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G = C20.17D4order 160 = 25·5

17th non-split extension by C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.17D4, C23.7D10, (C2×D4).6D5, (C4×Dic5)⋊5C2, (D4×C10).5C2, (C2×C4).50D10, C10.48(C2×D4), C4.7(C5⋊D4), C53(C4.4D4), C23.D59C2, (C2×Dic10)⋊10C2, C10.30(C4○D4), (C2×C20).33C22, (C2×C10).51C23, C2.16(D42D5), C22.58(C22×D5), (C22×C10).19C22, (C2×Dic5).18C22, C2.12(C2×C5⋊D4), SmallGroup(160,157)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.17D4
C1C5C10C2×C10C2×Dic5C4×Dic5 — C20.17D4
C5C2×C10 — C20.17D4
C1C22C2×D4

Generators and relations for C20.17D4
 G = < a,b,c | a20=b4=1, c2=a10, bab-1=a9, cac-1=a-1, cbc-1=a10b-1 >

Subgroups: 208 in 76 conjugacy classes, 33 normal (13 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, Dic5 [×4], C20 [×2], C2×C10, C2×C10 [×6], C4.4D4, Dic10 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C4×Dic5, C23.D5 [×4], C2×Dic10, D4×C10, C20.17D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C5⋊D4 [×2], C22×D5, D42D5 [×2], C2×C5⋊D4, C20.17D4

Smallest permutation representation of C20.17D4
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 67 36 52)(2 76 37 41)(3 65 38 50)(4 74 39 59)(5 63 40 48)(6 72 21 57)(7 61 22 46)(8 70 23 55)(9 79 24 44)(10 68 25 53)(11 77 26 42)(12 66 27 51)(13 75 28 60)(14 64 29 49)(15 73 30 58)(16 62 31 47)(17 71 32 56)(18 80 33 45)(19 69 34 54)(20 78 35 43)
(1 62 11 72)(2 61 12 71)(3 80 13 70)(4 79 14 69)(5 78 15 68)(6 77 16 67)(7 76 17 66)(8 75 18 65)(9 74 19 64)(10 73 20 63)(21 42 31 52)(22 41 32 51)(23 60 33 50)(24 59 34 49)(25 58 35 48)(26 57 36 47)(27 56 37 46)(28 55 38 45)(29 54 39 44)(30 53 40 43)

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

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

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

C20.17D4 is a maximal subgroup of
C23.D20  (C2×D4).F5  (D4×C10).C4  D20.1D4  C20⋊Q8⋊C2  Dic10.D4  (C8×Dic5)⋊C2  D20.D4  (C2×D8).D5  C4011D4  C40.22D4  (C5×Q8).D4  C40.31D4  C40.43D4  D20.38D4  2+ 1+4.D5  C42.106D10  C42.229D10  C42.114D10  C42.115D10  C24.32D10  C24.35D10  C245D10  Dic1019D4  C4⋊C4.178D10  C10.362+ 1+4  D2020D4  C10.422+ 1+4  C10.452+ 1+4  C10.742- 1+4  C10.812- 1+4  C10.622+ 1+4  C10.842- 1+4  C42.139D10  D5×C4.4D4  C42.141D10  C42.166D10  C4226D10  C42.238D10  C24.41D10  C24.42D10  C10.1052- 1+4  C10.1072- 1+4  (C2×C20)⋊17D4  C60.89D4  C60.69D4  C23.D5⋊S3  C60.17D4
C20.17D4 is a maximal quotient of
C24.4D10  C23⋊Dic10  C23.14D20  (C2×Dic5)⋊6Q8  C20.48(C4⋊C4)  (C2×C20).288D4  C42.62D10  C42.213D10  C20.16D8  C42.72D10  C20.Q16  C42.77D10  C24.19D10  C24.20D10  C60.89D4  C60.69D4  C23.D5⋊S3  C60.17D4

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B10A···10F10G···10N20A20B20C20D
order122222444444445510···1010···1020202020
size11114422101010102020222···24···44444

34 irreducible representations

dim111112222224
type+++++++++-
imageC1C2C2C2C2D4D5C4○D4D10D10C5⋊D4D42D5
kernelC20.17D4C4×Dic5C23.D5C2×Dic10D4×C10C20C2×D4C10C2×C4C23C4C2
# reps114112242484

Matrix representation of C20.17D4 in GL4(𝔽41) generated by

37000
01000
004018
0091
,
0100
40000
0092
00132
,
04000
40000
00320
00409
G:=sub<GL(4,GF(41))| [37,0,0,0,0,10,0,0,0,0,40,9,0,0,18,1],[0,40,0,0,1,0,0,0,0,0,9,1,0,0,2,32],[0,40,0,0,40,0,0,0,0,0,32,40,0,0,0,9] >;

C20.17D4 in GAP, Magma, Sage, TeX

C_{20}._{17}D_4
% in TeX

G:=Group("C20.17D4");
// GroupNames label

G:=SmallGroup(160,157);
// by ID

G=gap.SmallGroup(160,157);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,55,506,116,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=1,c^2=a^10,b*a*b^-1=a^9,c*a*c^-1=a^-1,c*b*c^-1=a^10*b^-1>;
// generators/relations

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