metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊20D10, C10.1262+ (1+4), (C2×Q8)⋊8D10, (C4×D20)⋊45C2, (C4×C20)⋊24C22, C22⋊C4⋊34D10, C4.4D4⋊12D5, C42⋊2D5⋊8C2, D10⋊D4⋊42C2, C23⋊D10⋊24C2, C22⋊D20⋊25C2, D10⋊3Q8⋊30C2, (C2×D4).110D10, C4⋊Dic5⋊41C22, (Q8×C10)⋊14C22, Dic5⋊4D4⋊31C2, D10.14(C4○D4), Dic5⋊D4⋊34C2, C20.23D4⋊22C2, (C2×C20).631C23, (C2×C10).222C24, C5⋊8(C22.32C24), (C4×Dic5)⋊36C22, D10.12D4⋊43C2, C2.50(D4⋊8D10), C2.75(D4⋊6D10), D10⋊C4⋊69C22, C23.44(C22×D5), (C2×D20).232C22, (D4×C10).210C22, C23.D10⋊39C2, C10.D4⋊36C22, C22.D20⋊25C2, (C22×C10).52C23, (C22×D5).96C23, (C23×D5).65C22, C22.243(C23×D5), C23.D5.56C22, (C2×Dic5).264C23, (C22×Dic5)⋊27C22, C2.78(D5×C4○D4), (C2×C4×D5)⋊52C22, (D5×C22⋊C4)⋊18C2, C10.189(C2×C4○D4), (C5×C4.4D4)⋊14C2, (C2×C5⋊D4)⋊24C22, (C5×C22⋊C4)⋊30C22, (C2×C4).197(C22×D5), SmallGroup(320,1350)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1070 in 250 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×20], C5, C2×C4 [×5], C2×C4 [×9], D4 [×9], Q8, C23 [×2], C23 [×7], D5 [×4], C10 [×3], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×6], C2×Q8, C24, Dic5 [×5], C20 [×5], D10 [×2], D10 [×12], C2×C10, C2×C10 [×6], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×2], C4×D5 [×3], D20 [×3], C2×Dic5 [×5], C2×Dic5, C5⋊D4 [×5], C2×C20 [×5], C5×D4, C5×Q8, C22×D5 [×3], C22×D5 [×4], C22×C10 [×2], C22.32C24, C4×Dic5, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5 [×3], C2×D20 [×2], C22×Dic5, C2×C5⋊D4 [×4], D4×C10, Q8×C10, C23×D5, C4×D20, C42⋊2D5, C23.D10, D5×C22⋊C4, Dic5⋊4D4, C22⋊D20, D10.12D4, D10⋊D4 [×2], C22.D20, C23⋊D10, Dic5⋊D4, D10⋊3Q8, C20.23D4, C5×C4.4D4, C42⋊20D10
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ (1+4) [×2], C22×D5 [×7], C22.32C24, C23×D5, D4⋊6D10, D5×C4○D4, D4⋊8D10, C42⋊20D10
Generators and relations
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=c-1 >
►On 80 points
(1 77 10 55)(2 73 6 51)(3 79 7 57)(4 75 8 53)(5 71 9 59)(11 76 18 54)(12 72 19 60)(13 78 20 56)(14 74 16 52)(15 80 17 58)(21 49 26 63)(22 69 27 45)(23 41 28 65)(24 61 29 47)(25 43 30 67)(31 48 36 62)(32 68 37 44)(33 50 38 64)(34 70 39 46)(35 42 40 66)
(1 27 12 38)(2 23 13 34)(3 29 14 40)(4 25 15 36)(5 21 11 32)(6 28 20 39)(7 24 16 35)(8 30 17 31)(9 26 18 37)(10 22 19 33)(41 78 70 73)(42 57 61 52)(43 80 62 75)(44 59 63 54)(45 72 64 77)(46 51 65 56)(47 74 66 79)(48 53 67 58)(49 76 68 71)(50 55 69 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)
(1 13)(2 12)(3 11)(4 15)(5 14)(6 19)(7 18)(8 17)(9 16)(10 20)(21 24)(22 23)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)(37 40)(38 39)(41 50)(42 49)(43 48)(44 47)(45 46)(51 55)(52 54)(56 60)(57 59)(61 68)(62 67)(63 66)(64 65)(69 70)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,77,10,55)(2,73,6,51)(3,79,7,57)(4,75,8,53)(5,71,9,59)(11,76,18,54)(12,72,19,60)(13,78,20,56)(14,74,16,52)(15,80,17,58)(21,49,26,63)(22,69,27,45)(23,41,28,65)(24,61,29,47)(25,43,30,67)(31,48,36,62)(32,68,37,44)(33,50,38,64)(34,70,39,46)(35,42,40,66), (1,27,12,38)(2,23,13,34)(3,29,14,40)(4,25,15,36)(5,21,11,32)(6,28,20,39)(7,24,16,35)(8,30,17,31)(9,26,18,37)(10,22,19,33)(41,78,70,73)(42,57,61,52)(43,80,62,75)(44,59,63,54)(45,72,64,77)(46,51,65,56)(47,74,66,79)(48,53,67,58)(49,76,68,71)(50,55,69,60), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,19)(7,18)(8,17)(9,16)(10,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,50)(42,49)(43,48)(44,47)(45,46)(51,55)(52,54)(56,60)(57,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,77,10,55)(2,73,6,51)(3,79,7,57)(4,75,8,53)(5,71,9,59)(11,76,18,54)(12,72,19,60)(13,78,20,56)(14,74,16,52)(15,80,17,58)(21,49,26,63)(22,69,27,45)(23,41,28,65)(24,61,29,47)(25,43,30,67)(31,48,36,62)(32,68,37,44)(33,50,38,64)(34,70,39,46)(35,42,40,66), (1,27,12,38)(2,23,13,34)(3,29,14,40)(4,25,15,36)(5,21,11,32)(6,28,20,39)(7,24,16,35)(8,30,17,31)(9,26,18,37)(10,22,19,33)(41,78,70,73)(42,57,61,52)(43,80,62,75)(44,59,63,54)(45,72,64,77)(46,51,65,56)(47,74,66,79)(48,53,67,58)(49,76,68,71)(50,55,69,60), (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), (1,13)(2,12)(3,11)(4,15)(5,14)(6,19)(7,18)(8,17)(9,16)(10,20)(21,24)(22,23)(25,30)(26,29)(27,28)(31,36)(32,35)(33,34)(37,40)(38,39)(41,50)(42,49)(43,48)(44,47)(45,46)(51,55)(52,54)(56,60)(57,59)(61,68)(62,67)(63,66)(64,65)(69,70)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([(1,77,10,55),(2,73,6,51),(3,79,7,57),(4,75,8,53),(5,71,9,59),(11,76,18,54),(12,72,19,60),(13,78,20,56),(14,74,16,52),(15,80,17,58),(21,49,26,63),(22,69,27,45),(23,41,28,65),(24,61,29,47),(25,43,30,67),(31,48,36,62),(32,68,37,44),(33,50,38,64),(34,70,39,46),(35,42,40,66)], [(1,27,12,38),(2,23,13,34),(3,29,14,40),(4,25,15,36),(5,21,11,32),(6,28,20,39),(7,24,16,35),(8,30,17,31),(9,26,18,37),(10,22,19,33),(41,78,70,73),(42,57,61,52),(43,80,62,75),(44,59,63,54),(45,72,64,77),(46,51,65,56),(47,74,66,79),(48,53,67,58),(49,76,68,71),(50,55,69,60)], [(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)], [(1,13),(2,12),(3,11),(4,15),(5,14),(6,19),(7,18),(8,17),(9,16),(10,20),(21,24),(22,23),(25,30),(26,29),(27,28),(31,36),(32,35),(33,34),(37,40),(38,39),(41,50),(42,49),(43,48),(44,47),(45,46),(51,55),(52,54),(56,60),(57,59),(61,68),(62,67),(63,66),(64,65),(69,70),(71,79),(72,78),(73,77),(74,76)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 4 | 5 | 0 | 0 | 0 | 0 |
| 38 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 9 | 0 | 0 |
| 0 | 0 | 32 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 9 |
| 0 | 0 | 0 | 0 | 32 | 30 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 34 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 7 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 23 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 0 | 0 |
| 0 | 0 | 40 | 34 | 0 | 0 |
| 0 | 0 | 0 | 0 | 34 | 34 |
| 0 | 0 | 0 | 0 | 1 | 7 |
G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[4,38,0,0,0,0,5,37,0,0,0,0,0,0,11,32,0,0,0,0,9,30,0,0,0,0,0,0,11,32,0,0,0,0,9,30],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,34,7,0,0,0,0,34,1],[1,23,0,0,0,0,0,40,0,0,0,0,0,0,7,40,0,0,0,0,7,34,0,0,0,0,0,0,34,1,0,0,0,0,34,7] >;
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D5 | C4○D4 | D10 | D10 | D10 | D10 | 2+ (1+4) | D4⋊6D10 | D5×C4○D4 | D4⋊8D10 |
| kernel | C42⋊20D10 | C4×D20 | C42⋊2D5 | C23.D10 | D5×C22⋊C4 | Dic5⋊4D4 | C22⋊D20 | D10.12D4 | D10⋊D4 | C22.D20 | C23⋊D10 | Dic5⋊D4 | D10⋊3Q8 | C20.23D4 | C5×C4.4D4 | C4.4D4 | D10 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 |
In GAP, Magma, Sage, TeX
C_4^2\rtimes_{20}D_{10} % in TeX
G:=Group("C4^2:20D10"); // GroupNames label
G:=SmallGroup(320,1350);
// by ID
G=gap.SmallGroup(320,1350);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,675,570,80,12550]);
// 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,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations