metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊28D10, C10.802+ (1+4), C4⋊1D4⋊9D5, (C2×D4)⋊13D10, (C4×C20)⋊37C22, C23⋊D10⋊28C2, (D4×C10)⋊34C22, C42⋊2D5⋊19C2, Dic5⋊D4⋊39C2, (C2×C20).638C23, (C2×C10).264C24, C2.84(D4⋊6D10), C23.D5⋊38C22, C23.70(C22×D5), C5⋊5(C22.54C24), C10.D4⋊37C22, (C22×C10).78C23, (C23×D5).73C22, C22.285(C23×D5), D10⋊C4.75C22, C23.18D10⋊28C2, (C2×Dic5).138C23, (C22×Dic5)⋊30C22, (C22×D5).118C23, (C5×C4⋊1D4)⋊15C2, (C2×C4).216(C22×D5), (C2×C5⋊D4).80C22, SmallGroup(320,1392)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×3], C2×C4 [×9], D4 [×12], C23, C23 [×3], C23 [×5], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×6], C2×D4 [×6], C24, Dic5 [×6], C20 [×3], D10 [×10], C2×C10, C2×C10 [×12], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C42⋊2C2 [×2], C4⋊1D4, C2×Dic5 [×6], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×6], C22×D5 [×2], C22×D5 [×3], C22×C10, C22×C10 [×3], C22.54C24, C10.D4 [×6], D10⋊C4 [×6], C23.D5 [×6], C4×C20, C22×Dic5 [×3], C2×C5⋊D4 [×6], D4×C10 [×6], C23×D5, C42⋊2D5 [×2], C23.18D10 [×3], C23⋊D10 [×3], Dic5⋊D4 [×6], C5×C4⋊1D4, C42⋊28D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ (1+4) [×3], C22×D5 [×7], C22.54C24, C23×D5, D4⋊6D10 [×3], C42⋊28D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >
►On 80 points
(1 61 16 66)(2 67 17 62)(3 63 18 68)(4 69 19 64)(5 65 20 70)(6 75 12 80)(7 71 13 76)(8 77 14 72)(9 73 15 78)(10 79 11 74)(21 33 50 52)(22 53 41 34)(23 35 42 54)(24 55 43 36)(25 37 44 56)(26 57 45 38)(27 39 46 58)(28 59 47 40)(29 31 48 60)(30 51 49 32)
(1 52 13 57)(2 58 14 53)(3 54 15 59)(4 60 11 55)(5 56 12 51)(6 32 20 37)(7 38 16 33)(8 34 17 39)(9 40 18 35)(10 36 19 31)(21 76 45 61)(22 62 46 77)(23 78 47 63)(24 64 48 79)(25 80 49 65)(26 66 50 71)(27 72 41 67)(28 68 42 73)(29 74 43 69)(30 70 44 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 6)(2 10)(3 9)(4 8)(5 7)(11 17)(12 16)(13 20)(14 19)(15 18)(21 44)(22 43)(23 42)(24 41)(25 50)(26 49)(27 48)(28 47)(29 46)(30 45)(31 34)(32 33)(35 40)(36 39)(37 38)(51 52)(53 60)(54 59)(55 58)(56 57)(61 65)(62 64)(66 70)(67 69)(71 75)(72 74)(76 80)(77 79)
G:=sub<Sym(80)| (1,61,16,66)(2,67,17,62)(3,63,18,68)(4,69,19,64)(5,65,20,70)(6,75,12,80)(7,71,13,76)(8,77,14,72)(9,73,15,78)(10,79,11,74)(21,33,50,52)(22,53,41,34)(23,35,42,54)(24,55,43,36)(25,37,44,56)(26,57,45,38)(27,39,46,58)(28,59,47,40)(29,31,48,60)(30,51,49,32), (1,52,13,57)(2,58,14,53)(3,54,15,59)(4,60,11,55)(5,56,12,51)(6,32,20,37)(7,38,16,33)(8,34,17,39)(9,40,18,35)(10,36,19,31)(21,76,45,61)(22,62,46,77)(23,78,47,63)(24,64,48,79)(25,80,49,65)(26,66,50,71)(27,72,41,67)(28,68,42,73)(29,74,43,69)(30,70,44,75), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,17)(12,16)(13,20)(14,19)(15,18)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,34)(32,33)(35,40)(36,39)(37,38)(51,52)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79)>;
G:=Group( (1,61,16,66)(2,67,17,62)(3,63,18,68)(4,69,19,64)(5,65,20,70)(6,75,12,80)(7,71,13,76)(8,77,14,72)(9,73,15,78)(10,79,11,74)(21,33,50,52)(22,53,41,34)(23,35,42,54)(24,55,43,36)(25,37,44,56)(26,57,45,38)(27,39,46,58)(28,59,47,40)(29,31,48,60)(30,51,49,32), (1,52,13,57)(2,58,14,53)(3,54,15,59)(4,60,11,55)(5,56,12,51)(6,32,20,37)(7,38,16,33)(8,34,17,39)(9,40,18,35)(10,36,19,31)(21,76,45,61)(22,62,46,77)(23,78,47,63)(24,64,48,79)(25,80,49,65)(26,66,50,71)(27,72,41,67)(28,68,42,73)(29,74,43,69)(30,70,44,75), (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), (1,6)(2,10)(3,9)(4,8)(5,7)(11,17)(12,16)(13,20)(14,19)(15,18)(21,44)(22,43)(23,42)(24,41)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,34)(32,33)(35,40)(36,39)(37,38)(51,52)(53,60)(54,59)(55,58)(56,57)(61,65)(62,64)(66,70)(67,69)(71,75)(72,74)(76,80)(77,79) );
G=PermutationGroup([(1,61,16,66),(2,67,17,62),(3,63,18,68),(4,69,19,64),(5,65,20,70),(6,75,12,80),(7,71,13,76),(8,77,14,72),(9,73,15,78),(10,79,11,74),(21,33,50,52),(22,53,41,34),(23,35,42,54),(24,55,43,36),(25,37,44,56),(26,57,45,38),(27,39,46,58),(28,59,47,40),(29,31,48,60),(30,51,49,32)], [(1,52,13,57),(2,58,14,53),(3,54,15,59),(4,60,11,55),(5,56,12,51),(6,32,20,37),(7,38,16,33),(8,34,17,39),(9,40,18,35),(10,36,19,31),(21,76,45,61),(22,62,46,77),(23,78,47,63),(24,64,48,79),(25,80,49,65),(26,66,50,71),(27,72,41,67),(28,68,42,73),(29,74,43,69),(30,70,44,75)], [(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)], [(1,6),(2,10),(3,9),(4,8),(5,7),(11,17),(12,16),(13,20),(14,19),(15,18),(21,44),(22,43),(23,42),(24,41),(25,50),(26,49),(27,48),(28,47),(29,46),(30,45),(31,34),(32,33),(35,40),(36,39),(37,38),(51,52),(53,60),(54,59),(55,58),(56,57),(61,65),(62,64),(66,70),(67,69),(71,75),(72,74),(76,80),(77,79)])
Matrix representation ►G ⊆ GL8(𝔽41)
| 18 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 26 | 0 | 7 | 7 |
| 0 | 0 | 0 | 0 | 39 | 30 | 36 | 12 |
| 0 | 0 | 0 | 0 | 13 | 12 | 13 | 2 |
| 0 | 0 | 0 | 0 | 25 | 29 | 2 | 13 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 24 | 34 | 38 | 3 |
| 0 | 0 | 0 | 0 | 6 | 17 | 38 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 | 18 | 6 |
| 0 | 0 | 0 | 0 | 13 | 28 | 35 | 23 |
| 1 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 35 | 6 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 35 | 3 | 16 |
| 0 | 0 | 0 | 0 | 7 | 35 | 16 | 28 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 7 |
| 0 | 0 | 0 | 0 | 0 | 0 | 34 | 7 |
| 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 35 | 1 | 40 |
| 0 | 0 | 0 | 0 | 40 | 35 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 7 | 34 |
G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;
47 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | ··· | 4I | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10N | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 4 | ··· | 4 |
47 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2+ (1+4) | D4⋊6D10 |
| kernel | C42⋊28D10 | C42⋊2D5 | C23.18D10 | C23⋊D10 | Dic5⋊D4 | C5×C4⋊1D4 | C4⋊1D4 | C42 | C2×D4 | C10 | C2 |
| # reps | 1 | 2 | 3 | 3 | 6 | 1 | 2 | 2 | 12 | 3 | 12 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{28}D_{10} % in TeX
G:=Group("C4^2:28D10"); // GroupNames label
G:=SmallGroup(320,1392);
// by ID
G=gap.SmallGroup(320,1392);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations