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G = C5×C4.4D4order 160 = 25·5

Direct product of C5 and C4.4D4

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

Aliases: C5×C4.4D4, C425C10, C20.39D4, C4.4(C5×D4), (C4×C20)⋊12C2, (Q8×C10)⋊9C2, (C2×Q8)⋊2C10, C2.8(D4×C10), C22⋊C45C10, (C2×D4).5C10, C10.71(C2×D4), (D4×C10).12C2, C23.2(C2×C10), C10.44(C4○D4), (C2×C20).66C22, (C2×C10).79C23, (C22×C10).2C22, C22.14(C22×C10), C2.7(C5×C4○D4), (C2×C4).6(C2×C10), (C5×C22⋊C4)⋊13C2, SmallGroup(160,185)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C4.4D4
C1C2C22C2×C10C22×C10C5×C22⋊C4 — C5×C4.4D4
C1C22 — C5×C4.4D4
C1C2×C10 — C5×C4.4D4

Generators and relations for C5×C4.4D4
 G = < a,b,c,d | a5=b4=c4=1, d2=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b2c-1 >

Subgroups: 116 in 76 conjugacy classes, 44 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C10, C10 [×2], C10 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C4.4D4, C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×C10 [×2], C4×C20, C5×C22⋊C4 [×4], D4×C10, Q8×C10, C5×C4.4D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C4○D4 [×2], C2×C10 [×7], C4.4D4, C5×D4 [×2], C22×C10, D4×C10, C5×C4○D4 [×2], C5×C4.4D4

Smallest permutation representation of C5×C4.4D4
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 16 31 76)(2 17 32 77)(3 18 33 78)(4 19 34 79)(5 20 35 80)(6 21 11 26)(7 22 12 27)(8 23 13 28)(9 24 14 29)(10 25 15 30)(36 61 51 66)(37 62 52 67)(38 63 53 68)(39 64 54 69)(40 65 55 70)(41 56 46 71)(42 57 47 72)(43 58 48 73)(44 59 49 74)(45 60 50 75)
(1 56 21 61)(2 57 22 62)(3 58 23 63)(4 59 24 64)(5 60 25 65)(6 36 76 41)(7 37 77 42)(8 38 78 43)(9 39 79 44)(10 40 80 45)(11 51 16 46)(12 52 17 47)(13 53 18 48)(14 54 19 49)(15 55 20 50)(26 66 31 71)(27 67 32 72)(28 68 33 73)(29 69 34 74)(30 70 35 75)
(1 51 31 36)(2 52 32 37)(3 53 33 38)(4 54 34 39)(5 55 35 40)(6 71 11 56)(7 72 12 57)(8 73 13 58)(9 74 14 59)(10 75 15 60)(16 61 76 66)(17 62 77 67)(18 63 78 68)(19 64 79 69)(20 65 80 70)(21 46 26 41)(22 47 27 42)(23 48 28 43)(24 49 29 44)(25 50 30 45)

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

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

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

C5×C4.4D4 is a maximal subgroup of
C42.7D10  C42⋊Dic5  C42.Dic5  C42.61D10  C42.62D10  C42.213D10  D20.23D4  C42.64D10  C42.214D10  C42.65D10  C425D10  D20.14D4  C42.233D10  C42.137D10  C42.138D10  C42.139D10  C42.140D10  C4218D10  C42.141D10  D2010D4  Dic1010D4  C4220D10  C4221D10  C42.234D10  C42.143D10  C42.144D10  C4222D10  C42.145D10

70 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H5A5B5C5D10A···10L10M···10T20A···20X20Y···20AF
order1222224···444555510···1010···1020···2020···20
size1111442···24411111···14···42···24···4

70 irreducible representations

dim11111111112222
type++++++
imageC1C2C2C2C2C5C10C10C10C10D4C4○D4C5×D4C5×C4○D4
kernelC5×C4.4D4C4×C20C5×C22⋊C4D4×C10Q8×C10C4.4D4C42C22⋊C4C2×D4C2×Q8C20C10C4C2
# reps1141144164424816

Matrix representation of C5×C4.4D4 in GL4(𝔽41) generated by

18000
01800
00100
00010
,
40000
04000
00923
00032
,
04000
1000
00320
00032
,
0100
1000
00139
00140
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,10,0,0,0,0,10],[40,0,0,0,0,40,0,0,0,0,9,0,0,0,23,32],[0,1,0,0,40,0,0,0,0,0,32,0,0,0,0,32],[0,1,0,0,1,0,0,0,0,0,1,1,0,0,39,40] >;

C5×C4.4D4 in GAP, Magma, Sage, TeX

C_5\times C_4._4D_4
% in TeX

G:=Group("C5xC4.4D4");
// GroupNames label

G:=SmallGroup(160,185);
// by ID

G=gap.SmallGroup(160,185);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,487,1514,194]);
// Polycyclic

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

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