metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D5, (C4×C20)⋊9C2, (C4×D5)⋊3C4, C4.22(C4×D5), C20.46(C2×C4), (C4×Dic5)⋊8C2, D10.8(C2×C4), (C2×C4).96D10, C5⋊2(C42⋊C2), C10.3(C4○D4), C2.2(C4○D20), D10⋊C4.7C2, C10.D4⋊17C2, (C2×C20).73C22, (C2×C10).13C23, C10.16(C22×C4), Dic5.11(C2×C4), C22.10(C22×D5), (C2×Dic5).27C22, (C22×D5).17C22, C2.5(C2×C4×D5), (C2×C4×D5).10C2, SmallGroup(160,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D5
G = < a,b,c,d | a4=b4=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 216 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, C23, D5, C10, C10, C42, C42, C22⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C2×C4×D5, C42⋊D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C4×D5, C22×D5, C2×C4×D5, C4○D20, C42⋊D5
►On 80 points
(1 49 9 44)(2 50 10 45)(3 46 6 41)(4 47 7 42)(5 48 8 43)(11 56 16 51)(12 57 17 52)(13 58 18 53)(14 59 19 54)(15 60 20 55)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(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)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(80)| (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,49,9,44)(2,50,10,45)(3,46,6,41)(4,47,7,42)(5,48,8,43)(11,56,16,51)(12,57,17,52)(13,58,18,53)(14,59,19,54)(15,60,20,55)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,71,51,61)(42,72,52,62)(43,73,53,63)(44,74,54,64)(45,75,55,65)(46,76,56,66)(47,77,57,67)(48,78,58,68)(49,79,59,69)(50,80,60,70), (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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,49,9,44),(2,50,10,45),(3,46,6,41),(4,47,7,42),(5,48,8,43),(11,56,16,51),(12,57,17,52),(13,58,18,53),(14,59,19,54),(15,60,20,55),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,71,51,61),(42,72,52,62),(43,73,53,63),(44,74,54,64),(45,75,55,65),(46,76,56,66),(47,77,57,67),(48,78,58,68),(49,79,59,69),(50,80,60,70)], [(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),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77)]])
C42⋊D5 is a maximal subgroup of
C42⋊3F5 D10.5C42 C42.243D10 D10.7C42 C42.185D10 C42⋊D10 C42.30D10 C42.31D10 C4×C4○D20 C42.277D10 D5×C42⋊C2 C42.188D10 C42⋊10D10 C42.93D10 C42.94D10 C42.96D10 C42.97D10 C42.102D10 C42.104D10 C42⋊11D10 C42.108D10 C42⋊12D10 C42⋊16D10 C42.115D10 C42.116D10 C42.122D10 C42.125D10 C42.126D10 C42.232D10 C42.131D10 C42.132D10 C42.133D10 C42.134D10 C42.138D10 C42⋊18D10 C42.141D10 C42⋊21D10 C42.144D10 C42.148D10 C42.151D10 C42.155D10 C42.156D10 C42.160D10 C42⋊24D10 C42.162D10 C42.163D10 C42⋊26D10 C42.168D10 C42.171D10 C42.174D10 C42.176D10 C42.178D10 (D5×C12)⋊C4 (C4×D15)⋊10C4 (D5×Dic3)⋊C4 D30.23(C2×C4) C42⋊2D15
C42⋊D5 is a maximal quotient of
C5⋊2(C42⋊8C4) C5⋊2(C42⋊5C4) C10.51(C4×D4) C22.58(D4×D5) D10⋊2(C4⋊C4) C10.54(C4×D4) C42.282D10 C42.243D10 C42.182D10 C42.185D10 C4×C10.D4 C42⋊4Dic5 C10.92(C4×D4) C4×D10⋊C4 (C2×C42)⋊D5 (D5×C12)⋊C4 (C4×D15)⋊10C4 (D5×Dic3)⋊C4 D30.23(C2×C4) C42⋊2D15
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | C4×D5 | C4○D20 |
| kernel | C42⋊D5 | C4×Dic5 | C10.D4 | D10⋊C4 | C4×C20 | C2×C4×D5 | C4×D5 | C42 | C10 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 2 | 4 | 6 | 8 | 16 |
Matrix representation of C42⋊D5 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 9 | 0 |
| 0 | 0 | 9 |
| 32 | 0 | 0 |
| 0 | 11 | 22 |
| 0 | 28 | 30 |
| 1 | 0 | 0 |
| 0 | 7 | 8 |
| 0 | 40 | 40 |
| 40 | 0 | 0 |
| 0 | 1 | 8 |
| 0 | 0 | 40 |
G:=sub<GL(3,GF(41))| [1,0,0,0,9,0,0,0,9],[32,0,0,0,11,28,0,22,30],[1,0,0,0,7,40,0,8,40],[40,0,0,0,1,0,0,8,40] >;
C42⋊D5 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_5
% in TeX
G:=Group("C4^2:D5"); // GroupNames label
G:=SmallGroup(160,93);
// by ID
G=gap.SmallGroup(160,93);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations