direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊7C4, C23.50D20, M4(2)⋊23D10, C10⋊4C4≀C2, C4○D20⋊13C4, D20⋊31(C2×C4), (C2×D20)⋊27C4, (C2×C20).177D4, C20.445(C2×D4), (C2×C4).155D20, (C2×Dic10)⋊26C4, Dic10⋊29(C2×C4), (C2×M4(2))⋊14D5, C22.16(C2×D20), (C10×M4(2))⋊22C2, (C2×C20).419C23, C20.128(C22×C4), C4○D20.42C22, (C4×Dic5)⋊62C22, (C22×C4).351D10, (C22×C10).106D4, C4.14(D10⋊C4), C20.115(C22⋊C4), (C5×M4(2))⋊35C22, (C22×C20).192C22, C22.52(D10⋊C4), C5⋊6(C2×C4≀C2), C4.54(C2×C4×D5), (C2×C4×Dic5)⋊2C2, (C2×C4).85(C4×D5), (C2×C10).32(C2×D4), C4.136(C2×C5⋊D4), (C2×C20).285(C2×C4), (C2×C4○D20).14C2, C2.34(C2×D10⋊C4), (C2×C4).278(C5⋊D4), C10.103(C2×C22⋊C4), (C2×C4).512(C22×D5), (C2×C10).132(C22⋊C4), SmallGroup(320,765)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D20⋊7C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b9, dcd-1=b3c >
Subgroups: 670 in 170 conjugacy classes, 63 normal (41 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C2×C10, C4≀C2, C2×C42, C2×M4(2), C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×C10, C2×C4≀C2, C4×Dic5, C4×Dic5, C2×C40, C5×M4(2), C5×M4(2), C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, D20⋊7C4, C2×C4×Dic5, C10×M4(2), C2×C4○D20, C2×D20⋊7C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C4≀C2, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C2×C4≀C2, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, D20⋊7C4, C2×D10⋊C4, C2×D20⋊7C4
►On 80 points
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 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 75)(2 74)(3 73)(4 72)(5 71)(6 70)(7 69)(8 68)(9 67)(10 66)(11 65)(12 64)(13 63)(14 62)(15 61)(16 80)(17 79)(18 78)(19 77)(20 76)(21 45)(22 44)(23 43)(24 42)(25 41)(26 60)(27 59)(28 58)(29 57)(30 56)(31 55)(32 54)(33 53)(34 52)(35 51)(36 50)(37 49)(38 48)(39 47)(40 46)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 26 31 36)(22 35 32 25)(23 24 33 34)(27 40 37 30)(28 29 38 39)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)(61 66 71 76)(62 75 72 65)(63 64 73 74)(67 80 77 70)(68 69 78 79)
G:=sub<Sym(80)| (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,80)(17,79)(18,78)(19,77)(20,76)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,51)(42,60)(43,49)(44,58)(45,47)(46,56)(48,54)(50,52)(53,59)(55,57)(61,66,71,76)(62,75,72,65)(63,64,73,74)(67,80,77,70)(68,69,78,79)>;
G:=Group( (1,51)(2,52)(3,53)(4,54)(5,55)(6,56)(7,57)(8,58)(9,59)(10,60)(11,41)(12,42)(13,43)(14,44)(15,45)(16,46)(17,47)(18,48)(19,49)(20,50)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (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), (1,75)(2,74)(3,73)(4,72)(5,71)(6,70)(7,69)(8,68)(9,67)(10,66)(11,65)(12,64)(13,63)(14,62)(15,61)(16,80)(17,79)(18,78)(19,77)(20,76)(21,45)(22,44)(23,43)(24,42)(25,41)(26,60)(27,59)(28,58)(29,57)(30,56)(31,55)(32,54)(33,53)(34,52)(35,51)(36,50)(37,49)(38,48)(39,47)(40,46), (1,11)(2,20)(3,9)(4,18)(5,7)(6,16)(8,14)(10,12)(13,19)(15,17)(21,26,31,36)(22,35,32,25)(23,24,33,34)(27,40,37,30)(28,29,38,39)(41,51)(42,60)(43,49)(44,58)(45,47)(46,56)(48,54)(50,52)(53,59)(55,57)(61,66,71,76)(62,75,72,65)(63,64,73,74)(67,80,77,70)(68,69,78,79) );
G=PermutationGroup([[(1,51),(2,52),(3,53),(4,54),(5,55),(6,56),(7,57),(8,58),(9,59),(10,60),(11,41),(12,42),(13,43),(14,44),(15,45),(16,46),(17,47),(18,48),(19,49),(20,50),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)], [(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)], [(1,75),(2,74),(3,73),(4,72),(5,71),(6,70),(7,69),(8,68),(9,67),(10,66),(11,65),(12,64),(13,63),(14,62),(15,61),(16,80),(17,79),(18,78),(19,77),(20,76),(21,45),(22,44),(23,43),(24,42),(25,41),(26,60),(27,59),(28,58),(29,57),(30,56),(31,55),(32,54),(33,53),(34,52),(35,51),(36,50),(37,49),(38,48),(39,47),(40,46)], [(1,11),(2,20),(3,9),(4,18),(5,7),(6,16),(8,14),(10,12),(13,19),(15,17),(21,26,31,36),(22,35,32,25),(23,24,33,34),(27,40,37,30),(28,29,38,39),(41,51),(42,60),(43,49),(44,58),(45,47),(46,56),(48,54),(50,52),(53,59),(55,57),(61,66,71,76),(62,75,72,65),(63,64,73,74),(67,80,77,70),(68,69,78,79)]])
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 1 | 1 | 1 | 1 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D4 | D5 | D10 | D10 | C4≀C2 | C4×D5 | D20 | C5⋊D4 | D20 | D20⋊7C4 |
| kernel | C2×D20⋊7C4 | D20⋊7C4 | C2×C4×Dic5 | C10×M4(2) | C2×C4○D20 | C2×Dic10 | C2×D20 | C4○D20 | C2×C20 | C22×C10 | C2×M4(2) | M4(2) | C22×C4 | C10 | C2×C4 | C2×C4 | C2×C4 | C23 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 2 | 4 | 3 | 1 | 2 | 4 | 2 | 8 | 8 | 4 | 8 | 4 | 8 |
Matrix representation of C2×D20⋊7C4 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 35 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 |
| 0 | 0 | 16 | 9 |
| 16 | 2 | 0 | 0 |
| 16 | 25 | 0 | 0 |
| 0 | 0 | 9 | 5 |
| 0 | 0 | 25 | 32 |
| 1 | 0 | 0 | 0 |
| 6 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 18 | 9 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[35,40,0,0,1,0,0,0,0,0,32,16,0,0,0,9],[16,16,0,0,2,25,0,0,0,0,9,25,0,0,5,32],[1,6,0,0,0,40,0,0,0,0,40,18,0,0,0,9] >;
C2×D20⋊7C4 in GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_7C_4 % in TeX
G:=Group("C2xD20:7C4"); // GroupNames label
G:=SmallGroup(320,765);
// by ID
G=gap.SmallGroup(320,765);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,58,136,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^9,d*c*d^-1=b^3*c>;
// generators/relations