metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊4D10, C4○D20⋊11C4, (C2×D20)⋊24C4, (C4×C20)⋊3C22, C4.83(C2×D20), D20⋊4C4⋊2C2, D20.36(C2×C4), (C2×C20).144D4, (C2×C4).147D20, C20.303(C2×D4), C42⋊C2⋊4D5, (C2×Dic10)⋊23C4, (C22×C10).78D4, C20.70(C22⋊C4), (C2×C20).794C23, C20.169(C22×C4), Dic10.38(C2×C4), C5⋊5(C42⋊C22), C4○D20.38C22, (C22×C4).113D10, C23.21(C5⋊D4), C4.Dic5⋊20C22, C4.10(D10⋊C4), (C22×C20).154C22, C22.25(D10⋊C4), C4.68(C2×C4×D5), (C2×C4).46(C4×D5), (C2×C4○D20).8C2, (C2×C20).267(C2×C4), (C5×C42⋊C2)⋊4C2, (C2×C10).461(C2×D4), (C2×C4).45(C5⋊D4), C10.88(C2×C22⋊C4), (C2×C4.Dic5)⋊10C2, C22.27(C2×C5⋊D4), C2.20(C2×D10⋊C4), (C2×C4).708(C22×D5), (C2×C10).81(C22⋊C4), SmallGroup(320,632)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊4D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=ab2, dad=ab-1, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 590 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊2C8, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C42⋊C22, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C2×C5⋊D4, C22×C20, D20⋊4C4, C2×C4.Dic5, C5×C42⋊C2, C2×C4○D20, C42⋊4D10
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C42⋊C22, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, C42⋊4D10
►On 80 points
(1 63 28 46)(2 69 29 42)(3 65 30 48)(4 61 26 44)(5 67 27 50)(6 64 24 47)(7 70 25 43)(8 66 21 49)(9 62 22 45)(10 68 23 41)(11 71 36 56)(12 77 37 52)(13 73 38 58)(14 79 39 54)(15 75 40 60)(16 76 31 51)(17 72 32 57)(18 78 33 53)(19 74 34 59)(20 80 35 55)
(1 18 10 13)(2 19 6 14)(3 20 7 15)(4 16 8 11)(5 17 9 12)(21 36 26 31)(22 37 27 32)(23 38 28 33)(24 39 29 34)(25 40 30 35)(41 58 46 53)(42 59 47 54)(43 60 48 55)(44 51 49 56)(45 52 50 57)(61 76 66 71)(62 77 67 72)(63 78 68 73)(64 79 69 74)(65 80 70 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 5)(2 4)(6 8)(9 10)(11 19)(12 18)(13 17)(14 16)(15 20)(21 29)(22 28)(23 27)(24 26)(25 30)(31 34)(32 33)(36 39)(37 38)(41 52)(42 51)(43 60)(44 59)(45 58)(46 57)(47 56)(48 55)(49 54)(50 53)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)
G:=sub<Sym(80)| (1,63,28,46)(2,69,29,42)(3,65,30,48)(4,61,26,44)(5,67,27,50)(6,64,24,47)(7,70,25,43)(8,66,21,49)(9,62,22,45)(10,68,23,41)(11,71,36,56)(12,77,37,52)(13,73,38,58)(14,79,39,54)(15,75,40,60)(16,76,31,51)(17,72,32,57)(18,78,33,53)(19,74,34,59)(20,80,35,55), (1,18,10,13)(2,19,6,14)(3,20,7,15)(4,16,8,11)(5,17,9,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,58,46,53)(42,59,47,54)(43,60,48,55)(44,51,49,56)(45,52,50,57)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (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), (1,5)(2,4)(6,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,29)(22,28)(23,27)(24,26)(25,30)(31,34)(32,33)(36,39)(37,38)(41,52)(42,51)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80)>;
G:=Group( (1,63,28,46)(2,69,29,42)(3,65,30,48)(4,61,26,44)(5,67,27,50)(6,64,24,47)(7,70,25,43)(8,66,21,49)(9,62,22,45)(10,68,23,41)(11,71,36,56)(12,77,37,52)(13,73,38,58)(14,79,39,54)(15,75,40,60)(16,76,31,51)(17,72,32,57)(18,78,33,53)(19,74,34,59)(20,80,35,55), (1,18,10,13)(2,19,6,14)(3,20,7,15)(4,16,8,11)(5,17,9,12)(21,36,26,31)(22,37,27,32)(23,38,28,33)(24,39,29,34)(25,40,30,35)(41,58,46,53)(42,59,47,54)(43,60,48,55)(44,51,49,56)(45,52,50,57)(61,76,66,71)(62,77,67,72)(63,78,68,73)(64,79,69,74)(65,80,70,75), (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), (1,5)(2,4)(6,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,29)(22,28)(23,27)(24,26)(25,30)(31,34)(32,33)(36,39)(37,38)(41,52)(42,51)(43,60)(44,59)(45,58)(46,57)(47,56)(48,55)(49,54)(50,53)(61,79)(62,78)(63,77)(64,76)(65,75)(66,74)(67,73)(68,72)(69,71)(70,80) );
G=PermutationGroup([[(1,63,28,46),(2,69,29,42),(3,65,30,48),(4,61,26,44),(5,67,27,50),(6,64,24,47),(7,70,25,43),(8,66,21,49),(9,62,22,45),(10,68,23,41),(11,71,36,56),(12,77,37,52),(13,73,38,58),(14,79,39,54),(15,75,40,60),(16,76,31,51),(17,72,32,57),(18,78,33,53),(19,74,34,59),(20,80,35,55)], [(1,18,10,13),(2,19,6,14),(3,20,7,15),(4,16,8,11),(5,17,9,12),(21,36,26,31),(22,37,27,32),(23,38,28,33),(24,39,29,34),(25,40,30,35),(41,58,46,53),(42,59,47,54),(43,60,48,55),(44,51,49,56),(45,52,50,57),(61,76,66,71),(62,77,67,72),(63,78,68,73),(64,79,69,74),(65,80,70,75)], [(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)], [(1,5),(2,4),(6,8),(9,10),(11,19),(12,18),(13,17),(14,16),(15,20),(21,29),(22,28),(23,27),(24,26),(25,30),(31,34),(32,33),(36,39),(37,38),(41,52),(42,51),(43,60),(44,59),(45,58),(46,57),(47,56),(48,55),(49,54),(50,53),(61,79),(62,78),(63,77),(64,76),(65,75),(66,74),(67,73),(68,72),(69,71),(70,80)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C5⋊D4 | C42⋊C22 | C42⋊4D10 |
| kernel | C42⋊4D10 | D20⋊4C4 | C2×C4.Dic5 | C5×C42⋊C2 | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C20 | C22×C10 | C42⋊C2 | C42 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 2 | 8 |
Matrix representation of C42⋊4D10 ►in GL4(𝔽41) generated by
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 17 | 1 | 0 | 0 |
| 40 | 24 | 0 | 0 |
| 11 | 32 | 0 | 0 |
| 9 | 30 | 0 | 0 |
| 0 | 0 | 11 | 32 |
| 0 | 0 | 9 | 30 |
| 40 | 7 | 0 | 0 |
| 34 | 7 | 0 | 0 |
| 0 | 0 | 1 | 34 |
| 0 | 0 | 7 | 34 |
| 7 | 40 | 0 | 0 |
| 7 | 34 | 0 | 0 |
| 0 | 0 | 14 | 11 |
| 0 | 0 | 27 | 27 |
G:=sub<GL(4,GF(41))| [0,0,17,40,0,0,1,24,40,0,0,0,0,40,0,0],[11,9,0,0,32,30,0,0,0,0,11,9,0,0,32,30],[40,34,0,0,7,7,0,0,0,0,1,7,0,0,34,34],[7,7,0,0,40,34,0,0,0,0,14,27,0,0,11,27] >;
C42⋊4D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_4D_{10} % in TeX
G:=Group("C4^2:4D10"); // GroupNames label
G:=SmallGroup(320,632);
// by ID
G=gap.SmallGroup(320,632);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,1123,1684,102,12550]);
// 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*b^2,d*a*d=a*b^-1,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations