direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×M4(2).8C22, (C2×D4).7C20, C4.50(D4×C10), (D4×C10).32C4, C4.D4⋊6C10, (C2×C20).517D4, C20.457(C2×D4), (C22×C4).6C20, C23.5(C2×C20), C4.10D4⋊6C10, (C2×M4(2))⋊9C10, (C22×C20).40C4, (C10×M4(2))⋊27C2, (C2×C20).608C23, M4(2).8(C2×C10), C20.162(C22⋊C4), (D4×C10).285C22, C22.10(C22×C20), (Q8×C10).249C22, (C22×C20).409C22, (C5×M4(2)).42C22, (C2×C4).6(C2×C20), (C2×C4○D4).3C10, (C2×C4).121(C5×D4), C4.22(C5×C22⋊C4), (C2×C20).192(C2×C4), (C10×C4○D4).17C2, (C2×D4).43(C2×C10), (C5×C4.D4)⋊13C2, C2.16(C10×C22⋊C4), (C2×C4).3(C22×C10), (C2×Q8).34(C2×C10), C22.3(C5×C22⋊C4), (C5×C4.10D4)⋊13C2, C10.145(C2×C22⋊C4), (C22×C10).37(C2×C4), (C22×C4).28(C2×C10), (C2×C10).94(C22⋊C4), (C2×C10).264(C22×C4), SmallGroup(320,914)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×M4(2).8C22
G = < a,b,c,d,e | a5=b8=c2=d2=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b5, dbd=bc, cd=dc, ece-1=b4c, ede-1=b4cd >
Subgroups: 242 in 150 conjugacy classes, 78 normal (26 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C20, C20, C20, C2×C10, C2×C10, C2×C10, C4.D4, C4.10D4, C2×M4(2), C2×C4○D4, C40, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, M4(2).8C22, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C4.D4, C5×C4.10D4, C10×M4(2), C10×C4○D4, C5×M4(2).8C22
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C23, C10, C22⋊C4, C22×C4, C2×D4, C20, C2×C10, C2×C22⋊C4, C2×C20, C5×D4, C22×C10, M4(2).8C22, C5×C22⋊C4, C22×C20, D4×C10, C10×C22⋊C4, C5×M4(2).8C22
►On 80 points
(1 24 65 51 30)(2 17 66 52 31)(3 18 67 53 32)(4 19 68 54 25)(5 20 69 55 26)(6 21 70 56 27)(7 22 71 49 28)(8 23 72 50 29)(9 80 62 37 42)(10 73 63 38 43)(11 74 64 39 44)(12 75 57 40 45)(13 76 58 33 46)(14 77 59 34 47)(15 78 60 35 48)(16 79 61 36 41)
(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 6)(4 8)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(50 54)(52 56)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)
(1 38)(2 35)(3 36)(4 33)(5 34)(6 39)(7 40)(8 37)(9 72)(10 65)(11 70)(12 71)(13 68)(14 69)(15 66)(16 67)(17 48)(18 41)(19 46)(20 47)(21 44)(22 45)(23 42)(24 43)(25 58)(26 59)(27 64)(28 57)(29 62)(30 63)(31 60)(32 61)(49 75)(50 80)(51 73)(52 78)(53 79)(54 76)(55 77)(56 74)
(1 8 3 2 5 4 7 6)(9 16 11 10 13 12 15 14)(17 20 19 22 21 24 23 18)(25 28 27 30 29 32 31 26)(33 40 35 34 37 36 39 38)(41 44 43 46 45 48 47 42)(49 56 51 50 53 52 55 54)(57 60 59 62 61 64 63 58)(65 72 67 66 69 68 71 70)(73 76 75 78 77 80 79 74)
G:=sub<Sym(80)| (1,24,65,51,30)(2,17,66,52,31)(3,18,67,53,32)(4,19,68,54,25)(5,20,69,55,26)(6,21,70,56,27)(7,22,71,49,28)(8,23,72,50,29)(9,80,62,37,42)(10,73,63,38,43)(11,74,64,39,44)(12,75,57,40,45)(13,76,58,33,46)(14,77,59,34,47)(15,78,60,35,48)(16,79,61,36,41), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80), (1,38)(2,35)(3,36)(4,33)(5,34)(6,39)(7,40)(8,37)(9,72)(10,65)(11,70)(12,71)(13,68)(14,69)(15,66)(16,67)(17,48)(18,41)(19,46)(20,47)(21,44)(22,45)(23,42)(24,43)(25,58)(26,59)(27,64)(28,57)(29,62)(30,63)(31,60)(32,61)(49,75)(50,80)(51,73)(52,78)(53,79)(54,76)(55,77)(56,74), (1,8,3,2,5,4,7,6)(9,16,11,10,13,12,15,14)(17,20,19,22,21,24,23,18)(25,28,27,30,29,32,31,26)(33,40,35,34,37,36,39,38)(41,44,43,46,45,48,47,42)(49,56,51,50,53,52,55,54)(57,60,59,62,61,64,63,58)(65,72,67,66,69,68,71,70)(73,76,75,78,77,80,79,74)>;
G:=Group( (1,24,65,51,30)(2,17,66,52,31)(3,18,67,53,32)(4,19,68,54,25)(5,20,69,55,26)(6,21,70,56,27)(7,22,71,49,28)(8,23,72,50,29)(9,80,62,37,42)(10,73,63,38,43)(11,74,64,39,44)(12,75,57,40,45)(13,76,58,33,46)(14,77,59,34,47)(15,78,60,35,48)(16,79,61,36,41), (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), (2,6)(4,8)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(33,37)(35,39)(42,46)(44,48)(50,54)(52,56)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80), (1,38)(2,35)(3,36)(4,33)(5,34)(6,39)(7,40)(8,37)(9,72)(10,65)(11,70)(12,71)(13,68)(14,69)(15,66)(16,67)(17,48)(18,41)(19,46)(20,47)(21,44)(22,45)(23,42)(24,43)(25,58)(26,59)(27,64)(28,57)(29,62)(30,63)(31,60)(32,61)(49,75)(50,80)(51,73)(52,78)(53,79)(54,76)(55,77)(56,74), (1,8,3,2,5,4,7,6)(9,16,11,10,13,12,15,14)(17,20,19,22,21,24,23,18)(25,28,27,30,29,32,31,26)(33,40,35,34,37,36,39,38)(41,44,43,46,45,48,47,42)(49,56,51,50,53,52,55,54)(57,60,59,62,61,64,63,58)(65,72,67,66,69,68,71,70)(73,76,75,78,77,80,79,74) );
G=PermutationGroup([[(1,24,65,51,30),(2,17,66,52,31),(3,18,67,53,32),(4,19,68,54,25),(5,20,69,55,26),(6,21,70,56,27),(7,22,71,49,28),(8,23,72,50,29),(9,80,62,37,42),(10,73,63,38,43),(11,74,64,39,44),(12,75,57,40,45),(13,76,58,33,46),(14,77,59,34,47),(15,78,60,35,48),(16,79,61,36,41)], [(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)], [(2,6),(4,8),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(33,37),(35,39),(42,46),(44,48),(50,54),(52,56),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80)], [(1,38),(2,35),(3,36),(4,33),(5,34),(6,39),(7,40),(8,37),(9,72),(10,65),(11,70),(12,71),(13,68),(14,69),(15,66),(16,67),(17,48),(18,41),(19,46),(20,47),(21,44),(22,45),(23,42),(24,43),(25,58),(26,59),(27,64),(28,57),(29,62),(30,63),(31,60),(32,61),(49,75),(50,80),(51,73),(52,78),(53,79),(54,76),(55,77),(56,74)], [(1,8,3,2,5,4,7,6),(9,16,11,10,13,12,15,14),(17,20,19,22,21,24,23,18),(25,28,27,30,29,32,31,26),(33,40,35,34,37,36,39,38),(41,44,43,46,45,48,47,42),(49,56,51,50,53,52,55,54),(57,60,59,62,61,64,63,58),(65,72,67,66,69,68,71,70),(73,76,75,78,77,80,79,74)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 10Q | ··· | 10X | 20A | ··· | 20H | 20I | ··· | 20T | 20U | ··· | 20AB | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C10 | C20 | C20 | D4 | C5×D4 | M4(2).8C22 | C5×M4(2).8C22 |
| kernel | C5×M4(2).8C22 | C5×C4.D4 | C5×C4.10D4 | C10×M4(2) | C10×C4○D4 | C22×C20 | D4×C10 | M4(2).8C22 | C4.D4 | C4.10D4 | C2×M4(2) | C2×C4○D4 | C22×C4 | C2×D4 | C2×C20 | C2×C4 | C5 | C1 |
| # reps | 1 | 2 | 2 | 2 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 4 | 16 | 16 | 4 | 16 | 2 | 8 |
Matrix representation of C5×M4(2).8C22 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 20 | 6 | 21 | 1 |
| 0 | 0 | 0 | 9 |
| 2 | 11 | 21 | 3 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 21 | 3 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 11 | 19 | 32 | 6 |
| 39 | 30 | 20 | 38 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 16 | 13 | 24 | 39 |
| 0 | 0 | 0 | 1 |
| 18 | 17 | 25 | 27 |
| 0 | 9 | 0 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[20,0,2,0,6,0,11,1,21,0,21,0,1,9,3,0],[1,0,0,0,0,1,0,0,21,0,40,0,3,0,0,40],[11,39,0,0,19,30,0,0,32,20,0,1,6,38,1,0],[16,0,18,0,13,0,17,9,24,0,25,0,39,1,27,0] >;
C5×M4(2).8C22 in GAP, Magma, Sage, TeX
C_5\times M_4(2)._8C_2^2
% in TeX
G:=Group("C5xM4(2).8C2^2"); // GroupNames label
G:=SmallGroup(320,914);
// by ID
G=gap.SmallGroup(320,914);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,856,7004,5052,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^8=c^2=d^2=1,e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e^-1=b^5,d*b*d=b*c,c*d=d*c,e*c*e^-1=b^4*c,e*d*e^-1=b^4*c*d>;
// generators/relations