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G = D4.10D10order 160 = 25·5

The non-split extension by D4 of D10 acting through Inn(D4)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.10D10, Q8.11D10, C522- 1+4, C10.13C24, C20.27C23, D10.8C23, D20.14C22, Dic5.8C23, Dic10.14C22, C4○D44D5, (Q8×D5)⋊5C2, C4○D209C2, C5⋊D4.C22, D42D55C2, (C2×C4).25D10, (C2×C10).5C23, (C4×D5).6C22, C2.14(C23×D5), C4.34(C22×D5), (C2×Dic10)⋊14C2, (C2×C20).49C22, (C5×D4).10C22, (C5×Q8).11C22, C22.4(C22×D5), (C2×Dic5).22C22, (C5×C4○D4)⋊5C2, SmallGroup(160,225)

Series: Derived Chief Lower central Upper central

C1C10 — D4.10D10
C1C5C10D10C4×D5Q8×D5 — D4.10D10
C5C10 — D4.10D10
C1C2C4○D4

Generators and relations for D4.10D10
 G = < a,b,c,d | a4=b2=1, c10=d2=a2, bab=cac-1=dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=c9 >

Subgroups: 376 in 146 conjugacy classes, 85 normal (12 characteristic)
C1, C2, C2 [×5], C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], C5, C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], D5 [×2], C10, C10 [×3], C2×Q8 [×5], C4○D4, C4○D4 [×9], Dic5 [×6], C20, C20 [×3], D10 [×2], C2×C10 [×3], 2- 1+4, Dic10 [×9], C4×D5 [×6], D20, C2×Dic5 [×6], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×3], C5×Q8, C2×Dic10 [×3], C4○D20 [×3], D42D5 [×6], Q8×D5 [×2], C5×C4○D4, D4.10D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2- 1+4, C22×D5 [×7], C23×D5, D4.10D10

Smallest permutation representation of D4.10D10
On 80 points
Generators in S80
(1 73 11 63)(2 64 12 74)(3 75 13 65)(4 66 14 76)(5 77 15 67)(6 68 16 78)(7 79 17 69)(8 70 18 80)(9 61 19 71)(10 72 20 62)(21 43 31 53)(22 54 32 44)(23 45 33 55)(24 56 34 46)(25 47 35 57)(26 58 36 48)(27 49 37 59)(28 60 38 50)(29 51 39 41)(30 42 40 52)
(1 57)(2 48)(3 59)(4 50)(5 41)(6 52)(7 43)(8 54)(9 45)(10 56)(11 47)(12 58)(13 49)(14 60)(15 51)(16 42)(17 53)(18 44)(19 55)(20 46)(21 79)(22 70)(23 61)(24 72)(25 63)(26 74)(27 65)(28 76)(29 67)(30 78)(31 69)(32 80)(33 71)(34 62)(35 73)(36 64)(37 75)(38 66)(39 77)(40 68)
(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 39 11 29)(2 28 12 38)(3 37 13 27)(4 26 14 36)(5 35 15 25)(6 24 16 34)(7 33 17 23)(8 22 18 32)(9 31 19 21)(10 40 20 30)(41 73 51 63)(42 62 52 72)(43 71 53 61)(44 80 54 70)(45 69 55 79)(46 78 56 68)(47 67 57 77)(48 76 58 66)(49 65 59 75)(50 74 60 64)

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

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

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

D4.10D10 is a maximal subgroup of
D4.9D20  D4.10D20  D20.38D4  D20.40D4  D4.11D20  D4.13D20  D811D10  D20.47D4  D86D10  D20.44D4  D20.33C23  D20.35C23  C10.C25  D20.37C23  D5×2- 1+4  D20.A4  D20.39D6  C30.C24  C15⋊2- 1+4  D12.29D10  D4.10D30
D4.10D10 is a maximal quotient of
C42.87D10  C42.89D10  C42.90D10  C42.91D10  C42.92D10  C42.94D10  C42.96D10  C42.98D10  C42.99D10  D4×Dic10  C42.105D10  C42.106D10  C42.108D10  D2024D4  Dic1023D4  D46D20  C42.115D10  C42.118D10  Dic1010Q8  Q86Dic10  C42.125D10  Q8×D20  C42.134D10  C42.135D10  C10.682- 1+4  Dic1019D4  C10.352+ 1+4  C10.362+ 1+4  C10.392+ 1+4  C10.732- 1+4  C10.742- 1+4  C10.502+ 1+4  C10.152- 1+4  C10.162- 1+4  Dic1021D4  C10.772- 1+4  C10.572+ 1+4  C10.582+ 1+4  C10.792- 1+4  C10.802- 1+4  C10.812- 1+4  C10.822- 1+4  C10.632+ 1+4  C10.842- 1+4  C10.852- 1+4  C10.692+ 1+4  C42.137D10  C42.139D10  C42.140D10  C42.141D10  Dic1010D4  C42.144D10  C42.145D10  Dic107Q8  C42.147D10  C42.148D10  C42.152D10  C42.154D10  C42.155D10  C42.157D10  C42.159D10  C42.160D10  C42.161D10  C42.162D10  C42.164D10  C42.165D10  C10.1042- 1+4  C10.1052- 1+4  C10.1062- 1+4  C10.1072- 1+4  C10.1472+ 1+4  D20.39D6  C30.C24  C15⋊2- 1+4  D12.29D10  D4.10D30

37 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E···4J5A5B10A10B10C···10H20A20B20C20D20E···20J
order122222244444···455101010···102020202020···20
size112221010222210···1022224···422224···4

37 irreducible representations

dim111111222244
type++++++++++--
imageC1C2C2C2C2C2D5D10D10D102- 1+4D4.10D10
kernelD4.10D10C2×Dic10C4○D20D42D5Q8×D5C5×C4○D4C4○D4C2×C4D4Q8C5C1
# reps133621266214

Matrix representation of D4.10D10 in GL4(𝔽41) generated by

2060
0206
60390
06039
,
25131923
28161822
19231628
18221325
,
0077
003440
343400
7100
,
001427
001127
271400
301400
G:=sub<GL(4,GF(41))| [2,0,6,0,0,2,0,6,6,0,39,0,0,6,0,39],[25,28,19,18,13,16,23,22,19,18,16,13,23,22,28,25],[0,0,34,7,0,0,34,1,7,34,0,0,7,40,0,0],[0,0,27,30,0,0,14,14,14,11,0,0,27,27,0,0] >;

D4.10D10 in GAP, Magma, Sage, TeX

D_4._{10}D_{10}
% in TeX

G:=Group("D4.10D10");
// GroupNames label

G:=SmallGroup(160,225);
// by ID

G=gap.SmallGroup(160,225);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=1,c^10=d^2=a^2,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^9>;
// generators/relations

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