metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊10D10, C10.962+ (1+4), C4⋊C4⋊44D10, (C4×D20)⋊8C2, (C4×C20)⋊6C22, D10⋊Q8⋊4C2, C4.D20⋊6C2, C42⋊D5⋊1C2, C42⋊2D5⋊3C2, C22⋊D20.2C2, C42⋊C2⋊11D5, (C2×C10).71C24, C4⋊Dic5⋊56C22, C22⋊C4.95D10, Dic5⋊4D4⋊43C2, D10.29(C4○D4), D10.13D4⋊4C2, C2.8(D4⋊8D10), (C2×C20).146C23, (C2×Dic10)⋊5C22, (C4×Dic5)⋊50C22, (C2×D20).25C22, (C22×C4).192D10, D10⋊C4⋊40C22, Dic5.14D4⋊4C2, C5⋊2(C22.45C24), C23.D5.4C22, C22.18(C4○D20), C10.D4⋊32C22, (C23×D5).37C22, (C22×D5).21C23, C22.100(C23×D5), C23.159(C22×D5), C23.23D10⋊27C2, (C22×C10).141C23, (C22×C20).435C22, (C2×Dic5).208C23, (C22×Dic5).88C22, C4⋊C4⋊D5⋊4C2, C2.10(D5×C4○D4), (C2×C4×D5)⋊45C22, (C5×C4⋊C4)⋊54C22, (D5×C22⋊C4)⋊26C2, C2.30(C2×C4○D20), C10.28(C2×C4○D4), (C2×D10⋊C4)⋊40C2, (C2×C5⋊D4).9C22, (C5×C42⋊C2)⋊13C2, (C2×C10).41(C4○D4), (C2×C4).274(C22×D5), (C5×C22⋊C4).138C22, SmallGroup(320,1199)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 998 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×6], C2×C4 [×12], D4 [×5], Q8, C23, C23 [×8], D5 [×4], C10 [×3], C10 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8, C24, Dic5 [×5], C20 [×6], D10 [×2], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×2], C2×C22⋊C4 [×2], C42⋊C2, C42⋊C2, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic10, C4×D5 [×4], D20 [×3], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×2], C2×C20 [×6], C2×C20 [×2], C22×D5 [×3], C22×D5 [×5], C22×C10, C22.45C24, C4×Dic5, C10.D4 [×5], C4⋊Dic5, D10⋊C4 [×11], C23.D5, C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×3], C2×D20 [×2], C22×Dic5, C2×C5⋊D4, C22×C20, C23×D5, C42⋊D5, C4×D20, C4.D20, C42⋊2D5, Dic5.14D4, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.13D4 [×2], D10⋊Q8, C4⋊C4⋊D5, C2×D10⋊C4, C23.23D10, C5×C42⋊C2, C42⋊10D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ (1+4), C22×D5 [×7], C22.45C24, C4○D20 [×2], C23×D5, C2×C4○D20, D5×C4○D4, D4⋊8D10, C42⋊10D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, bc=cb, dbd=a2b, dcd=c-1 >
►On 80 points
(1 53 13 44)(2 59 14 50)(3 55 15 46)(4 51 11 42)(5 57 12 48)(6 56 19 47)(7 52 20 43)(8 58 16 49)(9 54 17 45)(10 60 18 41)(21 71 34 61)(22 77 35 67)(23 73 31 63)(24 79 32 69)(25 75 33 65)(26 78 36 68)(27 74 37 64)(28 80 38 70)(29 76 39 66)(30 72 40 62)
(1 26 8 23)(2 27 9 24)(3 28 10 25)(4 29 6 21)(5 30 7 22)(11 39 19 34)(12 40 20 35)(13 36 16 31)(14 37 17 32)(15 38 18 33)(41 65 46 70)(42 66 47 61)(43 67 48 62)(44 68 49 63)(45 69 50 64)(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 7)(2 6)(3 10)(4 9)(5 8)(11 17)(12 16)(13 20)(14 19)(15 18)(21 37)(22 36)(23 40)(24 39)(25 38)(26 35)(27 34)(28 33)(29 32)(30 31)(42 50)(43 49)(44 48)(45 47)(51 59)(52 58)(53 57)(54 56)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)
G:=sub<Sym(80)| (1,53,13,44)(2,59,14,50)(3,55,15,46)(4,51,11,42)(5,57,12,48)(6,56,19,47)(7,52,20,43)(8,58,16,49)(9,54,17,45)(10,60,18,41)(21,71,34,61)(22,77,35,67)(23,73,31,63)(24,79,32,69)(25,75,33,65)(26,78,36,68)(27,74,37,64)(28,80,38,70)(29,76,39,66)(30,72,40,62), (1,26,8,23)(2,27,9,24)(3,28,10,25)(4,29,6,21)(5,30,7,22)(11,39,19,34)(12,40,20,35)(13,36,16,31)(14,37,17,32)(15,38,18,33)(41,65,46,70)(42,66,47,61)(43,67,48,62)(44,68,49,63)(45,69,50,64)(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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,17)(12,16)(13,20)(14,19)(15,18)(21,37)(22,36)(23,40)(24,39)(25,38)(26,35)(27,34)(28,33)(29,32)(30,31)(42,50)(43,49)(44,48)(45,47)(51,59)(52,58)(53,57)(54,56)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80)>;
G:=Group( (1,53,13,44)(2,59,14,50)(3,55,15,46)(4,51,11,42)(5,57,12,48)(6,56,19,47)(7,52,20,43)(8,58,16,49)(9,54,17,45)(10,60,18,41)(21,71,34,61)(22,77,35,67)(23,73,31,63)(24,79,32,69)(25,75,33,65)(26,78,36,68)(27,74,37,64)(28,80,38,70)(29,76,39,66)(30,72,40,62), (1,26,8,23)(2,27,9,24)(3,28,10,25)(4,29,6,21)(5,30,7,22)(11,39,19,34)(12,40,20,35)(13,36,16,31)(14,37,17,32)(15,38,18,33)(41,65,46,70)(42,66,47,61)(43,67,48,62)(44,68,49,63)(45,69,50,64)(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), (1,7)(2,6)(3,10)(4,9)(5,8)(11,17)(12,16)(13,20)(14,19)(15,18)(21,37)(22,36)(23,40)(24,39)(25,38)(26,35)(27,34)(28,33)(29,32)(30,31)(42,50)(43,49)(44,48)(45,47)(51,59)(52,58)(53,57)(54,56)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80) );
G=PermutationGroup([(1,53,13,44),(2,59,14,50),(3,55,15,46),(4,51,11,42),(5,57,12,48),(6,56,19,47),(7,52,20,43),(8,58,16,49),(9,54,17,45),(10,60,18,41),(21,71,34,61),(22,77,35,67),(23,73,31,63),(24,79,32,69),(25,75,33,65),(26,78,36,68),(27,74,37,64),(28,80,38,70),(29,76,39,66),(30,72,40,62)], [(1,26,8,23),(2,27,9,24),(3,28,10,25),(4,29,6,21),(5,30,7,22),(11,39,19,34),(12,40,20,35),(13,36,16,31),(14,37,17,32),(15,38,18,33),(41,65,46,70),(42,66,47,61),(43,67,48,62),(44,68,49,63),(45,69,50,64),(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)], [(1,7),(2,6),(3,10),(4,9),(5,8),(11,17),(12,16),(13,20),(14,19),(15,18),(21,37),(22,36),(23,40),(24,39),(25,38),(26,35),(27,34),(28,33),(29,32),(30,31),(42,50),(43,49),(44,48),(45,47),(51,59),(52,58),(53,57),(54,56),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(70,80)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 21 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 4 | 0 | 0 |
| 0 | 0 | 29 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 40 | 7 | 0 | 0 | 0 | 0 |
| 34 | 7 | 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 | 37 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 34 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 17 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,7,29,0,0,0,0,4,34,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,34,0,0,0,0,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,37,0,0,0,0,0,40],[40,34,0,0,0,0,0,1,0,0,0,0,0,0,1,17,0,0,0,0,0,40,0,0,0,0,0,0,40,4,0,0,0,0,0,1] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 10 | 10 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D5×C4○D4 | D4⋊8D10 |
| kernel | C42⋊10D10 | C42⋊D5 | C4×D20 | C4.D20 | C42⋊2D5 | Dic5.14D4 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.13D4 | D10⋊Q8 | C4⋊C4⋊D5 | C2×D10⋊C4 | C23.23D10 | C5×C42⋊C2 | C42⋊C2 | D10 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}D_{10} % in TeX
G:=Group("C4^2:10D10"); // GroupNames label
G:=SmallGroup(320,1199);
// by ID
G=gap.SmallGroup(320,1199);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,387,100,675,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations