metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊18D10, C10.1242+ 1+4, (C4×D5)⋊4D4, C4.32(D4×D5), (C2×Q8)⋊18D10, C20.61(C2×D4), C4.4D4⋊8D5, C20⋊D4⋊24C2, C20⋊4D4⋊14C2, (C4×C20)⋊22C22, C22⋊C4⋊20D10, D10.79(C2×D4), (C2×D20)⋊9C22, D10⋊D4⋊39C2, C22⋊D20⋊23C2, (C2×D4).171D10, C42⋊D5⋊19C2, Dic5.90(C2×D4), (Q8×C10)⋊12C22, C10.88(C22×D4), C20.23D4⋊21C2, (C2×C20).186C23, (C2×C10).218C24, C5⋊4(C22.29C24), (C4×Dic5)⋊35C22, C2.48(D4⋊8D10), D10⋊C4⋊23C22, C23.40(C22×D5), (D4×C10).153C22, C10.D4⋊55C22, (C22×C10).48C23, (C23×D5).63C22, C22.239(C23×D5), (C2×Dic5).113C23, (C22×D5).223C23, (C2×D4×D5)⋊16C2, C2.61(C2×D4×D5), (C2×C4×D5)⋊25C22, (C2×Q8⋊2D5)⋊10C2, (C5×C4.4D4)⋊10C2, (C2×C5⋊D4)⋊22C22, (C5×C22⋊C4)⋊28C22, (C2×C4).193(C22×D5), SmallGroup(320,1346)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊18D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=dad=a-1, cbc-1=a2b-1, dbd=b-1, dcd=c-1 >
Subgroups: 1598 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C22×D5, C22×D5, C22×D5, C22×C10, C22.29C24, C4×Dic5, C10.D4, D10⋊C4, C4×C20, C5×C22⋊C4, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, D4×D5, Q8⋊2D5, C2×C5⋊D4, D4×C10, Q8×C10, C23×D5, C42⋊D5, C20⋊4D4, C22⋊D20, D10⋊D4, C20⋊D4, C20.23D4, C5×C4.4D4, C2×D4×D5, C2×Q8⋊2D5, C42⋊18D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C22×D4, 2+ 1+4, C22×D5, C22.29C24, D4×D5, C23×D5, C2×D4×D5, D4⋊8D10, C42⋊18D10
►On 80 points
(1 53 18 58)(2 59 19 54)(3 55 20 60)(4 51 16 56)(5 57 17 52)(6 62 12 67)(7 68 13 63)(8 64 14 69)(9 70 15 65)(10 66 11 61)(21 34 41 77)(22 78 42 35)(23 36 43 79)(24 80 44 37)(25 38 45 71)(26 72 46 39)(27 40 47 73)(28 74 48 31)(29 32 49 75)(30 76 50 33)
(1 76 9 38)(2 72 10 34)(3 78 6 40)(4 74 7 36)(5 80 8 32)(11 77 19 39)(12 73 20 35)(13 79 16 31)(14 75 17 37)(15 71 18 33)(21 54 26 61)(22 67 27 60)(23 56 28 63)(24 69 29 52)(25 58 30 65)(41 59 46 66)(42 62 47 55)(43 51 48 68)(44 64 49 57)(45 53 50 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 17)(2 16)(3 20)(4 19)(5 18)(6 12)(7 11)(8 15)(9 14)(10 13)(21 28)(22 27)(23 26)(24 25)(29 30)(31 34)(32 33)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 50)(51 59)(52 58)(53 57)(54 56)(61 63)(64 70)(65 69)(66 68)(71 80)(72 79)(73 78)(74 77)(75 76)
G:=sub<Sym(80)| (1,53,18,58)(2,59,19,54)(3,55,20,60)(4,51,16,56)(5,57,17,52)(6,62,12,67)(7,68,13,63)(8,64,14,69)(9,70,15,65)(10,66,11,61)(21,34,41,77)(22,78,42,35)(23,36,43,79)(24,80,44,37)(25,38,45,71)(26,72,46,39)(27,40,47,73)(28,74,48,31)(29,32,49,75)(30,76,50,33), (1,76,9,38)(2,72,10,34)(3,78,6,40)(4,74,7,36)(5,80,8,32)(11,77,19,39)(12,73,20,35)(13,79,16,31)(14,75,17,37)(15,71,18,33)(21,54,26,61)(22,67,27,60)(23,56,28,63)(24,69,29,52)(25,58,30,65)(41,59,46,66)(42,62,47,55)(43,51,48,68)(44,64,49,57)(45,53,50,70), (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), (1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,28)(22,27)(23,26)(24,25)(29,30)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(51,59)(52,58)(53,57)(54,56)(61,63)(64,70)(65,69)(66,68)(71,80)(72,79)(73,78)(74,77)(75,76)>;
G:=Group( (1,53,18,58)(2,59,19,54)(3,55,20,60)(4,51,16,56)(5,57,17,52)(6,62,12,67)(7,68,13,63)(8,64,14,69)(9,70,15,65)(10,66,11,61)(21,34,41,77)(22,78,42,35)(23,36,43,79)(24,80,44,37)(25,38,45,71)(26,72,46,39)(27,40,47,73)(28,74,48,31)(29,32,49,75)(30,76,50,33), (1,76,9,38)(2,72,10,34)(3,78,6,40)(4,74,7,36)(5,80,8,32)(11,77,19,39)(12,73,20,35)(13,79,16,31)(14,75,17,37)(15,71,18,33)(21,54,26,61)(22,67,27,60)(23,56,28,63)(24,69,29,52)(25,58,30,65)(41,59,46,66)(42,62,47,55)(43,51,48,68)(44,64,49,57)(45,53,50,70), (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), (1,17)(2,16)(3,20)(4,19)(5,18)(6,12)(7,11)(8,15)(9,14)(10,13)(21,28)(22,27)(23,26)(24,25)(29,30)(31,34)(32,33)(35,40)(36,39)(37,38)(41,48)(42,47)(43,46)(44,45)(49,50)(51,59)(52,58)(53,57)(54,56)(61,63)(64,70)(65,69)(66,68)(71,80)(72,79)(73,78)(74,77)(75,76) );
G=PermutationGroup([[(1,53,18,58),(2,59,19,54),(3,55,20,60),(4,51,16,56),(5,57,17,52),(6,62,12,67),(7,68,13,63),(8,64,14,69),(9,70,15,65),(10,66,11,61),(21,34,41,77),(22,78,42,35),(23,36,43,79),(24,80,44,37),(25,38,45,71),(26,72,46,39),(27,40,47,73),(28,74,48,31),(29,32,49,75),(30,76,50,33)], [(1,76,9,38),(2,72,10,34),(3,78,6,40),(4,74,7,36),(5,80,8,32),(11,77,19,39),(12,73,20,35),(13,79,16,31),(14,75,17,37),(15,71,18,33),(21,54,26,61),(22,67,27,60),(23,56,28,63),(24,69,29,52),(25,58,30,65),(41,59,46,66),(42,62,47,55),(43,51,48,68),(44,64,49,57),(45,53,50,70)], [(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)], [(1,17),(2,16),(3,20),(4,19),(5,18),(6,12),(7,11),(8,15),(9,14),(10,13),(21,28),(22,27),(23,26),(24,25),(29,30),(31,34),(32,33),(35,40),(36,39),(37,38),(41,48),(42,47),(43,46),(44,45),(49,50),(51,59),(52,58),(53,57),(54,56),(61,63),(64,70),(65,69),(66,68),(71,80),(72,79),(73,78),(74,77),(75,76)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20L | 20M | 20N | 20O | 20P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 10 | 10 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | D10 | D10 | D10 | D10 | 2+ 1+4 | D4×D5 | D4⋊8D10 |
| kernel | C42⋊18D10 | C42⋊D5 | C20⋊4D4 | C22⋊D20 | D10⋊D4 | C20⋊D4 | C20.23D4 | C5×C4.4D4 | C2×D4×D5 | C2×Q8⋊2D5 | C4×D5 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | C10 | C4 | C2 |
| # reps | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C42⋊18D10 ►in GL8(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 39 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 39 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 | 40 | 0 |
| 0 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 6 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 23 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 40 |
| 6 | 34 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 35 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,40,0,0,1,0,0,0,0,39,0,0,1,0,0,0,0,0,39,1,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0],[0,6,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,40,23,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,40,0,0,0,0,0,1,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,5,0,0,0,0,0,0,34,35,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1] >;
C42⋊18D10 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{18}D_{10} % in TeX
G:=Group("C4^2:18D10"); // GroupNames label
G:=SmallGroup(320,1346);
// by ID
G=gap.SmallGroup(320,1346);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,675,570,297,192,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations