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G = C5×C22⋊SD16order 320 = 26·5

Direct product of C5 and C22⋊SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C22⋊SD16, D4.6(C5×D4), C22⋊C89C10, (C5×D4).40D4, C4.24(D4×C10), C22⋊Q81C10, D4⋊C49C10, (C2×C40)⋊34C22, (C2×SD16)⋊9C10, (C2×C10)⋊10SD16, C20.385(C2×D4), (C2×C20).319D4, C2.6(C10×SD16), C222(C5×SD16), C23.43(C5×D4), C10.97C22≀C2, (C10×SD16)⋊26C2, C10.86(C2×SD16), (Q8×C10)⋊26C22, (C22×D4).8C10, C22.80(D4×C10), (C2×C20).915C23, (C22×C10).165D4, C10.133(C8⋊C22), (D4×C10).295C22, (C22×C20).422C22, C4⋊C42(C2×C10), (C2×C8)⋊6(C2×C10), (C2×Q8)⋊1(C2×C10), (D4×C2×C10).20C2, (C2×C4).28(C5×D4), C2.8(C5×C8⋊C22), (C5×C22⋊C8)⋊26C2, (C5×C4⋊C4)⋊36C22, (C5×C22⋊Q8)⋊28C2, (C5×D4⋊C4)⋊33C2, C2.11(C5×C22≀C2), (C2×D4).53(C2×C10), (C2×C10).636(C2×D4), (C22×C4).40(C2×C10), (C2×C4).90(C22×C10), SmallGroup(320,951)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C22⋊SD16
C1C2C22C2×C4C2×C20Q8×C10C10×SD16 — C5×C22⋊SD16
C1C2C2×C4 — C5×C22⋊SD16
C1C2×C10C22×C20 — C5×C22⋊SD16

Generators and relations for C5×C22⋊SD16
 G = < a,b,c,d,e | a5=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d3 >

Subgroups: 402 in 188 conjugacy classes, 62 normal (30 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×3], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], D4 [×4], D4 [×6], Q8 [×2], C23, C23 [×10], C10 [×3], C10 [×6], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C24, C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×18], C22⋊C8, D4⋊C4 [×2], C22⋊Q8, C2×SD16 [×2], C22×D4, C40 [×2], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C5×Q8 [×2], C22×C10, C22×C10 [×10], C22⋊SD16, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×C40 [×2], C5×SD16 [×4], C22×C20, D4×C10 [×2], D4×C10 [×5], Q8×C10, C23×C10, C5×C22⋊C8, C5×D4⋊C4 [×2], C5×C22⋊Q8, C10×SD16 [×2], D4×C2×C10, C5×C22⋊SD16
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], SD16 [×2], C2×D4 [×3], C2×C10 [×7], C22≀C2, C2×SD16, C8⋊C22, C5×D4 [×6], C22×C10, C22⋊SD16, C5×SD16 [×2], D4×C10 [×3], C5×C22≀C2, C10×SD16, C5×C8⋊C22, C5×C22⋊SD16

Smallest permutation representation of C5×C22⋊SD16
On 80 points
Generators in S80
(1 42 75 40 67)(2 43 76 33 68)(3 44 77 34 69)(4 45 78 35 70)(5 46 79 36 71)(6 47 80 37 72)(7 48 73 38 65)(8 41 74 39 66)(9 25 60 17 52)(10 26 61 18 53)(11 27 62 19 54)(12 28 63 20 55)(13 29 64 21 56)(14 30 57 22 49)(15 31 58 23 50)(16 32 59 24 51)
(1 12)(2 6)(3 14)(4 8)(5 16)(7 10)(9 13)(11 15)(17 21)(18 38)(19 23)(20 40)(22 34)(24 36)(25 29)(26 48)(27 31)(28 42)(30 44)(32 46)(33 37)(35 39)(41 45)(43 47)(49 69)(50 54)(51 71)(52 56)(53 65)(55 67)(57 77)(58 62)(59 79)(60 64)(61 73)(63 75)(66 70)(68 72)(74 78)(76 80)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 73)(58 74)(59 75)(60 76)(61 77)(62 78)(63 79)(64 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)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)(17 35)(18 38)(19 33)(20 36)(21 39)(22 34)(23 37)(24 40)(25 45)(26 48)(27 43)(28 46)(29 41)(30 44)(31 47)(32 42)(49 69)(50 72)(51 67)(52 70)(53 65)(54 68)(55 71)(56 66)(57 77)(58 80)(59 75)(60 78)(61 73)(62 76)(63 79)(64 74)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E5A5B5C5D8A8B8C8D10A···10L10M···10T10U···10AJ20A···20H20I20J20K20L20M···20T40A···40P
order1222222222444445555888810···1010···1010···1020···202020202020···2040···40
size111122444422488111144441···12···24···42···244448···84···4

95 irreducible representations

dim1111111111112222222244
type++++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4D4SD16C5×D4C5×D4C5×D4C5×SD16C8⋊C22C5×C8⋊C22
kernelC5×C22⋊SD16C5×C22⋊C8C5×D4⋊C4C5×C22⋊Q8C10×SD16D4×C2×C10C22⋊SD16C22⋊C8D4⋊C4C22⋊Q8C2×SD16C22×D4C2×C20C5×D4C22×C10C2×C10C2×C4D4C23C22C10C2
# reps112121448484141441641614

Matrix representation of C5×C22⋊SD16 in GL4(𝔽41) generated by

10000
01000
0010
0001
,
40000
04000
00400
0001
,
1000
0100
00400
00040
,
152600
151500
00039
00200
,
1000
04000
00400
00040
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[15,15,0,0,26,15,0,0,0,0,0,20,0,0,39,0],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40] >;

C5×C22⋊SD16 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes {\rm SD}_{16}
% in TeX

G:=Group("C5xC2^2:SD16");
// GroupNames label

G:=SmallGroup(320,951);
// by ID

G=gap.SmallGroup(320,951);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,1766,7004,3511,172]);
// Polycyclic

G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
// generators/relations

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