direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C5×M5(2)⋊C2, D8.2C20, C20.62D8, C40.96D4, M5(2)⋊6C10, C8.2(C2×C20), (C5×D8).6C4, C4.11(C5×D8), C8.16(C5×D4), C40.84(C2×C4), (C2×D8).5C10, C8.C4⋊2C10, (C10×D8).12C2, (C2×C20).280D4, (C5×M5(2))⋊14C2, (C2×C10).24SD16, C22.3(C5×SD16), (C2×C40).266C22, C10.56(D4⋊C4), C20.120(C22⋊C4), (C2×C4).11(C5×D4), C4.5(C5×C22⋊C4), (C2×C8).13(C2×C10), (C5×C8.C4)⋊11C2, C2.10(C5×D4⋊C4), SmallGroup(320,166)
Series: Derived ►Chief ►Lower central ►Upper central
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
►On 80 points
(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