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G = C42⋊D10order 320 = 26·5

1st semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C421D10, M4(2)⋊16D10, C4≀C25D5, (D4×D5)⋊4C4, (Q8×D5)⋊4C4, D42D54C4, Q82D54C4, D4.11(C4×D5), C4.201(D4×D5), Q8.11(C4×D5), D204C44C2, D207C46C2, (C4×C20)⋊10C22, C4○D4.19D10, (C4×D5).104D4, D20.19(C2×C4), C20.360(C2×D4), C42⋊D59C2, (D5×M4(2))⋊9C2, C22.28(D4×D5), D42Dic52C2, C20.53(C22×C4), (C2×Dic5).45D4, (C4×Dic5)⋊3C22, (C22×D5).31D4, C4.Dic53C22, (C2×C20).261C23, Dic10.20(C2×C4), C54(C42⋊C22), C4○D20.10C22, D10.27(C22⋊C4), (C5×M4(2))⋊14C22, Dic5.39(C22⋊C4), (C5×C4≀C2)⋊6C2, C4.18(C2×C4×D5), (D5×C4○D4).2C2, (C4×D5).5(C2×C4), (C5×D4).19(C2×C4), (C2×C10).25(C2×D4), (C5×Q8).20(C2×C4), C2.26(D5×C22⋊C4), (C2×C4×D5).33C22, C10.66(C2×C22⋊C4), (C5×C4○D4).2C22, (C2×C4).368(C22×D5), SmallGroup(320,448)

Series: Derived Chief Lower central Upper central

C1C20 — C42⋊D10
C1C5C10C20C2×C20C2×C4×D5D5×C4○D4 — C42⋊D10
C5C10C20 — C42⋊D10
C1C4C2×C4C4≀C2

Generators and relations for C42⋊D10
 G = < a,b,c,d | a4=b4=c10=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 638 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], D5 [×3], C10, C10 [×2], C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2) [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×4], C2×C10, C2×C10, C4≀C2, C4≀C2 [×3], C42⋊C2, C2×M4(2), C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5 [×4], C4×D5 [×3], D20, D20, C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C22×D5, C22×D5, C42⋊C22, C8×D5, C8⋊D5, C4.Dic5, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×M4(2), C2×C4×D5, C2×C4×D5, C4○D20, C4○D20, D4×D5, D4×D5, D42D5, D42D5, Q8×D5, Q82D5, C5×C4○D4, D204C4, D207C4, D42Dic5, C5×C4≀C2, C42⋊D5, D5×M4(2), D5×C4○D4, C42⋊D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], C22×D5, C42⋊C22, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, C42⋊D10

Smallest permutation representation of C42⋊D10
On 80 points
Generators in S80
(1 78 63 27)(2 18)(3 80 65 29)(4 20)(5 72 67 21)(6 12)(7 74 69 23)(8 14)(9 76 61 25)(10 16)(11 59 45 34)(13 51 47 36)(15 53 49 38)(17 55 41 40)(19 57 43 32)(22 35)(24 37)(26 39)(28 31)(30 33)(42 64)(44 66)(46 68)(48 70)(50 62)(52 75)(54 77)(56 79)(58 71)(60 73)
(1 40 63 55)(2 56 64 31)(3 32 65 57)(4 58 66 33)(5 34 67 59)(6 60 68 35)(7 36 69 51)(8 52 70 37)(9 38 61 53)(10 54 62 39)(11 21 45 72)(12 73 46 22)(13 23 47 74)(14 75 48 24)(15 25 49 76)(16 77 50 26)(17 27 41 78)(18 79 42 28)(19 29 43 80)(20 71 44 30)
(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 26)(2 25)(3 24)(4 23)(5 22)(6 21)(7 30)(8 29)(9 28)(10 27)(11 60)(12 59)(13 58)(14 57)(15 56)(16 55)(17 54)(18 53)(19 52)(20 51)(31 49)(32 48)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 50)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(70 80)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K5A5B8A8B8C8D10A10B10C10D10E10F20A20B20C20D20E···20N20O20P40A40B40C40D
order1222222444444444445588881010101010102020202020···20202040404040
size112410102011244410102020202244202022448822224···4888888

50 irreducible representations

dim1111111111112222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4D4D4D5D10D10D10C4×D5C4×D5C42⋊C22D4×D5D4×D5C42⋊D10
kernelC42⋊D10D204C4D207C4D42Dic5C5×C4≀C2C42⋊D5D5×M4(2)D5×C4○D4D4×D5D42D5Q8×D5Q82D5C4×D5C2×Dic5C22×D5C4≀C2C42M4(2)C4○D4D4Q8C5C4C22C1
# reps1111111122222112222442228

Matrix representation of C42⋊D10 in GL6(𝔽41)

900000
090000
0040404037
0000320
000900
002139231
,
100000
010000
009000
0003200
0000320
00025259
,
35350000
6400000
000010
0040404037
001000
000001
,
35350000
4060000
000100
001000
0040404037
000001

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,21,0,0,40,0,9,39,0,0,40,32,0,23,0,0,37,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,25,0,0,0,0,32,25,0,0,0,0,0,9],[35,6,0,0,0,0,35,40,0,0,0,0,0,0,0,40,1,0,0,0,0,40,0,0,0,0,1,40,0,0,0,0,0,37,0,1],[35,40,0,0,0,0,35,6,0,0,0,0,0,0,0,1,40,0,0,0,1,0,40,0,0,0,0,0,40,0,0,0,0,0,37,1] >;

C42⋊D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes D_{10}
% in TeX

G:=Group("C4^2:D10");
// GroupNames label

G:=SmallGroup(320,448);
// by ID

G=gap.SmallGroup(320,448);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,58,136,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,c*a*c^-1=a*b=b*a,d*a*d=a*b^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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