direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8⋊2Dic5, C10⋊5C4≀C2, (D4×C10)⋊20C4, (Q8×C10)⋊18C4, C4○D4⋊3Dic5, (C2×Q8)⋊6Dic5, D4⋊5(C2×Dic5), (C2×D4)⋊8Dic5, Q8⋊5(C2×Dic5), C4○D4.36D10, (C2×C20).197D4, C20.452(C2×D4), (C2×C20).481C23, C20.144(C22×C4), (C4×Dic5)⋊63C22, (C22×C10).113D4, (C22×C4).357D10, C23.66(C5⋊D4), C4.Dic5⋊23C22, C4.33(C23.D5), C4.15(C22×Dic5), C20.146(C22⋊C4), C22.5(C23.D5), (C22×C20).207C22, C5⋊7(C2×C4≀C2), (C5×C4○D4)⋊9C4, (C2×C4×Dic5)⋊4C2, (C5×D4)⋊28(C2×C4), (C5×Q8)⋊26(C2×C4), (C2×C4○D4).4D5, (C10×C4○D4).4C2, (C2×C10).39(C2×D4), C4.143(C2×C5⋊D4), (C2×C20).298(C2×C4), (C2×C4.Dic5)⋊21C2, (C2×C4).54(C2×Dic5), C22.11(C2×C5⋊D4), C2.20(C2×C23.D5), (C2×C4).282(C5⋊D4), C10.125(C2×C22⋊C4), (C5×C4○D4).41C22, (C2×C4).566(C22×D5), (C2×C10).181(C22⋊C4), SmallGroup(320,862)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8⋊2Dic5
G = < a,b,c,d,e | a2=b4=d10=1, c2=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b2c, ece-1=b-1c, ede-1=d-1 >
Subgroups: 446 in 170 conjugacy classes, 71 normal (43 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×6], C22 [×3], C22 [×6], C5, C8 [×2], C2×C4 [×6], C2×C4 [×11], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C10, C10 [×2], C10 [×4], C42 [×3], C2×C8, M4(2) [×3], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], Dic5 [×4], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×6], C4≀C2 [×4], C2×C42, C2×M4(2), C2×C4○D4, C5⋊2C8 [×2], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×5], C5×D4 [×2], C5×D4 [×5], C5×Q8 [×2], C5×Q8, C22×C10, C22×C10, C2×C4≀C2, C2×C5⋊2C8, C4.Dic5 [×2], C4.Dic5, C4×Dic5 [×2], C4×Dic5, C22×Dic5, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4 [×4], C5×C4○D4 [×2], Q8⋊2Dic5 [×4], C2×C4.Dic5, C2×C4×Dic5, C10×C4○D4, C2×Q8⋊2Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C4≀C2 [×2], C2×C22⋊C4, C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C2×C4≀C2, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], Q8⋊2Dic5 [×2], C2×C23.D5, C2×Q8⋊2Dic5
►On 80 points
(1 22)(2 23)(3 24)(4 25)(5 21)(6 38)(7 39)(8 40)(9 36)(10 37)(11 35)(12 31)(13 32)(14 33)(15 34)(16 29)(17 30)(18 26)(19 27)(20 28)(41 54)(42 55)(43 56)(44 57)(45 58)(46 59)(47 60)(48 51)(49 52)(50 53)(61 79)(62 80)(63 71)(64 72)(65 73)(66 74)(67 75)(68 76)(69 77)(70 78)
(1 13 19 10)(2 14 20 6)(3 15 16 7)(4 11 17 8)(5 12 18 9)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)(41 76 46 71)(42 77 47 72)(43 78 48 73)(44 79 49 74)(45 80 50 75)(51 65 56 70)(52 66 57 61)(53 67 58 62)(54 68 59 63)(55 69 60 64)
(1 64 19 69)(2 70 20 65)(3 66 16 61)(4 62 17 67)(5 68 18 63)(6 51 14 56)(7 57 15 52)(8 53 11 58)(9 59 12 54)(10 55 13 60)(21 76 26 71)(22 72 27 77)(23 78 28 73)(24 74 29 79)(25 80 30 75)(31 41 36 46)(32 47 37 42)(33 43 38 48)(34 49 39 44)(35 45 40 50)
(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 4)(2 3)(6 7)(8 10)(11 13)(14 15)(16 20)(17 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)(41 71 46 76)(42 80 47 75)(43 79 48 74)(44 78 49 73)(45 77 50 72)(51 66 56 61)(52 65 57 70)(53 64 58 69)(54 63 59 68)(55 62 60 67)
G:=sub<Sym(80)| (1,22)(2,23)(3,24)(4,25)(5,21)(6,38)(7,39)(8,40)(9,36)(10,37)(11,35)(12,31)(13,32)(14,33)(15,34)(16,29)(17,30)(18,26)(19,27)(20,28)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78), (1,13,19,10)(2,14,20,6)(3,15,16,7)(4,11,17,8)(5,12,18,9)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,76,46,71)(42,77,47,72)(43,78,48,73)(44,79,49,74)(45,80,50,75)(51,65,56,70)(52,66,57,61)(53,67,58,62)(54,68,59,63)(55,69,60,64), (1,64,19,69)(2,70,20,65)(3,66,16,61)(4,62,17,67)(5,68,18,63)(6,51,14,56)(7,57,15,52)(8,53,11,58)(9,59,12,54)(10,55,13,60)(21,76,26,71)(22,72,27,77)(23,78,28,73)(24,74,29,79)(25,80,30,75)(31,41,36,46)(32,47,37,42)(33,43,38,48)(34,49,39,44)(35,45,40,50), (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), (1,4)(2,3)(6,7)(8,10)(11,13)(14,15)(16,20)(17,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(41,71,46,76)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,66,56,61)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67)>;
G:=Group( (1,22)(2,23)(3,24)(4,25)(5,21)(6,38)(7,39)(8,40)(9,36)(10,37)(11,35)(12,31)(13,32)(14,33)(15,34)(16,29)(17,30)(18,26)(19,27)(20,28)(41,54)(42,55)(43,56)(44,57)(45,58)(46,59)(47,60)(48,51)(49,52)(50,53)(61,79)(62,80)(63,71)(64,72)(65,73)(66,74)(67,75)(68,76)(69,77)(70,78), (1,13,19,10)(2,14,20,6)(3,15,16,7)(4,11,17,8)(5,12,18,9)(21,31,26,36)(22,32,27,37)(23,33,28,38)(24,34,29,39)(25,35,30,40)(41,76,46,71)(42,77,47,72)(43,78,48,73)(44,79,49,74)(45,80,50,75)(51,65,56,70)(52,66,57,61)(53,67,58,62)(54,68,59,63)(55,69,60,64), (1,64,19,69)(2,70,20,65)(3,66,16,61)(4,62,17,67)(5,68,18,63)(6,51,14,56)(7,57,15,52)(8,53,11,58)(9,59,12,54)(10,55,13,60)(21,76,26,71)(22,72,27,77)(23,78,28,73)(24,74,29,79)(25,80,30,75)(31,41,36,46)(32,47,37,42)(33,43,38,48)(34,49,39,44)(35,45,40,50), (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), (1,4)(2,3)(6,7)(8,10)(11,13)(14,15)(16,20)(17,19)(22,25)(23,24)(27,30)(28,29)(32,35)(33,34)(37,40)(38,39)(41,71,46,76)(42,80,47,75)(43,79,48,74)(44,78,49,73)(45,77,50,72)(51,66,56,61)(52,65,57,70)(53,64,58,69)(54,63,59,68)(55,62,60,67) );
G=PermutationGroup([(1,22),(2,23),(3,24),(4,25),(5,21),(6,38),(7,39),(8,40),(9,36),(10,37),(11,35),(12,31),(13,32),(14,33),(15,34),(16,29),(17,30),(18,26),(19,27),(20,28),(41,54),(42,55),(43,56),(44,57),(45,58),(46,59),(47,60),(48,51),(49,52),(50,53),(61,79),(62,80),(63,71),(64,72),(65,73),(66,74),(67,75),(68,76),(69,77),(70,78)], [(1,13,19,10),(2,14,20,6),(3,15,16,7),(4,11,17,8),(5,12,18,9),(21,31,26,36),(22,32,27,37),(23,33,28,38),(24,34,29,39),(25,35,30,40),(41,76,46,71),(42,77,47,72),(43,78,48,73),(44,79,49,74),(45,80,50,75),(51,65,56,70),(52,66,57,61),(53,67,58,62),(54,68,59,63),(55,69,60,64)], [(1,64,19,69),(2,70,20,65),(3,66,16,61),(4,62,17,67),(5,68,18,63),(6,51,14,56),(7,57,15,52),(8,53,11,58),(9,59,12,54),(10,55,13,60),(21,76,26,71),(22,72,27,77),(23,78,28,73),(24,74,29,79),(25,80,30,75),(31,41,36,46),(32,47,37,42),(33,43,38,48),(34,49,39,44),(35,45,40,50)], [(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)], [(1,4),(2,3),(6,7),(8,10),(11,13),(14,15),(16,20),(17,19),(22,25),(23,24),(27,30),(28,29),(32,35),(33,34),(37,40),(38,39),(41,71,46,76),(42,80,47,75),(43,79,48,74),(44,78,49,73),(45,77,50,72),(51,66,56,61),(52,65,57,70),(53,64,58,69),(54,63,59,68),(55,62,60,67)])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | ··· | 10 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | - | - | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | Dic5 | Dic5 | Dic5 | D10 | C4≀C2 | C5⋊D4 | C5⋊D4 | Q8⋊2Dic5 |
| kernel | C2×Q8⋊2Dic5 | Q8⋊2Dic5 | C2×C4.Dic5 | C2×C4×Dic5 | C10×C4○D4 | D4×C10 | Q8×C10 | C5×C4○D4 | C2×C20 | C22×C10 | C2×C4○D4 | C22×C4 | C2×D4 | C2×Q8 | C4○D4 | C4○D4 | C10 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 4 | 8 |
Matrix representation of C2×Q8⋊2Dic5 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 32 |
| 17 | 35 | 0 | 0 |
| 7 | 24 | 0 | 0 |
| 0 | 0 | 0 | 9 |
| 0 | 0 | 9 | 0 |
| 40 | 40 | 0 | 0 |
| 8 | 7 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 7 | 6 | 0 | 0 |
| 33 | 34 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,32],[17,7,0,0,35,24,0,0,0,0,0,9,0,0,9,0],[40,8,0,0,40,7,0,0,0,0,40,0,0,0,0,1],[7,33,0,0,6,34,0,0,0,0,32,0,0,0,0,1] >;
C2×Q8⋊2Dic5 in GAP, Magma, Sage, TeX
C_2\times Q_8\rtimes_2{\rm Dic}_5 % in TeX
G:=Group("C2xQ8:2Dic5"); // GroupNames label
G:=SmallGroup(320,862);
// by ID
G=gap.SmallGroup(320,862);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,136,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^10=1,c^2=b^2,e^2=d^5,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,d*c*d^-1=b^2*c,e*c*e^-1=b^-1*c,e*d*e^-1=d^-1>;
// generators/relations