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## G = C5×M5(2)⋊C2order 320 = 26·5

### Direct product of C5 and M5(2)⋊C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C5×M5(2)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C2×C40 — C5×C8.C4 — C5×M5(2)⋊C2
 Lower central C1 — C2 — C4 — C8 — C5×M5(2)⋊C2
 Upper central C1 — C10 — C2×C20 — C2×C40 — C5×M5(2)⋊C2

Generators and relations for C5×M5(2)⋊C2
G = < a,b,c,d | a5=b16=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b9, dbd=b3c, cd=dc >

Subgroups: 162 in 62 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, C10, C10, C16, C2×C8, M4(2), D8, D8, C2×D4, C20, C2×C10, C2×C10, C8.C4, M5(2), C2×D8, C40, C40, C2×C20, C5×D4, C22×C10, M5(2)⋊C2, C80, C2×C40, C5×M4(2), C5×D8, C5×D8, D4×C10, C5×C8.C4, C5×M5(2), C10×D8, C5×M5(2)⋊C2
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, D8, SD16, C20, C2×C10, D4⋊C4, C2×C20, C5×D4, M5(2)⋊C2, C5×C22⋊C4, C5×D8, C5×SD16, C5×D4⋊C4, C5×M5(2)⋊C2

Smallest permutation representation of C5×M5(2)⋊C2
On 80 points
Generators in S80
(1 32 47 57 67)(2 17 48 58 68)(3 18 33 59 69)(4 19 34 60 70)(5 20 35 61 71)(6 21 36 62 72)(7 22 37 63 73)(8 23 38 64 74)(9 24 39 49 75)(10 25 40 50 76)(11 26 41 51 77)(12 27 42 52 78)(13 28 43 53 79)(14 29 44 54 80)(15 30 45 55 65)(16 31 46 56 66)
(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 10)(4 12)(6 14)(8 16)(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)
(2 12)(3 15)(4 10)(5 13)(6 8)(7 11)(14 16)(17 27)(18 30)(19 25)(20 28)(21 23)(22 26)(29 31)(33 45)(34 40)(35 43)(36 38)(37 41)(42 48)(44 46)(50 60)(51 63)(52 58)(53 61)(54 56)(55 59)(62 64)(65 69)(66 80)(68 78)(70 76)(71 79)(72 74)(73 77)

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

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

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

80 conjugacy classes

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

80 irreducible representations

 dim 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 C4 C5 C10 C10 C10 C20 D4 D4 D8 SD16 C5×D4 C5×D4 C5×D8 C5×SD16 M5(2)⋊C2 C5×M5(2)⋊C2 kernel C5×M5(2)⋊C2 C5×C8.C4 C5×M5(2) C10×D8 C5×D8 M5(2)⋊C2 C8.C4 M5(2) C2×D8 D8 C40 C2×C20 C20 C2×C10 C8 C2×C4 C4 C22 C5 C1 # reps 1 1 1 1 4 4 4 4 4 16 1 1 2 2 4 4 8 8 2 8

Matrix representation of C5×M5(2)⋊C2 in GL4(𝔽241) generated by

 87 0 0 0 0 87 0 0 0 0 87 0 0 0 0 87
,
 185 214 239 0 181 126 1 240 209 30 163 134 95 152 168 8
,
 1 0 0 0 0 1 0 0 185 214 240 0 65 225 0 240
,
 1 0 0 0 240 240 0 0 169 74 11 230 90 133 230 230
G:=sub<GL(4,GF(241))| [87,0,0,0,0,87,0,0,0,0,87,0,0,0,0,87],[185,181,209,95,214,126,30,152,239,1,163,168,0,240,134,8],[1,0,185,65,0,1,214,225,0,0,240,0,0,0,0,240],[1,240,169,90,0,240,74,133,0,0,11,230,0,0,230,230] >;

C5×M5(2)⋊C2 in GAP, Magma, Sage, TeX

C_5\times M_5(2)\rtimes C_2
% in TeX

G:=Group("C5xM5(2):C2");
// GroupNames label

G:=SmallGroup(320,166);
// by ID

G=gap.SmallGroup(320,166);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,3650,136,3511,172,10085,5052,124]);
// 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^9,d*b*d=b^3*c,c*d=d*c>;
// generators/relations

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