direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8○D4, D4.C20, Q8.C20, M4(2)⋊5C10, C20.54C23, C40.30C22, C40○(C5×D4), C40○(C5×Q8), (C2×C8)⋊7C10, (C2×C40)⋊15C2, C4.5(C2×C20), C8.7(C2×C10), (C5×D4).3C4, (C5×Q8).3C4, C20.53(C2×C4), C4○D4.3C10, C40○(C5×M4(2)), C2.7(C22×C20), C22.1(C2×C20), (C5×M4(2))⋊11C2, C10.48(C22×C4), C4.12(C22×C10), (C2×C20).128C22, C40○(C5×C4○D4), (C5×C4○D4).6C2, (C2×C10).28(C2×C4), (C2×C4).24(C2×C10), SmallGroup(160,192)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8○D4
G = < a,b,c,d | a5=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >
Subgroups: 68 in 62 conjugacy classes, 56 normal (14 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, D4, Q8, C10, C10, C2×C8, M4(2), C4○D4, C20, C20, C2×C10, C8○D4, C40, C40, C2×C20, C5×D4, C5×Q8, C2×C40, C5×M4(2), C5×C4○D4, C5×C8○D4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C22×C4, C20, C2×C10, C8○D4, C2×C20, C22×C10, C22×C20, C5×C8○D4
►On 80 points
(1 63 71 23 31)(2 64 72 24 32)(3 57 65 17 25)(4 58 66 18 26)(5 59 67 19 27)(6 60 68 20 28)(7 61 69 21 29)(8 62 70 22 30)(9 74 50 34 46)(10 75 51 35 47)(11 76 52 36 48)(12 77 53 37 41)(13 78 54 38 42)(14 79 55 39 43)(15 80 56 40 44)(16 73 49 33 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)
(1 35 5 39)(2 36 6 40)(3 37 7 33)(4 38 8 34)(9 66 13 70)(10 67 14 71)(11 68 15 72)(12 69 16 65)(17 77 21 73)(18 78 22 74)(19 79 23 75)(20 80 24 76)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)(41 61 45 57)(42 62 46 58)(43 63 47 59)(44 64 48 60)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 66)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 65)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)
G:=sub<Sym(80)| (1,63,71,23,31)(2,64,72,24,32)(3,57,65,17,25)(4,58,66,18,26)(5,59,67,19,27)(6,60,68,20,28)(7,61,69,21,29)(8,62,70,22,30)(9,74,50,34,46)(10,75,51,35,47)(11,76,52,36,48)(12,77,53,37,41)(13,78,54,38,42)(14,79,55,39,43)(15,80,56,40,44)(16,73,49,33,45), (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), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,66,13,70)(10,67,14,71)(11,68,15,72)(12,69,16,65)(17,77,21,73)(18,78,22,74)(19,79,23,75)(20,80,24,76)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(41,61,45,57)(42,62,46,58)(43,63,47,59)(44,64,48,60), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60)>;
G:=Group( (1,63,71,23,31)(2,64,72,24,32)(3,57,65,17,25)(4,58,66,18,26)(5,59,67,19,27)(6,60,68,20,28)(7,61,69,21,29)(8,62,70,22,30)(9,74,50,34,46)(10,75,51,35,47)(11,76,52,36,48)(12,77,53,37,41)(13,78,54,38,42)(14,79,55,39,43)(15,80,56,40,44)(16,73,49,33,45), (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), (1,35,5,39)(2,36,6,40)(3,37,7,33)(4,38,8,34)(9,66,13,70)(10,67,14,71)(11,68,15,72)(12,69,16,65)(17,77,21,73)(18,78,22,74)(19,79,23,75)(20,80,24,76)(25,53,29,49)(26,54,30,50)(27,55,31,51)(28,56,32,52)(41,61,45,57)(42,62,46,58)(43,63,47,59)(44,64,48,60), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,66)(10,67)(11,68)(12,69)(13,70)(14,71)(15,72)(16,65)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(41,61)(42,62)(43,63)(44,64)(45,57)(46,58)(47,59)(48,60) );
G=PermutationGroup([[(1,63,71,23,31),(2,64,72,24,32),(3,57,65,17,25),(4,58,66,18,26),(5,59,67,19,27),(6,60,68,20,28),(7,61,69,21,29),(8,62,70,22,30),(9,74,50,34,46),(10,75,51,35,47),(11,76,52,36,48),(12,77,53,37,41),(13,78,54,38,42),(14,79,55,39,43),(15,80,56,40,44),(16,73,49,33,45)], [(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)], [(1,35,5,39),(2,36,6,40),(3,37,7,33),(4,38,8,34),(9,66,13,70),(10,67,14,71),(11,68,15,72),(12,69,16,65),(17,77,21,73),(18,78,22,74),(19,79,23,75),(20,80,24,76),(25,53,29,49),(26,54,30,50),(27,55,31,51),(28,56,32,52),(41,61,45,57),(42,62,46,58),(43,63,47,59),(44,64,48,60)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,66),(10,67),(11,68),(12,69),(13,70),(14,71),(15,72),(16,65),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(41,61),(42,62),(43,63),(44,64),(45,57),(46,58),(47,59),(48,60)]])
C5×C8○D4 is a maximal subgroup of
C40.92D4 C40.70C23 D4.3D20 D4.4D20 D4.5D20 C40.93D4 C40.50D4 C20.72C24 D4.11D20 D4.12D20 D4.13D20
C5×C8○D4 is a maximal quotient of
D4×C40 Q8×C40
100 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 10A | 10B | 10C | 10D | 10E | ··· | 10P | 20A | ··· | 20H | 20I | ··· | 20T | 40A | ··· | 40P | 40Q | ··· | 40AN |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | C8○D4 | C5×C8○D4 |
| kernel | C5×C8○D4 | C2×C40 | C5×M4(2) | C5×C4○D4 | C5×D4 | C5×Q8 | C8○D4 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C5 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 12 | 12 | 4 | 24 | 8 | 4 | 16 |
Matrix representation of C5×C8○D4 ►in GL2(𝔽41) generated by
| 10 | 0 |
| 0 | 10 |
| 3 | 0 |
| 0 | 3 |
| 1 | 39 |
| 1 | 40 |
| 1 | 39 |
| 0 | 40 |
G:=sub<GL(2,GF(41))| [10,0,0,10],[3,0,0,3],[1,1,39,40],[1,0,39,40] >;
C5×C8○D4 in GAP, Magma, Sage, TeX
C_5\times C_8\circ D_4
% in TeX
G:=Group("C5xC8oD4"); // GroupNames label
G:=SmallGroup(160,192);
// by ID
G=gap.SmallGroup(160,192);
# by ID
G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,764,88]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations