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## G = C5×C16⋊C22order 320 = 26·5

### Direct product of C5 and C16⋊C22

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×C16⋊C22
 Chief series C1 — C2 — C4 — C8 — C40 — C5×D8 — C5×D16 — C5×C16⋊C22
 Lower central C1 — C2 — C4 — C8 — C5×C16⋊C22
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×C16⋊C22

Generators and relations for C5×C16⋊C22
G = < a,b,c,d | a5=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b7, dbd=b9, cd=dc >

Subgroups: 226 in 90 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C16, C2×C8, D8, D8, D8, SD16, Q16, C2×D4, C4○D4, C20, C20, C2×C10, C2×C10, M5(2), D16, SD32, C2×D8, C4○D8, C40, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C16⋊C22, C80, C2×C40, C5×D8, C5×D8, C5×D8, C5×SD16, C5×Q16, D4×C10, C5×C4○D4, C5×M5(2), C5×D16, C5×SD32, C10×D8, C5×C4○D8, C5×C16⋊C22
Quotients: C1, C2, C22, C5, D4, C23, C10, D8, C2×D4, C2×C10, C2×D8, C5×D4, C22×C10, C16⋊C22, C5×D8, D4×C10, C10×D8, C5×C16⋊C22

Smallest permutation representation of C5×C16⋊C22
On 80 points
Generators in S80
(1 56 70 31 46)(2 57 71 32 47)(3 58 72 17 48)(4 59 73 18 33)(5 60 74 19 34)(6 61 75 20 35)(7 62 76 21 36)(8 63 77 22 37)(9 64 78 23 38)(10 49 79 24 39)(11 50 80 25 40)(12 51 65 26 41)(13 52 66 27 42)(14 53 67 28 43)(15 54 68 29 44)(16 55 69 30 45)
(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 29)(18 20)(19 27)(21 25)(22 32)(24 30)(26 28)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)(49 55)(50 62)(51 53)(52 60)(54 58)(57 63)(59 61)(65 67)(66 74)(68 72)(69 79)(71 77)(73 75)(76 80)
(1 9)(3 11)(5 13)(7 15)(17 25)(19 27)(21 29)(23 31)(34 42)(36 44)(38 46)(40 48)(50 58)(52 60)(54 62)(56 64)(66 74)(68 76)(70 78)(72 80)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 5A 5B 5C 5D 8A 8B 8C 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10T 16A 16B 16C 16D 20A ··· 20H 20I 20J 20K 20L 40A ··· 40H 40I 40J 40K 40L 80A ··· 80P order 1 2 2 2 2 2 4 4 4 5 5 5 5 8 8 8 10 10 10 10 10 10 10 10 10 ··· 10 16 16 16 16 20 ··· 20 20 20 20 20 40 ··· 40 40 40 40 40 80 ··· 80 size 1 1 2 8 8 8 2 2 8 1 1 1 1 2 2 4 1 1 1 1 2 2 2 2 8 ··· 8 4 4 4 4 2 ··· 2 8 8 8 8 2 ··· 2 4 4 4 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D4 D4 D8 D8 C5×D4 C5×D4 C5×D8 C5×D8 C16⋊C22 C5×C16⋊C22 kernel C5×C16⋊C22 C5×M5(2) C5×D16 C5×SD32 C10×D8 C5×C4○D8 C16⋊C22 M5(2) D16 SD32 C2×D8 C4○D8 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 2 2 1 1 4 4 8 8 4 4 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×C16⋊C22 in GL4(𝔽241) generated by

 91 0 0 0 0 91 0 0 0 0 91 0 0 0 0 91
,
 68 0 214 227 22 0 105 174 22 11 83 162 219 219 180 90
,
 1 1 16 215 0 240 24 1 0 0 22 11 0 0 219 219
,
 240 0 100 50 0 240 4 2 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,91,0,0,0,0,91],[68,22,22,219,0,0,11,219,214,105,83,180,227,174,162,90],[1,0,0,0,1,240,0,0,16,24,22,219,215,1,11,219],[240,0,0,0,0,240,0,0,100,4,1,0,50,2,0,1] >;

C5×C16⋊C22 in GAP, Magma, Sage, TeX

C_5\times C_{16}\rtimes C_2^2
% in TeX

G:=Group("C5xC16:C2^2");
// GroupNames label

G:=SmallGroup(320,1010);
// by ID

G=gap.SmallGroup(320,1010);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,3446,4204,2111,242,10085,5052,124]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^16=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^7,d*b*d=b^9,c*d=d*c>;
// generators/relations

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