direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8.C22, Q16⋊2C10, C20.64D4, SD16⋊2C10, M4(2)⋊2C10, C20.49C23, C40.13C22, C8.(C2×C10), (C2×Q8)⋊4C10, (C5×Q16)⋊6C2, C4.15(C5×D4), (Q8×C10)⋊11C2, (C5×SD16)⋊6C2, C4○D4.2C10, D4.3(C2×C10), (C2×C10).25D4, C2.16(D4×C10), C10.79(C2×D4), Q8.3(C2×C10), C22.6(C5×D4), (C5×M4(2))⋊6C2, C4.6(C22×C10), (C2×C20).70C22, (C5×D4).13C22, (C5×Q8).14C22, (C5×C4○D4).5C2, (C2×C4).11(C2×C10), SmallGroup(160,198)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8.C22
G = < a,b,c,d | a5=b8=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b3, dbd=b5, dcd=b4c >
Subgroups: 84 in 60 conjugacy classes, 40 normal (24 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, Q8, C10, C10, M4(2), SD16, Q16, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C8.C22, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C5×Q8, C5×M4(2), C5×SD16, C5×Q16, Q8×C10, C5×C4○D4, C5×C8.C22
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C8.C22, C5×D4, C22×C10, D4×C10, C5×C8.C22
►On 80 points
(1 22 69 49 26)(2 23 70 50 27)(3 24 71 51 28)(4 17 72 52 29)(5 18 65 53 30)(6 19 66 54 31)(7 20 67 55 32)(8 21 68 56 25)(9 78 58 35 48)(10 79 59 36 41)(11 80 60 37 42)(12 73 61 38 43)(13 74 62 39 44)(14 75 63 40 45)(15 76 64 33 46)(16 77 57 34 47)
(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 4)(3 7)(6 8)(9 15)(11 13)(12 16)(17 23)(19 21)(20 24)(25 31)(27 29)(28 32)(33 35)(34 38)(37 39)(42 44)(43 47)(46 48)(50 52)(51 55)(54 56)(57 61)(58 64)(60 62)(66 68)(67 71)(70 72)(73 77)(74 80)(76 78)
(1 38)(2 35)(3 40)(4 37)(5 34)(6 39)(7 36)(8 33)(9 70)(10 67)(11 72)(12 69)(13 66)(14 71)(15 68)(16 65)(17 42)(18 47)(19 44)(20 41)(21 46)(22 43)(23 48)(24 45)(25 64)(26 61)(27 58)(28 63)(29 60)(30 57)(31 62)(32 59)(49 73)(50 78)(51 75)(52 80)(53 77)(54 74)(55 79)(56 76)
G:=sub<Sym(80)| (1,22,69,49,26)(2,23,70,50,27)(3,24,71,51,28)(4,17,72,52,29)(5,18,65,53,30)(6,19,66,54,31)(7,20,67,55,32)(8,21,68,56,25)(9,78,58,35,48)(10,79,59,36,41)(11,80,60,37,42)(12,73,61,38,43)(13,74,62,39,44)(14,75,63,40,45)(15,76,64,33,46)(16,77,57,34,47), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56)(57,61)(58,64)(60,62)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78), (1,38)(2,35)(3,40)(4,37)(5,34)(6,39)(7,36)(8,33)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(17,42)(18,47)(19,44)(20,41)(21,46)(22,43)(23,48)(24,45)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76)>;
G:=Group( (1,22,69,49,26)(2,23,70,50,27)(3,24,71,51,28)(4,17,72,52,29)(5,18,65,53,30)(6,19,66,54,31)(7,20,67,55,32)(8,21,68,56,25)(9,78,58,35,48)(10,79,59,36,41)(11,80,60,37,42)(12,73,61,38,43)(13,74,62,39,44)(14,75,63,40,45)(15,76,64,33,46)(16,77,57,34,47), (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), (2,4)(3,7)(6,8)(9,15)(11,13)(12,16)(17,23)(19,21)(20,24)(25,31)(27,29)(28,32)(33,35)(34,38)(37,39)(42,44)(43,47)(46,48)(50,52)(51,55)(54,56)(57,61)(58,64)(60,62)(66,68)(67,71)(70,72)(73,77)(74,80)(76,78), (1,38)(2,35)(3,40)(4,37)(5,34)(6,39)(7,36)(8,33)(9,70)(10,67)(11,72)(12,69)(13,66)(14,71)(15,68)(16,65)(17,42)(18,47)(19,44)(20,41)(21,46)(22,43)(23,48)(24,45)(25,64)(26,61)(27,58)(28,63)(29,60)(30,57)(31,62)(32,59)(49,73)(50,78)(51,75)(52,80)(53,77)(54,74)(55,79)(56,76) );
G=PermutationGroup([[(1,22,69,49,26),(2,23,70,50,27),(3,24,71,51,28),(4,17,72,52,29),(5,18,65,53,30),(6,19,66,54,31),(7,20,67,55,32),(8,21,68,56,25),(9,78,58,35,48),(10,79,59,36,41),(11,80,60,37,42),(12,73,61,38,43),(13,74,62,39,44),(14,75,63,40,45),(15,76,64,33,46),(16,77,57,34,47)], [(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)], [(2,4),(3,7),(6,8),(9,15),(11,13),(12,16),(17,23),(19,21),(20,24),(25,31),(27,29),(28,32),(33,35),(34,38),(37,39),(42,44),(43,47),(46,48),(50,52),(51,55),(54,56),(57,61),(58,64),(60,62),(66,68),(67,71),(70,72),(73,77),(74,80),(76,78)], [(1,38),(2,35),(3,40),(4,37),(5,34),(6,39),(7,36),(8,33),(9,70),(10,67),(11,72),(12,69),(13,66),(14,71),(15,68),(16,65),(17,42),(18,47),(19,44),(20,41),(21,46),(22,43),(23,48),(24,45),(25,64),(26,61),(27,58),(28,63),(29,60),(30,57),(31,62),(32,59),(49,73),(50,78),(51,75),(52,80),(53,77),(54,74),(55,79),(56,76)]])
C5×C8.C22 is a maximal subgroup of
D20.39D4 M4(2).15D10 M4(2).16D10 D20.40D4 D40⋊C22 C40.C23 D20.44D4
55 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
55 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | C5×D4 | C5×D4 | C8.C22 | C5×C8.C22 |
| kernel | C5×C8.C22 | C5×M4(2) | C5×SD16 | C5×Q16 | Q8×C10 | C5×C4○D4 | C8.C22 | M4(2) | SD16 | Q16 | C2×Q8 | C4○D4 | C20 | C2×C10 | C4 | C22 | C5 | C1 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 4 | 4 | 8 | 8 | 4 | 4 | 1 | 1 | 4 | 4 | 1 | 4 |
Matrix representation of C5×C8.C22 ►in GL4(𝔽41) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 10 |
| 16 | 25 | 16 | 16 |
| 16 | 16 | 25 | 16 |
| 25 | 25 | 25 | 16 |
| 16 | 25 | 25 | 25 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
G:=sub<GL(4,GF(41))| [10,0,0,0,0,10,0,0,0,0,10,0,0,0,0,10],[16,16,25,16,25,16,25,25,16,25,25,25,16,16,16,25],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0] >;
C5×C8.C22 in GAP, Magma, Sage, TeX
C_5\times C_8.C_2^2
% in TeX
G:=Group("C5xC8.C2^2"); // GroupNames label
G:=SmallGroup(160,198);
// by ID
G=gap.SmallGroup(160,198);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,487,1514,3604,1810,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^3,d*b*d=b^5,d*c*d=b^4*c>;
// generators/relations