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G = Dic54D4order 160 = 25·5

1st semidirect product of Dic5 and D4 acting through Inn(Dic5)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic54D4, C23.14D10, C53(C4×D4), C5⋊D42C4, C2.2(D4×D5), D103(C2×C4), C22⋊C47D5, C221(C4×D5), Dic52(C2×C4), (C2×C4).28D10, C10.18(C2×D4), (C4×Dic5)⋊11C2, C10.D49C2, D10⋊C410C2, C10.22(C4○D4), C2.2(D42D5), (C2×C10).22C23, (C2×C20).51C22, C10.20(C22×C4), (C22×Dic5)⋊1C2, C22.14(C22×D5), (C22×C10).11C22, (C2×Dic5).61C22, (C22×D5).19C22, (C2×C4×D5)⋊9C2, C2.9(C2×C4×D5), (C2×C10)⋊5(C2×C4), (C5×C22⋊C4)⋊9C2, (C2×C5⋊D4).2C2, SmallGroup(160,102)

Series: Derived Chief Lower central Upper central

C1C10 — Dic54D4
C1C5C10C2×C10C22×D5C2×C5⋊D4 — Dic54D4
C5C10 — Dic54D4
C1C22C22⋊C4

Generators and relations for Dic54D4
 G = < a,b,c,d | a10=c4=d2=1, b2=a5, bab-1=cac-1=a-1, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 280 in 94 conjugacy classes, 43 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×7], C22, C22 [×2], C22 [×6], C5, C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4 [×2], C2×D4, Dic5 [×4], Dic5, C20 [×2], D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×3], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C22×D5, C22×C10, C4×Dic5, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C22×Dic5, C2×C5⋊D4, Dic54D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C2×C4×D5, D4×D5, D42D5, Dic54D4

Smallest permutation representation of Dic54D4
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 40 6 35)(2 39 7 34)(3 38 8 33)(4 37 9 32)(5 36 10 31)(11 66 16 61)(12 65 17 70)(13 64 18 69)(14 63 19 68)(15 62 20 67)(21 45 26 50)(22 44 27 49)(23 43 28 48)(24 42 29 47)(25 41 30 46)(51 79 56 74)(52 78 57 73)(53 77 58 72)(54 76 59 71)(55 75 60 80)
(1 63 28 55)(2 62 29 54)(3 61 30 53)(4 70 21 52)(5 69 22 51)(6 68 23 60)(7 67 24 59)(8 66 25 58)(9 65 26 57)(10 64 27 56)(11 46 77 38)(12 45 78 37)(13 44 79 36)(14 43 80 35)(15 42 71 34)(16 41 72 33)(17 50 73 32)(18 49 74 31)(19 48 75 40)(20 47 76 39)
(1 60)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

Dic54D4 is a maximal subgroup of
C5⋊C88D4  C5⋊C8⋊D4  D10⋊M4(2)  Dic5⋊M4(2)  C24.24D10  C24.27D10  C24.31D10  C42.188D10  C42.91D10  C4210D10  C42.96D10  C4×D42D5  C42.104D10  C4×D4×D5  C4211D10  C42.108D10  Dic1023D4  C42.119D10  C24.56D10  C244D10  C24.33D10  C24.35D10  C20⋊(C4○D4)  Dic1020D4  C10.342+ 1+4  C4⋊C421D10  C10.402+ 1+4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C22⋊Q825D5  C4⋊C426D10  Dic1021D4  C10.1182+ 1+4  C10.522+ 1+4  C10.562+ 1+4  C10.572+ 1+4  C4⋊C4.197D10  C10.1212+ 1+4  C10.822- 1+4  C4⋊C428D10  C10.1222+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.662+ 1+4  C10.672+ 1+4  C10.852- 1+4  C42.233D10  C42.137D10  C42.138D10  Dic1010D4  C4220D10  C4221D10  C42.234D10  C42.143D10  C42.160D10  C42.189D10  C42.161D10  C42.162D10  C42.163D10  C42.164D10  Dic54D12  Dic1514D4  D6⋊(C4×D5)  Dic159D4  C1526(C4×D4)  Dic1516D4  Dic1519D4  Dic52S4
Dic54D4 is a maximal quotient of
C10.49(C4×D4)  Dic52C42  C10.51(C4×D4)  C10.52(C4×D4)  D102C42  D103(C4⋊C4)  C10.54(C4×D4)  C10.55(C4×D4)  C55(C8×D4)  D104M4(2)  Dic52M4(2)  C52C826D4  Dic54D8  D4.D55C4  Dic56SD16  D4⋊D56C4  Dic57SD16  C5⋊Q165C4  Dic54Q16  Q8⋊D56C4  M4(2).22D10  C42.196D10  C22⋊C4×Dic5  C24.3D10  C24.4D10  C24.46D10  C24.12D10  C24.13D10  C23.45D20  Dic54D12  Dic1514D4  D6⋊(C4×D5)  Dic159D4  C1526(C4×D4)  Dic1516D4  Dic1519D4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10F10G10H10I10J20A···20H
order122222224444444444445510···101010101020···20
size11112210102222555510101010222···244444···4

40 irreducible representations

dim11111111122222244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10C4×D5D4×D5D42D5
kernelDic54D4C4×Dic5C10.D4D10⋊C4C5×C22⋊C4C2×C4×D5C22×Dic5C2×C5⋊D4C5⋊D4Dic5C22⋊C4C10C2×C4C23C22C2C2
# reps11111111822242822

Matrix representation of Dic54D4 in GL4(𝔽41) generated by

04000
13500
00400
00040
,
9000
133200
00320
00032
,
1000
64000
0014
002040
,
40000
04000
004037
0001
G:=sub<GL(4,GF(41))| [0,1,0,0,40,35,0,0,0,0,40,0,0,0,0,40],[9,13,0,0,0,32,0,0,0,0,32,0,0,0,0,32],[1,6,0,0,0,40,0,0,0,0,1,20,0,0,4,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,37,1] >;

Dic54D4 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_4D_4
% in TeX

G:=Group("Dic5:4D4");
// GroupNames label

G:=SmallGroup(160,102);
// by ID

G=gap.SmallGroup(160,102);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,188,50,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^10=c^4=d^2=1,b^2=a^5,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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