direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D5×C4.10D4, M4(2).20D10, (C4×D5).2D4, C4.150(D4×D5), C20.95(C2×D4), (C2×C20).7C23, (C2×Q8).94D10, C4.12D20⋊7C2, C20.10D4⋊3C2, (C2×Dic10).7C4, (D5×M4(2)).7C2, (Q8×C10).5C22, C4.Dic5.4C22, D10.54(C22⋊C4), Dic5.22(C22⋊C4), (C2×Dic10).50C22, (C5×M4(2)).12C22, (C2×C4×D5).2C4, (C2×Q8×D5).1C2, (C2×C4).6(C4×D5), C5⋊3(C2×C4.10D4), (C2×C4×D5).7C22, C22.16(C2×C4×D5), (C2×C20).20(C2×C4), C2.15(D5×C22⋊C4), (C2×C4).7(C22×D5), (C5×C4.10D4)⋊5C2, C10.55(C2×C22⋊C4), (C2×Dic5).3(C2×C4), (C2×C10).111(C22×C4), (C22×D5).100(C2×C4), SmallGroup(320,377)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D5×C4.10D4
G = < a,b,c,d,e | a5=b2=c4=1, d4=c2, e2=dcd-1=c-1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ce=ec, ede-1=c-1d3 >
Subgroups: 526 in 146 conjugacy classes, 53 normal (21 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C23, D5, D5, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C2×Q8, C2×Q8, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4.10D4, C4.10D4, C2×M4(2), C22×Q8, C5⋊2C8, C40, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C2×C4.10D4, C8×D5, C8⋊D5, C4.Dic5, C5×M4(2), C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5, Q8×C10, C4.12D20, C20.10D4, C5×C4.10D4, D5×M4(2), C2×Q8×D5, D5×C4.10D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4.10D4, C2×C22⋊C4, C4×D5, C22×D5, C2×C4.10D4, C2×C4×D5, D4×D5, D5×C22⋊C4, D5×C4.10D4
►On 80 points
(1 59 51 46 65)(2 60 52 47 66)(3 61 53 48 67)(4 62 54 41 68)(5 63 55 42 69)(6 64 56 43 70)(7 57 49 44 71)(8 58 50 45 72)(9 27 38 21 75)(10 28 39 22 76)(11 29 40 23 77)(12 30 33 24 78)(13 31 34 17 79)(14 32 35 18 80)(15 25 36 19 73)(16 26 37 20 74)
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(33 78)(34 79)(35 80)(36 73)(37 74)(38 75)(39 76)(40 77)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)
(1 3 5 7)(2 8 6 4)(9 15 13 11)(10 12 14 16)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 35 37 39)(34 40 38 36)(41 47 45 43)(42 44 46 48)(49 51 53 55)(50 56 54 52)(57 59 61 63)(58 64 62 60)(65 67 69 71)(66 72 70 68)(73 79 77 75)(74 76 78 80)
(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 29 7 31 5 25 3 27)(2 30 4 28 6 26 8 32)(9 65 11 71 13 69 15 67)(10 70 16 72 14 66 12 68)(17 55 19 53 21 51 23 49)(18 52 24 54 22 56 20 50)(33 62 39 64 37 58 35 60)(34 63 36 61 38 59 40 57)(41 76 43 74 45 80 47 78)(42 73 48 75 46 77 44 79)
G:=sub<Sym(80)| (1,59,51,46,65)(2,60,52,47,66)(3,61,53,48,67)(4,62,54,41,68)(5,63,55,42,69)(6,64,56,43,70)(7,57,49,44,71)(8,58,50,45,72)(9,27,38,21,75)(10,28,39,22,76)(11,29,40,23,77)(12,30,33,24,78)(13,31,34,17,79)(14,32,35,18,80)(15,25,36,19,73)(16,26,37,20,74), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,79,77,75)(74,76,78,80), (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), (1,29,7,31,5,25,3,27)(2,30,4,28,6,26,8,32)(9,65,11,71,13,69,15,67)(10,70,16,72,14,66,12,68)(17,55,19,53,21,51,23,49)(18,52,24,54,22,56,20,50)(33,62,39,64,37,58,35,60)(34,63,36,61,38,59,40,57)(41,76,43,74,45,80,47,78)(42,73,48,75,46,77,44,79)>;
G:=Group( (1,59,51,46,65)(2,60,52,47,66)(3,61,53,48,67)(4,62,54,41,68)(5,63,55,42,69)(6,64,56,43,70)(7,57,49,44,71)(8,58,50,45,72)(9,27,38,21,75)(10,28,39,22,76)(11,29,40,23,77)(12,30,33,24,78)(13,31,34,17,79)(14,32,35,18,80)(15,25,36,19,73)(16,26,37,20,74), (1,65)(2,66)(3,67)(4,68)(5,69)(6,70)(7,71)(8,72)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26)(33,78)(34,79)(35,80)(36,73)(37,74)(38,75)(39,76)(40,77)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61), (1,3,5,7)(2,8,6,4)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,31,29,27)(26,28,30,32)(33,35,37,39)(34,40,38,36)(41,47,45,43)(42,44,46,48)(49,51,53,55)(50,56,54,52)(57,59,61,63)(58,64,62,60)(65,67,69,71)(66,72,70,68)(73,79,77,75)(74,76,78,80), (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), (1,29,7,31,5,25,3,27)(2,30,4,28,6,26,8,32)(9,65,11,71,13,69,15,67)(10,70,16,72,14,66,12,68)(17,55,19,53,21,51,23,49)(18,52,24,54,22,56,20,50)(33,62,39,64,37,58,35,60)(34,63,36,61,38,59,40,57)(41,76,43,74,45,80,47,78)(42,73,48,75,46,77,44,79) );
G=PermutationGroup([[(1,59,51,46,65),(2,60,52,47,66),(3,61,53,48,67),(4,62,54,41,68),(5,63,55,42,69),(6,64,56,43,70),(7,57,49,44,71),(8,58,50,45,72),(9,27,38,21,75),(10,28,39,22,76),(11,29,40,23,77),(12,30,33,24,78),(13,31,34,17,79),(14,32,35,18,80),(15,25,36,19,73),(16,26,37,20,74)], [(1,65),(2,66),(3,67),(4,68),(5,69),(6,70),(7,71),(8,72),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26),(33,78),(34,79),(35,80),(36,73),(37,74),(38,75),(39,76),(40,77),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61)], [(1,3,5,7),(2,8,6,4),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,31,29,27),(26,28,30,32),(33,35,37,39),(34,40,38,36),(41,47,45,43),(42,44,46,48),(49,51,53,55),(50,56,54,52),(57,59,61,63),(58,64,62,60),(65,67,69,71),(66,72,70,68),(73,79,77,75),(74,76,78,80)], [(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)], [(1,29,7,31,5,25,3,27),(2,30,4,28,6,26,8,32),(9,65,11,71,13,69,15,67),(10,70,16,72,14,66,12,68),(17,55,19,53,21,51,23,49),(18,52,24,54,22,56,20,50),(33,62,39,64,37,58,35,60),(34,63,36,61,38,59,40,57),(41,76,43,74,45,80,47,78),(42,73,48,75,46,77,44,79)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | 10B | 10C | 10D | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 5 | 5 | 10 | 2 | 2 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | C4.10D4 | D4×D5 | D5×C4.10D4 |
| kernel | D5×C4.10D4 | C4.12D20 | C20.10D4 | C5×C4.10D4 | D5×M4(2) | C2×Q8×D5 | C2×Dic10 | C2×C4×D5 | C4×D5 | C4.10D4 | M4(2) | C2×Q8 | C2×C4 | D5 | C4 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 4 | 2 | 8 | 2 | 4 | 2 |
Matrix representation of D5×C4.10D4 ►in GL6(𝔽41)
| 35 | 1 | 0 | 0 | 0 | 0 |
| 5 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 40 | 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 9 | 40 | 23 |
| 0 | 0 | 0 | 40 | 32 | 1 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 32 | 9 | 40 | 23 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 | 32 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 23 | 37 | 36 |
| 0 | 0 | 13 | 28 | 8 | 26 |
| 0 | 0 | 39 | 35 | 0 | 0 |
| 0 | 0 | 36 | 28 | 6 | 36 |
G:=sub<GL(6,GF(41))| [35,5,0,0,0,0,1,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,32,0,0,0,1,0,9,40,0,0,0,0,40,32,0,0,0,0,23,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,32,0,0,0,9,40,0,0,0,40,40,0,0,0,0,0,23,0,32],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,18,13,39,36,0,0,23,28,35,28,0,0,37,8,0,6,0,0,36,26,0,36] >;
D5×C4.10D4 in GAP, Magma, Sage, TeX
D_5\times C_4._{10}D_4 % in TeX
G:=Group("D5xC4.10D4"); // GroupNames label
G:=SmallGroup(320,377);
// by ID
G=gap.SmallGroup(320,377);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,219,58,570,136,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^4=1,d^4=c^2,e^2=d*c*d^-1=c^-1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*e=e*c,e*d*e^-1=c^-1*d^3>;
// generators/relations