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G = D5×C4.10D4order 320 = 26·5

Direct product of D5 and C4.10D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×C4.10D4, M4(2).20D10, (C4×D5).2D4, C4.150(D4×D5), C20.95(C2×D4), (C2×C20).7C23, (C2×Q8).94D10, C4.12D207C2, C20.10D43C2, (C2×Dic10).7C4, (D5×M4(2)).7C2, (Q8×C10).5C22, C4.Dic5.4C22, D10.54(C22⋊C4), Dic5.22(C22⋊C4), (C2×Dic10).50C22, (C5×M4(2)).12C22, (C2×C4×D5).2C4, (C2×Q8×D5).1C2, (C2×C4).6(C4×D5), C53(C2×C4.10D4), (C2×C4×D5).7C22, C22.16(C2×C4×D5), (C2×C20).20(C2×C4), C2.15(D5×C22⋊C4), (C2×C4).7(C22×D5), (C5×C4.10D4)⋊5C2, C10.55(C2×C22⋊C4), (C2×Dic5).3(C2×C4), (C2×C10).111(C22×C4), (C22×D5).100(C2×C4), SmallGroup(320,377)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D5×C4.10D4
C1C5C10C20C2×C20C2×C4×D5C2×Q8×D5 — D5×C4.10D4
C5C10C2×C10 — D5×C4.10D4
C1C2C2×C4C4.10D4

Generators and relations for D5×C4.10D4
 G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >

Subgroups: 526 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], Q8 [×8], C23, D5 [×2], D5, C10, C10, C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4 [×3], C2×Q8, C2×Q8 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.10D4, C4.10D4 [×3], C2×M4(2) [×2], C22×Q8, C52C8 [×2], C40 [×2], Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C2×C4.10D4, C8×D5 [×2], C8⋊D5 [×2], C4.Dic5 [×2], C5×M4(2) [×2], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], Q8×C10, C4.12D20 [×2], C20.10D4, C5×C4.10D4, D5×M4(2) [×2], C2×Q8×D5, D5×C4.10D4
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], C4.10D4 [×2], C2×C22⋊C4, C4×D5 [×2], C22×D5, C2×C4.10D4, C2×C4×D5, D4×D5 [×2], D5×C22⋊C4, D5×C4.10D4

Smallest permutation representation of D5×C4.10D4
On 80 points
Generators in S80
(1 59 51 46 65)(2 60 52 47 66)(3 61 53 48 67)(4 62 54 41 68)(5 63 55 42 69)(6 64 56 43 70)(7 57 49 44 71)(8 58 50 45 72)(9 27 38 21 75)(10 28 39 22 76)(11 29 40 23 77)(12 30 33 24 78)(13 31 34 17 79)(14 32 35 18 80)(15 25 36 19 73)(16 26 37 20 74)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 79 77 75)(74 76 78 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 29 7 31 5 25 3 27)(2 30 4 28 6 26 8 32)(9 65 11 71 13 69 15 67)(10 70 16 72 14 66 12 68)(17 55 19 53 21 51 23 49)(18 52 24 54 22 56 20 50)(33 62 39 64 37 58 35 60)(34 63 36 61 38 59 40 57)(41 76 43 74 45 80 47 78)(42 73 48 75 46 77 44 79)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12222244444444558888888810101010202020202020202040···40
size1125510224410102020224444202020202244444488888···8

44 irreducible representations

dim1111111122222448
type++++++++++-+-
imageC1C2C2C2C2C2C4C4D4D5D10D10C4×D5C4.10D4D4×D5D5×C4.10D4
kernelD5×C4.10D4C4.12D20C20.10D4C5×C4.10D4D5×M4(2)C2×Q8×D5C2×Dic10C2×C4×D5C4×D5C4.10D4M4(2)C2×Q8C2×C4D5C4C1
# reps1211214442428242

Matrix representation of D5×C4.10D4 in GL6(𝔽41)

3510000
5400000
001000
000100
000010
000001
,
40400000
010000
0040000
0004000
0000400
0000040
,
100000
010000
000100
0040000
003294023
00040321
,
4000000
0400000
0000400
003294023
0004000
00320032
,
4000000
0400000
0018233736
001328826
00393500
003628636

G:=sub<GL(6,GF(41))| [35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,32,0,0,0,1,0,9,40,0,0,0,0,40,32,0,0,0,0,23,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,32,0,0,0,9,40,0,0,0,40,40,0,0,0,0,0,23,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,13,39,36,0,0,23,28,35,28,0,0,37,8,0,6,0,0,36,26,0,36] >;

D5×C4.10D4 in GAP, Magma, Sage, TeX

D_5\times C_4._{10}D_4
% in TeX

G:=Group("D5xC4.10D4");
// GroupNames label

G:=SmallGroup(320,377);
// by ID

G=gap.SmallGroup(320,377);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,58,570,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=d*c*d^-1=c^-1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations

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