4,69,9,64)(5,70,10,65)(11,76,16,71)(12,77,17,72)(13,78,18,73)(14,79,19,74)(15,80,20,75)(21,41,26,46)(22,42,27,47)(23,43,28,48)(24,44,29,49)(25,45,30,50)(31,51,36,56)(32,52,37,57)(33,53,38,58)(34,54,39,59)(35,55,40,60), (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), (1,5)(2,4)(6,10)(7,9)(11,20)(12,19)(13,18)(14,17)(15,16)(21,35)(22,34)(23,33)(24,32)(25,31)(26,40)(27,39)(28,38)(29,37)(30,36)(41,45)(42,44)(46,50)(47,49)(51,60)(52,59)(53,58)(54,57)(55,56)(61,75)(62,74)(63,73)(64,72)(65,71)(66,80)(67,79)(68,78)(69,77)(70,76) );
G=PermutationGroup([[(1,41),(2,42),(3,43),(4,44),(5,45),(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)], [(1,16,6,11),(2,17,7,12),(3,18,8,13),(4,19,9,14),(5,20,10,15),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,56,46,51),(42,57,47,52),(43,58,48,53),(44,59,49,54),(45,60,50,55),(61,71,66,76),(62,72,67,77),(63,73,68,78),(64,74,69,79),(65,75,70,80)], [(1,66,6,61),(2,67,7,62),(3,68,8,63),(4,69,9,64),(5,70,10,65),(11,76,16,71),(12,77,17,72),(13,78,18,73),(14,79,19,74),(15,80,20,75),(21,41,26,46),(22,42,27,47),(23,43,28,48),(24,44,29,49),(25,45,30,50),(31,51,36,56),(32,52,37,57),(33,53,38,58),(34,54,39,59),(35,55,40,60)], [(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)], [(1,5),(2,4),(6,10),(7,9),(11,20),(12,19),(13,18),(14,17),(15,16),(21,35),(22,34),(23,33),(24,32),(25,31),(26,40),(27,39),(28,38),(29,37),(30,36),(41,45),(42,44),(46,50),(47,49),(51,60),(52,59),(53,58),(54,57),(55,56),(61,75),(62,74),(63,73),(64,72),(65,71),(66,80),(67,79),(68,78),(69,77),(70,76)]])
C2×Q8⋊D5 is a maximal subgroup of
D20.6D4 Dic5⋊7SD16 Q8⋊2D20 D10⋊2SD16 D20⋊4D4 C5⋊(C8⋊D4) Q8⋊D5⋊6C4 Dic5⋊SD16 D20.12D4 C42.56D10 Q8⋊D20 Q8.1D20 D20.36D4 D20.37D4 C5⋊2C8⋊24D4 C22⋊Q8⋊D5 D20.23D4 C42.64D10 C42.214D10 C20⋊5SD16 C20⋊6SD16 C42.80D10 Dic5⋊5SD16 D10⋊6SD16 C40⋊15D4 C40⋊9D4 (C2×Q16)⋊D5 D20.17D4 C40.37D4 C40.28D4 M4(2).15D10 (C5×Q8)⋊13D4 (C5×D4)⋊14D4 C2×D5×SD16 C40.C23 D20.34C23
C2×Q8⋊D5 is a maximal quotient of
C4⋊C4.228D10 C20.48SD16 Q8⋊D20 C22⋊Q8.D5 D20.36D4 C5⋊2C8⋊24D4 C20.SD16 C20.Q16 C20⋊5SD16 D20⋊5Q8 C20⋊6SD16 C20.D8 (C5×Q8)⋊13D4
34 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
34 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | SD16 | D10 | D10 | C5⋊D4 | C5⋊D4 | Q8⋊D5 |
| kernel | C2×Q8⋊D5 | C2×C5⋊2C8 | Q8⋊D5 | C2×D20 | Q8×C10 | C20 | C2×C10 | C2×Q8 | C10 | C2×C4 | Q8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 4 |
Matrix representation of C2×Q8⋊D5 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 30 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 23 | 1 |
| 0 | 0 | 5 | 18 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 35 | 40 |
| 0 | 0 | 36 | 40 |
| 1 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 |
| 0 | 0 | 0 | 34 |
| 0 | 0 | 35 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,1,0,0,39,1,0,0,0,0,40,0,0,0,0,40],[0,15,0,0,30,0,0,0,0,0,23,5,0,0,1,18],[1,0,0,0,0,1,0,0,0,0,35,36,0,0,40,40],[1,40,0,0,0,40,0,0,0,0,0,35,0,0,34,0] >;
C2×Q8⋊D5 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes D_5
% in TeX
G:=Group("C2xQ8:D5"); // GroupNames label
G:=SmallGroup(160,162);
// by ID
G=gap.SmallGroup(160,162);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,86,579,159,69,4613]);
// 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=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations