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G = C42⋊D5order 160 = 25·5

1st semidirect product of C42 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C421D5, (C4×C20)⋊9C2, (C4×D5)⋊3C4, C4.22(C4×D5), C20.46(C2×C4), (C4×Dic5)⋊8C2, D10.8(C2×C4), (C2×C4).96D10, C52(C42⋊C2), C10.3(C4○D4), C2.2(C4○D20), D10⋊C4.7C2, C10.D417C2, (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

C1C10 — C42⋊D5
C1C5C10C2×C10C22×D5C2×C4×D5 — C42⋊D5
C5C10 — C42⋊D5
C1C2×C4C42

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 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C2×C4, C2×C4 [×2], C2×C4 [×7], C23, D5 [×2], C10, C10 [×2], C42, C42, C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C42⋊C2, C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C4×Dic5, C10.D4 [×2], D10⋊C4 [×2], C4×C20, C2×C4×D5, C42⋊D5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C4×D5 [×2], C22×D5, C2×C4×D5, C4○D20 [×2], C42⋊D5

Smallest permutation representation of C42⋊D5
On 80 points
Generators in S80
(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
C423F5  D10.5C42  C42.243D10  D10.7C42  C42.185D10  C42⋊D10  C42.30D10  C42.31D10  C4×C4○D20  C42.277D10  D5×C42⋊C2  C42.188D10  C4210D10  C42.93D10  C42.94D10  C42.96D10  C42.97D10  C42.102D10  C42.104D10  C4211D10  C42.108D10  C4212D10  C4216D10  C42.115D10  C42.116D10  C42.122D10  C42.125D10  C42.126D10  C42.232D10  C42.131D10  C42.132D10  C42.133D10  C42.134D10  C42.138D10  C4218D10  C42.141D10  C4221D10  C42.144D10  C42.148D10  C42.151D10  C42.155D10  C42.156D10  C42.160D10  C4224D10  C42.162D10  C42.163D10  C4226D10  C42.168D10  C42.171D10  C42.174D10  C42.176D10  C42.178D10  (D5×C12)⋊C4  (C4×D15)⋊10C4  (D5×Dic3)⋊C4  D30.23(C2×C4)  C422D15
C42⋊D5 is a maximal quotient of
C52(C428C4)  C52(C425C4)  C10.51(C4×D4)  C22.58(D4×D5)  D102(C4⋊C4)  C10.54(C4×D4)  C42.282D10  C42.243D10  C42.182D10  C42.185D10  C4×C10.D4  C424Dic5  C10.92(C4×D4)  C4×D10⋊C4  (C2×C42)⋊D5  (D5×C12)⋊C4  (C4×D15)⋊10C4  (D5×Dic3)⋊C4  D30.23(C2×C4)  C422D15

52 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B10A···10F20A···20X
order122222444444444···45510···1020···20
size111110101111222210···10222···22···2

52 irreducible representations

dim111111122222
type++++++++
imageC1C2C2C2C2C2C4D5C4○D4D10C4×D5C4○D20
kernelC42⋊D5C4×Dic5C10.D4D10⋊C4C4×C20C2×C4×D5C4×D5C42C10C2×C4C4C2
# reps1122118246816

Matrix representation of C42⋊D5 in GL3(𝔽41) generated by

100
090
009
,
3200
01122
02830
,
100
078
04040
,
4000
018
0040
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

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