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G = C422D5order 160 = 25·5

2nd semidirect product of C42 and D5 acting via D5/C5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C422D5, (C4×C20)⋊1C2, (C2×C4).63D10, C51(C422C2), C2.8(C4○D20), C10.6(C4○D4), C10.D41C2, D10⋊C4.1C2, (C2×C20).75C22, (C2×C10).17C23, (C2×Dic5).4C22, (C22×D5).3C22, C22.38(C22×D5), SmallGroup(160,97)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C422D5
C1C5C10C2×C10C22×D5D10⋊C4 — C422D5
C5C2×C10 — C422D5
C1C22C42

Generators and relations for C422D5
 G = < a,b,c,d | a4=b4=c5=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >

Subgroups: 192 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2 [×3], C2, C4 [×6], C22, C22 [×3], C5, C2×C4 [×3], C2×C4 [×3], C23, D5, C10 [×3], C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×3], C20 [×3], D10 [×3], C2×C10, C422C2, C2×Dic5 [×3], C2×C20 [×3], C22×D5, C10.D4 [×3], D10⋊C4 [×3], C4×C20, C422D5
Quotients: C1, C2 [×7], C22 [×7], C23, D5, C4○D4 [×3], D10 [×3], C422C2, C22×D5, C4○D20 [×3], C422D5

Smallest permutation representation of C422D5
On 80 points
Generators in S80
(1 54 14 44)(2 55 15 45)(3 51 11 41)(4 52 12 42)(5 53 13 43)(6 56 16 46)(7 57 17 47)(8 58 18 48)(9 59 19 49)(10 60 20 50)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 65)(26 76 36 66)(27 77 37 67)(28 78 38 68)(29 79 39 69)(30 80 40 70)
(1 29 9 24)(2 30 10 25)(3 26 6 21)(4 27 7 22)(5 28 8 23)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 66 46 61)(42 67 47 62)(43 68 48 63)(44 69 49 64)(45 70 50 65)(51 76 56 71)(52 77 57 72)(53 78 58 73)(54 79 59 74)(55 80 60 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)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 36)(22 40)(23 39)(24 38)(25 37)(26 31)(27 35)(28 34)(29 33)(30 32)(41 46)(42 50)(43 49)(44 48)(45 47)(51 56)(52 60)(53 59)(54 58)(55 57)(61 71)(62 75)(63 74)(64 73)(65 72)(66 76)(67 80)(68 79)(69 78)(70 77)

G:=sub<Sym(80)| (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77)>;

G:=Group( (1,54,14,44)(2,55,15,45)(3,51,11,41)(4,52,12,42)(5,53,13,43)(6,56,16,46)(7,57,17,47)(8,58,18,48)(9,59,19,49)(10,60,20,50)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)(26,76,36,66)(27,77,37,67)(28,78,38,68)(29,79,39,69)(30,80,40,70), (1,29,9,24)(2,30,10,25)(3,26,6,21)(4,27,7,22)(5,28,8,23)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,66,46,61)(42,67,47,62)(43,68,48,63)(44,69,49,64)(45,70,50,65)(51,76,56,71)(52,77,57,72)(53,78,58,73)(54,79,59,74)(55,80,60,75), (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)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,36)(22,40)(23,39)(24,38)(25,37)(26,31)(27,35)(28,34)(29,33)(30,32)(41,46)(42,50)(43,49)(44,48)(45,47)(51,56)(52,60)(53,59)(54,58)(55,57)(61,71)(62,75)(63,74)(64,73)(65,72)(66,76)(67,80)(68,79)(69,78)(70,77) );

G=PermutationGroup([(1,54,14,44),(2,55,15,45),(3,51,11,41),(4,52,12,42),(5,53,13,43),(6,56,16,46),(7,57,17,47),(8,58,18,48),(9,59,19,49),(10,60,20,50),(21,71,31,61),(22,72,32,62),(23,73,33,63),(24,74,34,64),(25,75,35,65),(26,76,36,66),(27,77,37,67),(28,78,38,68),(29,79,39,69),(30,80,40,70)], [(1,29,9,24),(2,30,10,25),(3,26,6,21),(4,27,7,22),(5,28,8,23),(11,36,16,31),(12,37,17,32),(13,38,18,33),(14,39,19,34),(15,40,20,35),(41,66,46,61),(42,67,47,62),(43,68,48,63),(44,69,49,64),(45,70,50,65),(51,76,56,71),(52,77,57,72),(53,78,58,73),(54,79,59,74),(55,80,60,75)], [(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),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,36),(22,40),(23,39),(24,38),(25,37),(26,31),(27,35),(28,34),(29,33),(30,32),(41,46),(42,50),(43,49),(44,48),(45,47),(51,56),(52,60),(53,59),(54,58),(55,57),(61,71),(62,75),(63,74),(64,73),(65,72),(66,76),(67,80),(68,79),(69,78),(70,77)])

C422D5 is a maximal subgroup of
C42.277D10  C4210D10  C42.95D10  C42.96D10  C42.98D10  C42.104D10  C4216D10  C4217D10  C42.118D10  C42.122D10  C42.132D10  C42.133D10  C42.134D10  C42.137D10  C4220D10  C42.150D10  C42.154D10  D5×C422C2  C42.189D10  C4225D10  C42.165D10  C4228D10  C42.180D10  (C4×C20)⋊C6  (C4×Dic3)⋊D5  C10.D4⋊S3  C423D15
C422D5 is a maximal quotient of
(C2×Dic5).Q8  (C22×C4).D10  C10.(C4⋊D4)  (C22×D5).Q8  C10.92(C4×D4)  C425Dic5  (C2×C42)⋊D5  (C4×Dic3)⋊D5  C10.D4⋊S3  C423D15

46 conjugacy classes

class 1 2A2B2C2D4A···4F4G4H4I5A5B10A···10F20A···20X
order122224···44445510···1020···20
size1111202···2202020222···22···2

46 irreducible representations

dim11112222
type++++++
imageC1C2C2C2D5C4○D4D10C4○D20
kernelC422D5C10.D4D10⋊C4C4×C20C42C10C2×C4C2
# reps133126624

Matrix representation of C422D5 in GL4(𝔽41) generated by

15500
42600
0090
0009
,
9000
0900
001740
00124
,
1000
0100
0001
004034
,
11300
04000
0010
003440
G:=sub<GL(4,GF(41))| [15,4,0,0,5,26,0,0,0,0,9,0,0,0,0,9],[9,0,0,0,0,9,0,0,0,0,17,1,0,0,40,24],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,34],[1,0,0,0,13,40,0,0,0,0,1,34,0,0,0,40] >;

C422D5 in GAP, Magma, Sage, TeX

C_4^2\rtimes_2D_5
% in TeX

G:=Group("C4^2:2D5");
// GroupNames label

G:=SmallGroup(160,97);
// by ID

G=gap.SmallGroup(160,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,55,506,86,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^5=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations

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