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G = C10×C4.D4order 320 = 26·5

Direct product of C10 and C4.D4

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C10×C4.D4, C24.2C20, (C2×D4).6C20, C4.48(D4×C10), (D4×C10).31C4, (C2×C20).515D4, C20.455(C2×D4), (C23×C10).2C4, C23.4(C2×C20), M4(2)⋊8(C2×C10), (C2×M4(2))⋊8C10, (C22×D4).5C10, (C10×M4(2))⋊26C2, (C2×C20).606C23, C22.8(C22×C20), C20.125(C22⋊C4), (D4×C10).284C22, (C5×M4(2))⋊37C22, (C22×C20).407C22, (D4×C2×C10).17C2, (C2×C4).21(C5×D4), (C2×C4).19(C2×C20), C4.10(C5×C22⋊C4), (C2×C20).365(C2×C4), (C2×D4).42(C2×C10), C2.14(C10×C22⋊C4), (C2×C4).1(C22×C10), C10.143(C2×C22⋊C4), (C22×C10).36(C2×C4), (C22×C4).26(C2×C10), C22.18(C5×C22⋊C4), (C2×C10).262(C22×C4), (C2×C10).200(C22⋊C4), SmallGroup(320,912)

Series: Derived Chief Lower central Upper central

C1C22 — C10×C4.D4
C1C2C4C2×C4C2×C20C5×M4(2)C5×C4.D4 — C10×C4.D4
C1C2C22 — C10×C4.D4
C1C2×C10C22×C20 — C10×C4.D4

Generators and relations for C10×C4.D4
 G = < a,b,c,d | a10=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 370 in 186 conjugacy classes, 82 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C22 [×3], C22 [×18], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], D4 [×8], C23, C23 [×4], C23 [×8], C10, C10 [×2], C10 [×6], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C2×D4 [×4], C2×D4 [×4], C24 [×2], C20 [×4], C2×C10 [×3], C2×C10 [×18], C4.D4 [×4], C2×M4(2) [×2], C22×D4, C40 [×4], C2×C20 [×2], C2×C20 [×4], C5×D4 [×8], C22×C10, C22×C10 [×4], C22×C10 [×8], C2×C4.D4, C2×C40 [×2], C5×M4(2) [×4], C5×M4(2) [×2], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C5×C4.D4 [×4], C10×M4(2) [×2], D4×C2×C10, C10×C4.D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C5, C2×C4 [×6], D4 [×4], C23, C10 [×7], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C20 [×4], C2×C10 [×7], C4.D4 [×2], C2×C22⋊C4, C2×C20 [×6], C5×D4 [×4], C22×C10, C2×C4.D4, C5×C22⋊C4 [×4], C22×C20, D4×C10 [×2], C5×C4.D4 [×2], C10×C22⋊C4, C10×C4.D4

Smallest permutation representation of C10×C4.D4
On 80 points
Generators in S80
(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 66 52 50)(2 67 53 41)(3 68 54 42)(4 69 55 43)(5 70 56 44)(6 61 57 45)(7 62 58 46)(8 63 59 47)(9 64 60 48)(10 65 51 49)(11 31 73 22)(12 32 74 23)(13 33 75 24)(14 34 76 25)(15 35 77 26)(16 36 78 27)(17 37 79 28)(18 38 80 29)(19 39 71 30)(20 40 72 21)
(1 74 66 32 52 12 50 23)(2 75 67 33 53 13 41 24)(3 76 68 34 54 14 42 25)(4 77 69 35 55 15 43 26)(5 78 70 36 56 16 44 27)(6 79 61 37 57 17 45 28)(7 80 62 38 58 18 46 29)(8 71 63 39 59 19 47 30)(9 72 64 40 60 20 48 21)(10 73 65 31 51 11 49 22)
(1 28 66 17 52 37 50 79)(2 29 67 18 53 38 41 80)(3 30 68 19 54 39 42 71)(4 21 69 20 55 40 43 72)(5 22 70 11 56 31 44 73)(6 23 61 12 57 32 45 74)(7 24 62 13 58 33 46 75)(8 25 63 14 59 34 47 76)(9 26 64 15 60 35 48 77)(10 27 65 16 51 36 49 78)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D5A5B5C5D8A···8H10A···10L10M···10T10U···10AJ20A···20P40A···40AF
order1222222222444455558···810···1010···1010···1020···2040···40
size1111224444222211114···41···12···24···42···24···4

110 irreducible representations

dim1111111111112244
type++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D4C5×D4C4.D4C5×C4.D4
kernelC10×C4.D4C5×C4.D4C10×M4(2)D4×C2×C10D4×C10C23×C10C2×C4.D4C4.D4C2×M4(2)C22×D4C2×D4C24C2×C20C2×C4C10C2
# reps14214441684161641628

Matrix representation of C10×C4.D4 in GL6(𝔽41)

4000000
0400000
0010000
0001000
0000100
0000010
,
100000
010000
000100
0040000
000001
0000400
,
010000
4000000
000001
000010
0040000
000100
,
100000
0400000
0000400
0000040
0004000
001000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,40,0,0] >;

C10×C4.D4 in GAP, Magma, Sage, TeX

C_{10}\times C_4.D_4
% in TeX

G:=Group("C10xC4.D4");
// GroupNames label

G:=SmallGroup(320,912);
// by ID

G=gap.SmallGroup(320,912);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,7004,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^4=1,c^4=b^2,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^3>;
// generators/relations

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