metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊15D4, (C2×D4)⋊43D10, (C2×Q8)⋊32D10, D10⋊6(C4○D4), C20⋊2D4⋊46C2, C20.265(C2×D4), (C22×C4)⋊29D10, C23⋊D10⋊34C2, D10⋊3Q8⋊47C2, (D4×C10)⋊47C22, C4⋊Dic5⋊80C22, (Q8×C10)⋊39C22, (C2×C20).651C23, (C2×C10).314C24, (C22×C20)⋊42C22, C5⋊8(C22.19C24), (C4×Dic5)⋊60C22, C10.164(C22×D4), C23.D5⋊65C22, C10.D4⋊76C22, C22.323(C23×D5), C23.137(C22×D5), C23.18D10⋊34C2, C23.21D10⋊38C2, (C22×C10).240C23, (C2×Dic5).303C23, (C23×D5).127C22, (C22×D5).253C23, D10⋊C4.159C22, (C22×Dic5).259C22, (C2×C4○D4)⋊6D5, (D5×C22×C4)⋊7C2, (C10×C4○D4)⋊6C2, (C4×C5⋊D4)⋊60C2, (C2×C4)⋊14(C5⋊D4), C2.103(D5×C4○D4), (C2×C10).80(C2×D4), C4.100(C2×C5⋊D4), C10.215(C2×C4○D4), (C2×C4×D5).333C22, C22.23(C2×C5⋊D4), C2.37(C22×C5⋊D4), (C2×C4).639(C22×D5), (C2×C5⋊D4).148C22, SmallGroup(320,1500)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×C20)⋊15D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, cac-1=ab10, ad=da, cbc-1=dbd=b9, dcd=c-1 >
Subgroups: 1118 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×C10, C22×C10, C22.19C24, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C2×C4×D5, C2×C4×D5, C22×Dic5, C2×C5⋊D4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C23×D5, C23.21D10, C4×C5⋊D4, C23.18D10, C23⋊D10, C20⋊2D4, D10⋊3Q8, D5×C22×C4, C10×C4○D4, (C2×C20)⋊15D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C5⋊D4, C22×D5, C22.19C24, C2×C5⋊D4, C23×D5, D5×C4○D4, C22×C5⋊D4, (C2×C20)⋊15D4
►On 80 points
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(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 79 27 47)(2 68 28 56)(3 77 29 45)(4 66 30 54)(5 75 31 43)(6 64 32 52)(7 73 33 41)(8 62 34 50)(9 71 35 59)(10 80 36 48)(11 69 37 57)(12 78 38 46)(13 67 39 55)(14 76 40 44)(15 65 21 53)(16 74 22 42)(17 63 23 51)(18 72 24 60)(19 61 25 49)(20 70 26 58)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(41 75)(42 64)(43 73)(44 62)(45 71)(46 80)(47 69)(48 78)(49 67)(50 76)(51 65)(52 74)(53 63)(54 72)(55 61)(56 70)(57 79)(58 68)(59 77)(60 66)
G:=sub<Sym(80)| (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,79,27,47)(2,68,28,56)(3,77,29,45)(4,66,30,54)(5,75,31,43)(6,64,32,52)(7,73,33,41)(8,62,34,50)(9,71,35,59)(10,80,36,48)(11,69,37,57)(12,78,38,46)(13,67,39,55)(14,76,40,44)(15,65,21,53)(16,74,22,42)(17,63,23,51)(18,72,24,60)(19,61,25,49)(20,70,26,58), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,75)(42,64)(43,73)(44,62)(45,71)(46,80)(47,69)(48,78)(49,67)(50,76)(51,65)(52,74)(53,63)(54,72)(55,61)(56,70)(57,79)(58,68)(59,77)(60,66)>;
G:=Group( (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (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), (1,79,27,47)(2,68,28,56)(3,77,29,45)(4,66,30,54)(5,75,31,43)(6,64,32,52)(7,73,33,41)(8,62,34,50)(9,71,35,59)(10,80,36,48)(11,69,37,57)(12,78,38,46)(13,67,39,55)(14,76,40,44)(15,65,21,53)(16,74,22,42)(17,63,23,51)(18,72,24,60)(19,61,25,49)(20,70,26,58), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,75)(42,64)(43,73)(44,62)(45,71)(46,80)(47,69)(48,78)(49,67)(50,76)(51,65)(52,74)(53,63)(54,72)(55,61)(56,70)(57,79)(58,68)(59,77)(60,66) );
G=PermutationGroup([[(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(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)], [(1,79,27,47),(2,68,28,56),(3,77,29,45),(4,66,30,54),(5,75,31,43),(6,64,32,52),(7,73,33,41),(8,62,34,50),(9,71,35,59),(10,80,36,48),(11,69,37,57),(12,78,38,46),(13,67,39,55),(14,76,40,44),(15,65,21,53),(16,74,22,42),(17,63,23,51),(18,72,24,60),(19,61,25,49),(20,70,26,58)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38),(41,75),(42,64),(43,73),(44,62),(45,71),(46,80),(47,69),(48,78),(49,67),(50,76),(51,65),(52,74),(53,63),(54,72),(55,61),(56,70),(57,79),(58,68),(59,77),(60,66)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | ··· | 10R | 20A | ··· | 20H | 20I | ··· | 20T |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 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 | 4 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D5×C4○D4 |
| kernel | (C2×C20)⋊15D4 | C23.21D10 | C4×C5⋊D4 | C23.18D10 | C23⋊D10 | C20⋊2D4 | D10⋊3Q8 | D5×C22×C4 | C10×C4○D4 | C2×C20 | C2×C4○D4 | D10 | C22×C4 | C2×D4 | C2×Q8 | C2×C4 | C2 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 2 | 8 | 6 | 6 | 2 | 16 | 8 |
Matrix representation of (C2×C20)⋊15D4 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 40 | 40 |
| 1 | 1 | 0 | 0 |
| 5 | 6 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 0 | 32 |
| 20 | 3 | 0 | 0 |
| 3 | 21 | 0 | 0 |
| 0 | 0 | 40 | 39 |
| 0 | 0 | 1 | 1 |
| 6 | 7 | 0 | 0 |
| 36 | 35 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 1 | 1 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,40,0,0,0,40],[1,5,0,0,1,6,0,0,0,0,32,0,0,0,0,32],[20,3,0,0,3,21,0,0,0,0,40,1,0,0,39,1],[6,36,0,0,7,35,0,0,0,0,40,1,0,0,0,1] >;
(C2×C20)⋊15D4 in GAP, Magma, Sage, TeX
(C_2\times C_{20})\rtimes_{15}D_4 % in TeX
G:=Group("(C2xC20):15D4"); // GroupNames label
G:=SmallGroup(320,1500);
// by ID
G=gap.SmallGroup(320,1500);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,c*a*c^-1=a*b^10,a*d=d*a,c*b*c^-1=d*b*d=b^9,d*c*d=c^-1>;
// generators/relations