metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C40.7D4, D40.6C4, C8.17D20, C20.48D8, C8.2(C4×D5), C40.63(C2×C4), C8.C4⋊1D5, (C2×C8).43D10, (C2×C20).95D4, C20.4C8⋊5C2, C4.21(D4⋊D5), (C2×D40).12C2, C5⋊2(M5(2)⋊C2), (C2×C10).4SD16, C22.3(Q8⋊D5), C4.4(D10⋊C4), C2.9(D20⋊6C4), C20.51(C22⋊C4), (C2×C40).100C22, C10.22(D4⋊C4), (C5×C8.C4)⋊9C2, (C2×C4).18(C5⋊D4), SmallGroup(320,53)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D40.6C4
G = < a,b,c | a40=b2=1, c4=a20, bab=a-1, cac-1=a11, cbc-1=a15b >
Subgroups: 414 in 62 conjugacy classes, 25 normal (23 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C8, C2×C4, D4, C23, D5, C10, C10, C16, C2×C8, M4(2), D8, C2×D4, C20, D10, C2×C10, C8.C4, M5(2), C2×D8, C40, C40, D20, C2×C20, C22×D5, M5(2)⋊C2, C5⋊2C16, D40, D40, C2×C40, C5×M4(2), C2×D20, C20.4C8, C5×C8.C4, C2×D40, D40.6C4
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D8, SD16, D10, D4⋊C4, C4×D5, D20, C5⋊D4, M5(2)⋊C2, D10⋊C4, D4⋊D5, Q8⋊D5, D20⋊6C4, D40.6C4
►On 80 points
(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 10)(2 9)(3 8)(4 7)(5 6)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)(22 29)(23 28)(24 27)(25 26)(41 71)(42 70)(43 69)(44 68)(45 67)(46 66)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(72 80)(73 79)(74 78)(75 77)
(1 69 11 59 21 49 31 79)(2 80 12 70 22 60 32 50)(3 51 13 41 23 71 33 61)(4 62 14 52 24 42 34 72)(5 73 15 63 25 53 35 43)(6 44 16 74 26 64 36 54)(7 55 17 45 27 75 37 65)(8 66 18 56 28 46 38 76)(9 77 19 67 29 57 39 47)(10 48 20 78 30 68 40 58)
G:=sub<Sym(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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(72,80)(73,79)(74,78)(75,77), (1,69,11,59,21,49,31,79)(2,80,12,70,22,60,32,50)(3,51,13,41,23,71,33,61)(4,62,14,52,24,42,34,72)(5,73,15,63,25,53,35,43)(6,44,16,74,26,64,36,54)(7,55,17,45,27,75,37,65)(8,66,18,56,28,46,38,76)(9,77,19,67,29,57,39,47)(10,48,20,78,30,68,40,58)>;
G:=Group( (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(41,71)(42,70)(43,69)(44,68)(45,67)(46,66)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(72,80)(73,79)(74,78)(75,77), (1,69,11,59,21,49,31,79)(2,80,12,70,22,60,32,50)(3,51,13,41,23,71,33,61)(4,62,14,52,24,42,34,72)(5,73,15,63,25,53,35,43)(6,44,16,74,26,64,36,54)(7,55,17,45,27,75,37,65)(8,66,18,56,28,46,38,76)(9,77,19,67,29,57,39,47)(10,48,20,78,30,68,40,58) );
G=PermutationGroup([[(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30),(22,29),(23,28),(24,27),(25,26),(41,71),(42,70),(43,69),(44,68),(45,67),(46,66),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(72,80),(73,79),(74,78),(75,77)], [(1,69,11,59,21,49,31,79),(2,80,12,70,22,60,32,50),(3,51,13,41,23,71,33,61),(4,62,14,52,24,42,34,72),(5,73,15,63,25,53,35,43),(6,44,16,74,26,64,36,54),(7,55,17,45,27,75,37,65),(8,66,18,56,28,46,38,76),(9,77,19,67,29,57,39,47),(10,48,20,78,30,68,40,58)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 10A | 10B | 10C | 10D | 16A | 16B | 16C | 16D | 20A | 20B | 20C | 20D | 20E | 20F | 40A | ··· | 40H | 40I | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 16 | 16 | 16 | 16 | 20 | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 40 | 40 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 8 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D8 | SD16 | D10 | C4×D5 | D20 | C5⋊D4 | M5(2)⋊C2 | D4⋊D5 | Q8⋊D5 | D40.6C4 |
| kernel | D40.6C4 | C20.4C8 | C5×C8.C4 | C2×D40 | D40 | C40 | C2×C20 | C8.C4 | C20 | C2×C10 | C2×C8 | C8 | C8 | C2×C4 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 8 |
Matrix representation of D40.6C4 ►in GL4(𝔽241) generated by
| 29 | 185 | 0 | 0 |
| 76 | 194 | 0 | 0 |
| 7 | 182 | 56 | 27 |
| 35 | 172 | 214 | 228 |
| 227 | 27 | 0 | 0 |
| 207 | 14 | 0 | 0 |
| 13 | 224 | 190 | 190 |
| 94 | 109 | 240 | 51 |
| 0 | 0 | 240 | 1 |
| 126 | 196 | 239 | 51 |
| 19 | 61 | 45 | 0 |
| 219 | 58 | 45 | 0 |
G:=sub<GL(4,GF(241))| [29,76,7,35,185,194,182,172,0,0,56,214,0,0,27,228],[227,207,13,94,27,14,224,109,0,0,190,240,0,0,190,51],[0,126,19,219,0,196,61,58,240,239,45,45,1,51,0,0] >;
D40.6C4 in GAP, Magma, Sage, TeX
D_{40}._6C_4 % in TeX
G:=Group("D40.6C4"); // GroupNames label
G:=SmallGroup(320,53);
// by ID
G=gap.SmallGroup(320,53);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,184,675,794,192,1684,851,102,12550]);
// Polycyclic
G:=Group<a,b,c|a^40=b^2=1,c^4=a^20,b*a*b=a^-1,c*a*c^-1=a^11,c*b*c^-1=a^15*b>;
// generators/relations