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G = D5×SD32order 320 = 26·5

Direct product of D5 and SD32

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

Aliases: D5×SD32, C165D10, C805C22, Q161D10, D8.2D10, D10.25D8, Dic5.8D8, C40.16C23, D40.2C22, Dic205C22, C4.4(D4×D5), C52(C2×SD32), (D5×C16)⋊4C2, D8.D53C2, (D5×Q16)⋊3C2, (D5×D8).1C2, C16⋊D55C2, C2.19(D5×D8), (C5×SD32)⋊3C2, (C4×D5).59D4, C10.35(C2×D8), C20.10(C2×D4), C5⋊SD321C2, C52C8.25D4, C52C166C22, (C5×Q16)⋊4C22, (C5×D8).2C22, C8.22(C22×D5), (C8×D5).40C22, SmallGroup(320,540)

Series: Derived Chief Lower central Upper central

C1C40 — D5×SD32
C1C5C10C20C40C8×D5D5×D8 — D5×SD32
C5C10C20C40 — D5×SD32
C1C2C4C8SD32

Generators and relations for D5×SD32
 G = < a,b,c,d | a5=b2=c16=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c7 >

Subgroups: 518 in 90 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C2 [×4], C4, C4 [×3], C22 [×5], C5, C8, C8, C2×C4 [×2], D4 [×3], Q8 [×3], C23, D5 [×2], D5, C10, C10, C16, C16, C2×C8, D8, D8 [×2], Q16, Q16 [×2], C2×D4, C2×Q8, Dic5, Dic5, C20, C20, D10, D10 [×3], C2×C10, C2×C16, SD32, SD32 [×3], C2×D8, C2×Q16, C52C8, C40, Dic10 [×2], C4×D5, C4×D5, D20, C5⋊D4, C5×D4, C5×Q8, C22×D5, C2×SD32, C52C16, C80, C8×D5, D40, Dic20, D4⋊D5, C5⋊Q16, C5×D8, C5×Q16, D4×D5, Q8×D5, D5×C16, C16⋊D5, D8.D5, C5⋊SD32, C5×SD32, D5×D8, D5×Q16, D5×SD32
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], SD32 [×2], C2×D8, C22×D5, C2×SD32, D4×D5, D5×D8, D5×SD32

Smallest permutation representation of D5×SD32
On 80 points
Generators in S80
(1 45 26 53 71)(2 46 27 54 72)(3 47 28 55 73)(4 48 29 56 74)(5 33 30 57 75)(6 34 31 58 76)(7 35 32 59 77)(8 36 17 60 78)(9 37 18 61 79)(10 38 19 62 80)(11 39 20 63 65)(12 40 21 64 66)(13 41 22 49 67)(14 42 23 50 68)(15 43 24 51 69)(16 44 25 52 70)
(1 79)(2 80)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 73)(12 74)(13 75)(14 76)(15 77)(16 78)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 27)(19 25)(20 32)(21 23)(22 30)(24 28)(29 31)(33 41)(34 48)(35 39)(36 46)(38 44)(40 42)(43 47)(49 57)(50 64)(51 55)(52 62)(54 60)(56 58)(59 63)(65 77)(66 68)(67 75)(69 73)(70 80)(72 78)(74 76)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A10B10C10D16A16B16C16D16E16F16G16H20A20B20C20D40A40B40C40D80A···80H
order1222224444558888101010101616161616161616202020204040404080···80
size11558402810402222101022161622221010101044161644444···4

44 irreducible representations

dim11111111222222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D5D8D8D10D10D10SD32D4×D5D5×D8D5×SD32
kernelD5×SD32D5×C16C16⋊D5D8.D5C5⋊SD32C5×SD32D5×D8D5×Q16C52C8C4×D5SD32Dic5D10C16D8Q16D5C4C2C1
# reps11111111112222228248

Matrix representation of D5×SD32 in GL4(𝔽241) generated by

0100
24018900
0010
0001
,
0100
1000
002400
000240
,
1000
0100
0062222
0010297
,
1000
0100
0024054
0001
G:=sub<GL(4,GF(241))| [0,240,0,0,1,189,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,240,0,0,0,0,240],[1,0,0,0,0,1,0,0,0,0,62,102,0,0,222,97],[1,0,0,0,0,1,0,0,0,0,240,0,0,0,54,1] >;

D5×SD32 in GAP, Magma, Sage, TeX

D_5\times {\rm SD}_{32}
% in TeX

G:=Group("D5xSD32");
// GroupNames label

G:=SmallGroup(320,540);
// by ID

G=gap.SmallGroup(320,540);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,135,184,346,185,192,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^16=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^7>;
// generators/relations

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