direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×D4.3D4, C40.103D4, C8○D4⋊1C10, C8⋊C22.C10, D4.3(C5×D4), C8.23(C5×D4), Q8.3(C5×D4), (C5×D4).28D4, C4.37(D4×C10), (C5×Q8).28D4, C8.C4⋊5C10, (C2×SD16)⋊2C10, C4.D4⋊3C10, C20.398(C2×D4), C8.C22⋊3C10, (C10×SD16)⋊13C2, C4.10D4⋊3C10, (C2×C40).272C22, (C2×C20).613C23, M4(2).3(C2×C10), C10.154(C4⋊D4), (D4×C10).193C22, (Q8×C10).167C22, (C5×M4(2)).47C22, (C5×C8○D4)⋊10C2, (C2×C8).24(C2×C10), (C5×C4.D4)⋊9C2, C2.23(C5×C4⋊D4), (C5×C8⋊C22).2C2, (C5×C8.C4)⋊14C2, C22.6(C5×C4○D4), C4○D4.10(C2×C10), (C2×D4).16(C2×C10), (C5×C4.10D4)⋊9C2, (C5×C8.C22)⋊10C2, (C2×C4).8(C22×C10), (C2×Q8).11(C2×C10), (C5×C4○D4).55C22, (C2×C10).115(C4○D4), SmallGroup(320,972)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×D4.3D4
G = < a,b,c,d,e | a5=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >
Subgroups: 194 in 104 conjugacy classes, 50 normal (all characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C10, C10, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C20, C20, C2×C10, C2×C10, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C40, C40, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C5×Q8, C22×C10, D4.3D4, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C5×D8, C5×SD16, C5×Q16, D4×C10, Q8×C10, C5×C4○D4, C5×C4.D4, C5×C4.10D4, C5×C8.C4, C5×C8○D4, C10×SD16, C5×C8⋊C22, C5×C8.C22, C5×D4.3D4
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C4○D4, C2×C10, C4⋊D4, C5×D4, C22×C10, D4.3D4, D4×C10, C5×C4○D4, C5×C4⋊D4, C5×D4.3D4
►On 80 points
(1 59 67 19 27)(2 60 68 20 28)(3 61 69 21 29)(4 62 70 22 30)(5 63 71 23 31)(6 64 72 24 32)(7 57 65 17 25)(8 58 66 18 26)(9 38 41 50 78)(10 39 42 51 79)(11 40 43 52 80)(12 33 44 53 73)(13 34 45 54 74)(14 35 46 55 75)(15 36 47 56 76)(16 37 48 49 77)
(1 7 5 3)(2 8 6 4)(9 11 13 15)(10 12 14 16)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 63 61 59)(58 64 62 60)(65 71 69 67)(66 72 70 68)(73 75 77 79)(74 76 78 80)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 79)(18 80)(19 73)(20 74)(21 75)(22 76)(23 77)(24 78)(41 64)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)(49 71)(50 72)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)
(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 4 5 8)(2 7 6 3)(9 12 13 16)(10 15 14 11)(17 24 21 20)(18 19 22 23)(25 32 29 28)(26 27 30 31)(33 34 37 38)(35 40 39 36)(41 44 45 48)(42 47 46 43)(49 50 53 54)(51 56 55 52)(57 64 61 60)(58 59 62 63)(65 72 69 68)(66 67 70 71)(73 74 77 78)(75 80 79 76)
G:=sub<Sym(80)| (1,59,67,19,27)(2,60,68,20,28)(3,61,69,21,29)(4,62,70,22,30)(5,63,71,23,31)(6,64,72,24,32)(7,57,65,17,25)(8,58,66,18,26)(9,38,41,50,78)(10,39,42,51,79)(11,40,43,52,80)(12,33,44,53,73)(13,34,45,54,74)(14,35,46,55,75)(15,36,47,56,76)(16,37,48,49,77), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,71)(50,72)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,24,21,20)(18,19,22,23)(25,32,29,28)(26,27,30,31)(33,34,37,38)(35,40,39,36)(41,44,45,48)(42,47,46,43)(49,50,53,54)(51,56,55,52)(57,64,61,60)(58,59,62,63)(65,72,69,68)(66,67,70,71)(73,74,77,78)(75,80,79,76)>;
G:=Group( (1,59,67,19,27)(2,60,68,20,28)(3,61,69,21,29)(4,62,70,22,30)(5,63,71,23,31)(6,64,72,24,32)(7,57,65,17,25)(8,58,66,18,26)(9,38,41,50,78)(10,39,42,51,79)(11,40,43,52,80)(12,33,44,53,73)(13,34,45,54,74)(14,35,46,55,75)(15,36,47,56,76)(16,37,48,49,77), (1,7,5,3)(2,8,6,4)(9,11,13,15)(10,12,14,16)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,63,61,59)(58,64,62,60)(65,71,69,67)(66,72,70,68)(73,75,77,79)(74,76,78,80), (1,33)(2,34)(3,35)(4,36)(5,37)(6,38)(7,39)(8,40)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,79)(18,80)(19,73)(20,74)(21,75)(22,76)(23,77)(24,78)(41,64)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)(49,71)(50,72)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70), (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), (1,4,5,8)(2,7,6,3)(9,12,13,16)(10,15,14,11)(17,24,21,20)(18,19,22,23)(25,32,29,28)(26,27,30,31)(33,34,37,38)(35,40,39,36)(41,44,45,48)(42,47,46,43)(49,50,53,54)(51,56,55,52)(57,64,61,60)(58,59,62,63)(65,72,69,68)(66,67,70,71)(73,74,77,78)(75,80,79,76) );
G=PermutationGroup([[(1,59,67,19,27),(2,60,68,20,28),(3,61,69,21,29),(4,62,70,22,30),(5,63,71,23,31),(6,64,72,24,32),(7,57,65,17,25),(8,58,66,18,26),(9,38,41,50,78),(10,39,42,51,79),(11,40,43,52,80),(12,33,44,53,73),(13,34,45,54,74),(14,35,46,55,75),(15,36,47,56,76),(16,37,48,49,77)], [(1,7,5,3),(2,8,6,4),(9,11,13,15),(10,12,14,16),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,63,61,59),(58,64,62,60),(65,71,69,67),(66,72,70,68),(73,75,77,79),(74,76,78,80)], [(1,33),(2,34),(3,35),(4,36),(5,37),(6,38),(7,39),(8,40),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,79),(18,80),(19,73),(20,74),(21,75),(22,76),(23,77),(24,78),(41,64),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63),(49,71),(50,72),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70)], [(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)], [(1,4,5,8),(2,7,6,3),(9,12,13,16),(10,15,14,11),(17,24,21,20),(18,19,22,23),(25,32,29,28),(26,27,30,31),(33,34,37,38),(35,40,39,36),(41,44,45,48),(42,47,46,43),(49,50,53,54),(51,56,55,52),(57,64,61,60),(58,59,62,63),(65,72,69,68),(66,67,70,71),(73,74,77,78),(75,80,79,76)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 5A | 5B | 5C | 5D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 10O | 10P | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 20M | 20N | 20O | 20P | 40A | ··· | 40H | 40I | ··· | 40T | 40U | ··· | 40AB |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 8 | 2 | 2 | 4 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | D4 | D4 | C4○D4 | C5×D4 | C5×D4 | C5×D4 | C5×C4○D4 | D4.3D4 | C5×D4.3D4 |
| kernel | C5×D4.3D4 | C5×C4.D4 | C5×C4.10D4 | C5×C8.C4 | C5×C8○D4 | C10×SD16 | C5×C8⋊C22 | C5×C8.C22 | D4.3D4 | C4.D4 | C4.10D4 | C8.C4 | C8○D4 | C2×SD16 | C8⋊C22 | C8.C22 | C40 | C5×D4 | C5×Q8 | C2×C10 | C8 | D4 | Q8 | C22 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 1 | 1 | 2 | 8 | 4 | 4 | 8 | 2 | 8 |
Matrix representation of C5×D4.3D4 ►in GL4(𝔽41) generated by
| 18 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 18 |
| 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 40 | 40 | 1 | 2 |
| 0 | 1 | 40 | 40 |
| 40 | 40 | 1 | 2 |
| 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 40 | 1 | 1 |
| 15 | 15 | 0 | 0 |
| 26 | 15 | 0 | 0 |
| 15 | 15 | 0 | 11 |
| 26 | 0 | 15 | 30 |
| 15 | 15 | 0 | 0 |
| 15 | 26 | 0 | 0 |
| 15 | 15 | 11 | 11 |
| 0 | 26 | 15 | 30 |
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[0,40,40,0,1,0,40,1,0,0,1,40,0,0,2,40],[40,0,0,0,40,0,1,40,1,1,0,1,2,0,0,1],[15,26,15,26,15,15,15,0,0,0,0,15,0,0,11,30],[15,15,15,0,15,26,15,26,0,0,11,15,0,0,11,30] >;
C5×D4.3D4 in GAP, Magma, Sage, TeX
C_5\times D_4._3D_4
% in TeX
G:=Group("C5xD4.3D4"); // GroupNames label
G:=SmallGroup(320,972);
// by ID
G=gap.SmallGroup(320,972);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,7004,172,10085,2539,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=c^2=1,d^4=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^-1,b*d=d*b,c*d=d*c,e*c*e^-1=b*c,e*d*e^-1=d^3>;
// generators/relations