metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊16D10, C10.182+ (1+4), C4⋊C4⋊49D10, (C4×D4)⋊18D5, (D4×C20)⋊20C2, (C22×C4)⋊5D10, (C4×C20)⋊32C22, D10⋊Q8⋊8C2, C22⋊C4⋊48D10, C4⋊Dic5⋊9C22, (C2×D4).217D10, C23⋊D10.5C2, C42⋊D5⋊32C2, C42⋊2D5⋊16C2, D10.31(C4○D4), D10.12D4⋊7C2, C23.D5⋊9C22, C20.48D4⋊11C2, (C2×C10).100C24, (C2×C20).699C23, (C22×C20)⋊37C22, Dic5.5D4⋊7C2, (C4×Dic5)⋊52C22, (C2×Dic10)⋊6C22, C2.19(D4⋊6D10), C5⋊3(C22.45C24), (D4×C10).307C22, C22.12(C4○D20), C10.D4⋊42C22, (C22×D5).35C23, (C23×D5).41C22, C23.174(C22×D5), C22.125(C23×D5), D10⋊C4.85C22, C23.11D10⋊29C2, C23.23D10⋊16C2, C23.18D10⋊18C2, (C22×C10).170C23, (C2×Dic5).217C23, (C22×Dic5).98C22, C4⋊C4⋊D5⋊7C2, (C4×C5⋊D4)⋊43C2, C2.23(D5×C4○D4), (C5×C4⋊C4)⋊61C22, (D5×C22⋊C4)⋊29C2, C2.49(C2×C4○D20), C10.140(C2×C4○D4), (C2×D10⋊C4)⋊22C2, (C2×C4×D5).252C22, (C2×C10).16(C4○D4), (C5×C22⋊C4)⋊57C22, (C2×C4).284(C22×D5), (C2×C5⋊D4).16C22, SmallGroup(320,1228)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 934 in 248 conjugacy classes, 97 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×11], C22, C22 [×2], C22 [×16], C5, C2×C4 [×5], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×3], C10 [×3], C10 [×3], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×12], C4⋊C4, C4⋊C4 [×7], C22×C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×2], D10 [×9], C2×C10, C2×C10 [×2], C2×C10 [×5], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4, C4×D4, C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C42⋊2C2 [×2], Dic10, C4×D5 [×3], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×3], C2×C20 [×5], C2×C20 [×2], C5×D4 [×2], C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5, D10⋊C4 [×8], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C22×Dic5, C2×C5⋊D4 [×2], C22×C20 [×2], D4×C10, C23×D5, C42⋊D5, C42⋊2D5, C23.11D10, D5×C22⋊C4, D10.12D4, Dic5.5D4, D10⋊Q8, C4⋊C4⋊D5, C20.48D4, C2×D10⋊C4, C4×C5⋊D4, C23.23D10, C23.18D10, C23⋊D10, D4×C20, C42⋊16D10
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, D4⋊6D10, D5×C4○D4, C42⋊16D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >
►On 80 points
(1 62 20 53)(2 68 16 59)(3 64 17 55)(4 70 18 51)(5 66 19 57)(6 54 11 63)(7 60 12 69)(8 56 13 65)(9 52 14 61)(10 58 15 67)(21 44 36 79)(22 50 37 75)(23 46 38 71)(24 42 39 77)(25 48 40 73)(26 74 31 49)(27 80 32 45)(28 76 33 41)(29 72 34 47)(30 78 35 43)
(1 30 10 25)(2 26 6 21)(3 27 7 22)(4 28 8 23)(5 29 9 24)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 65 71 51)(42 66 72 52)(43 67 73 53)(44 68 74 54)(45 69 75 55)(46 70 76 56)(47 61 77 57)(48 62 78 58)(49 63 79 59)(50 64 80 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)
(2 5)(3 4)(6 9)(7 8)(11 14)(12 13)(16 19)(17 18)(21 39)(22 38)(23 37)(24 36)(25 40)(26 34)(27 33)(28 32)(29 31)(30 35)(41 75)(42 74)(43 73)(44 72)(45 71)(46 80)(47 79)(48 78)(49 77)(50 76)(51 60)(52 59)(53 58)(54 57)(55 56)(61 68)(62 67)(63 66)(64 65)(69 70)
G:=sub<Sym(80)| (1,62,20,53)(2,68,16,59)(3,64,17,55)(4,70,18,51)(5,66,19,57)(6,54,11,63)(7,60,12,69)(8,56,13,65)(9,52,14,61)(10,58,15,67)(21,44,36,79)(22,50,37,75)(23,46,38,71)(24,42,39,77)(25,48,40,73)(26,74,31,49)(27,80,32,45)(28,76,33,41)(29,72,34,47)(30,78,35,43), (1,30,10,25)(2,26,6,21)(3,27,7,22)(4,28,8,23)(5,29,9,24)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,65,71,51)(42,66,72,52)(43,67,73,53)(44,68,74,54)(45,69,75,55)(46,70,76,56)(47,61,77,57)(48,62,78,58)(49,63,79,59)(50,64,80,60), (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), (2,5)(3,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,39)(22,38)(23,37)(24,36)(25,40)(26,34)(27,33)(28,32)(29,31)(30,35)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,70)>;
G:=Group( (1,62,20,53)(2,68,16,59)(3,64,17,55)(4,70,18,51)(5,66,19,57)(6,54,11,63)(7,60,12,69)(8,56,13,65)(9,52,14,61)(10,58,15,67)(21,44,36,79)(22,50,37,75)(23,46,38,71)(24,42,39,77)(25,48,40,73)(26,74,31,49)(27,80,32,45)(28,76,33,41)(29,72,34,47)(30,78,35,43), (1,30,10,25)(2,26,6,21)(3,27,7,22)(4,28,8,23)(5,29,9,24)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,65,71,51)(42,66,72,52)(43,67,73,53)(44,68,74,54)(45,69,75,55)(46,70,76,56)(47,61,77,57)(48,62,78,58)(49,63,79,59)(50,64,80,60), (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), (2,5)(3,4)(6,9)(7,8)(11,14)(12,13)(16,19)(17,18)(21,39)(22,38)(23,37)(24,36)(25,40)(26,34)(27,33)(28,32)(29,31)(30,35)(41,75)(42,74)(43,73)(44,72)(45,71)(46,80)(47,79)(48,78)(49,77)(50,76)(51,60)(52,59)(53,58)(54,57)(55,56)(61,68)(62,67)(63,66)(64,65)(69,70) );
G=PermutationGroup([(1,62,20,53),(2,68,16,59),(3,64,17,55),(4,70,18,51),(5,66,19,57),(6,54,11,63),(7,60,12,69),(8,56,13,65),(9,52,14,61),(10,58,15,67),(21,44,36,79),(22,50,37,75),(23,46,38,71),(24,42,39,77),(25,48,40,73),(26,74,31,49),(27,80,32,45),(28,76,33,41),(29,72,34,47),(30,78,35,43)], [(1,30,10,25),(2,26,6,21),(3,27,7,22),(4,28,8,23),(5,29,9,24),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,65,71,51),(42,66,72,52),(43,67,73,53),(44,68,74,54),(45,69,75,55),(46,70,76,56),(47,61,77,57),(48,62,78,58),(49,63,79,59),(50,64,80,60)], [(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)], [(2,5),(3,4),(6,9),(7,8),(11,14),(12,13),(16,19),(17,18),(21,39),(22,38),(23,37),(24,36),(25,40),(26,34),(27,33),(28,32),(29,31),(30,35),(41,75),(42,74),(43,73),(44,72),(45,71),(46,80),(47,79),(48,78),(49,77),(50,76),(51,60),(52,59),(53,58),(54,57),(55,56),(61,68),(62,67),(63,66),(64,65),(69,70)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 14 | 18 |
| 0 | 0 | 0 | 0 | 37 | 27 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 37 | 0 | 0 |
| 0 | 0 | 36 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 35 | 6 | 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 | 3 | 40 |
| 35 | 7 | 0 | 0 | 0 | 0 |
| 36 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 40 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,14,37,0,0,0,0,18,27],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,36,0,0,0,0,37,26,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[0,35,0,0,0,0,7,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,40],[35,36,0,0,0,0,7,6,0,0,0,0,0,0,1,28,0,0,0,0,0,40,0,0,0,0,0,0,1,3,0,0,0,0,0,40] >;
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20H | 20I | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 10 | 10 | 20 | 2 | ··· | 2 | 4 | 4 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 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 | 1 | 2 | 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 | C2 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D10 | C4○D20 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 |
| kernel | C42⋊16D10 | C42⋊D5 | C42⋊2D5 | C23.11D10 | D5×C22⋊C4 | D10.12D4 | Dic5.5D4 | D10⋊Q8 | C4⋊C4⋊D5 | C20.48D4 | C2×D10⋊C4 | C4×C5⋊D4 | C23.23D10 | C23.18D10 | C23⋊D10 | D4×C20 | C4×D4 | D10 | C2×C10 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C22 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 2 | 4 | 2 | 4 | 2 | 16 | 1 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{16}D_{10} % in TeX
G:=Group("C4^2:16D10"); // GroupNames label
G:=SmallGroup(320,1228);
// by ID
G=gap.SmallGroup(320,1228);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,136,12550]);
// 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^-1*b^2,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations