direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊2D5, Q8⋊5D10, D20⋊9C22, C10.9C24, C20.23C23, D10.4C23, Dic5.16C23, (C2×Q8)⋊6D5, (Q8×C10)⋊6C2, (C2×D20)⋊12C2, C10⋊3(C4○D4), (C2×C4).62D10, (C4×D5)⋊5C22, (C5×Q8)⋊6C22, C4.23(C22×D5), C2.10(C23×D5), (C2×C20).47C22, (C2×C10).67C23, C22.32(C22×D5), (C2×Dic5).65C22, (C22×D5).33C22, (C2×C4×D5)⋊5C2, C5⋊3(C2×C4○D4), SmallGroup(160,221)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊2D5
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, ce=ec, ede=d-1 >
Subgroups: 488 in 164 conjugacy classes, 89 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, D10, C2×C10, C2×C4○D4, C4×D5, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, C2×C4×D5, C2×D20, Q8⋊2D5, Q8×C10, C2×Q8⋊2D5
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, Q8⋊2D5, C23×D5, C2×Q8⋊2D5
►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 26 6 21)(2 27 7 22)(3 28 8 23)(4 29 9 24)(5 30 10 25)(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 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 45)(2 44)(3 43)(4 42)(5 41)(6 50)(7 49)(8 48)(9 47)(10 46)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(21 65)(22 64)(23 63)(24 62)(25 61)(26 70)(27 69)(28 68)(29 67)(30 66)(31 80)(32 79)(33 78)(34 77)(35 76)(36 75)(37 74)(38 73)(39 72)(40 71)
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), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(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), (1,2,3,4,5)(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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,50)(7,49)(8,48)(9,47)(10,46)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,65)(22,64)(23,63)(24,62)(25,61)(26,70)(27,69)(28,68)(29,67)(30,66)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71)>;
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), (1,26,6,21)(2,27,7,22)(3,28,8,23)(4,29,9,24)(5,30,10,25)(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), (1,2,3,4,5)(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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,50)(7,49)(8,48)(9,47)(10,46)(11,60)(12,59)(13,58)(14,57)(15,56)(16,55)(17,54)(18,53)(19,52)(20,51)(21,65)(22,64)(23,63)(24,62)(25,61)(26,70)(27,69)(28,68)(29,67)(30,66)(31,80)(32,79)(33,78)(34,77)(35,76)(36,75)(37,74)(38,73)(39,72)(40,71) );
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)], [(1,26,6,21),(2,27,7,22),(3,28,8,23),(4,29,9,24),(5,30,10,25),(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)], [(1,2,3,4,5),(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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,50),(7,49),(8,48),(9,47),(10,46),(11,60),(12,59),(13,58),(14,57),(15,56),(16,55),(17,54),(18,53),(19,52),(20,51),(21,65),(22,64),(23,63),(24,62),(25,61),(26,70),(27,69),(28,68),(29,67),(30,66),(31,80),(32,79),(33,78),(34,77),(35,76),(36,75),(37,74),(38,73),(39,72),(40,71)]])
C2×Q8⋊2D5 is a maximal subgroup of
M4(2).21D10 Q8⋊(C4×D5) Q8⋊2D5⋊C4 Q8.D20 D20⋊4D4 D20⋊7D4 D20.17D4 (C2×Q8)⋊6F5 (C2×Q8)⋊7F5 (C2×Q8).5F5 C42.126D10 Q8⋊5D20 Q8⋊6D20 C4⋊C4⋊26D10 C10.172- 1+4 D20⋊21D4 D20⋊22D4 C42.233D10 C42⋊18D10 D20⋊10D4 C42.171D10 C42.240D10 D20⋊12D4 D40⋊C22 C10.452- 1+4 C10.1482+ 1+4 Dic5.20C24 C2×D5×C4○D4 D20.39C23
C2×Q8⋊2D5 is a maximal quotient of
C10.112+ 1+4 Q8⋊6Dic10 Q8⋊6D20 C42.131D10 C42.135D10 C42.136D10 C22⋊Q8⋊25D5 C4⋊C4⋊26D10 D20⋊21D4 C10.532+ 1+4 C10.772- 1+4 C10.562+ 1+4 C10.572+ 1+4 C42.237D10 C42.152D10 C42.153D10 C42.155D10 C42.156D10 C42.240D10 D20⋊12D4 C42.241D10 D20⋊9Q8 C42.177D10 C42.178D10 C42.179D10 C2×Q8×Dic5 C10.452- 1+4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 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 | D5 | C4○D4 | D10 | D10 | Q8⋊2D5 |
| kernel | C2×Q8⋊2D5 | C2×C4×D5 | C2×D20 | Q8⋊2D5 | Q8×C10 | C2×Q8 | C10 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 2 | 4 | 6 | 8 | 4 |
Matrix representation of C2×Q8⋊2D5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 9 | 9 |
| 6 | 40 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 35 | 1 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,1,0,0,39,1],[1,0,0,0,0,1,0,0,0,0,32,9,0,0,0,9],[6,1,0,0,40,0,0,0,0,0,1,0,0,0,0,1],[35,6,0,0,1,6,0,0,0,0,1,40,0,0,0,40] >;
C2×Q8⋊2D5 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_2D_5
% in TeX
G:=Group("C2xQ8:2D5"); // GroupNames label
G:=SmallGroup(160,221);
// by ID
G=gap.SmallGroup(160,221);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,579,159,69,4613]);
// 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,c*e=e*c,e*d*e=d^-1>;
// generators/relations