metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊18D10, C10.1242+ (1+4), (C4×D5)⋊4D4, C4.32(D4×D5), (C2×Q8)⋊18D10, C4.4D4⋊8D5, C20.61(C2×D4), C20⋊D4⋊24C2, C4⋊D20⋊14C2, (C4×C20)⋊22C22, C22⋊C4⋊20D10, D10.79(C2×D4), (C2×D20)⋊9C22, C22⋊D20⋊23C2, D10⋊D4⋊39C2, (C2×D4).171D10, C42⋊D5⋊19C2, Dic5.90(C2×D4), (Q8×C10)⋊12C22, C10.88(C22×D4), C20.23D4⋊21C2, (C2×C10).218C24, (C2×C20).186C23, 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
Subgroups: 1598 in 334 conjugacy classes, 103 normal (29 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×30], C5, C2×C4, C2×C4 [×4], C2×C4 [×11], D4 [×22], Q8 [×2], C23 [×2], C23 [×13], D5 [×6], C10, C10 [×2], C10 [×2], C42, C42, C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×3], C2×D4, C2×D4 [×18], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×4], D10 [×2], D10 [×22], C2×C10, C2×C10 [×6], C42⋊C2, C22≀C2 [×4], C4⋊D4 [×4], C4.4D4, C4.4D4, C4⋊1D4 [×2], C22×D4, C2×C4○D4, C4×D5 [×4], C4×D5 [×4], D20 [×12], C2×Dic5, C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×4], C5×D4 [×2], C5×Q8 [×2], C22×D5, C22×D5 [×4], C22×D5 [×8], C22×C10 [×2], C22.29C24, C4×Dic5, C10.D4 [×2], D10⋊C4 [×6], C4×C20, C5×C22⋊C4 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20 [×2], C2×D20 [×6], D4×D5 [×4], Q8⋊2D5 [×4], C2×C5⋊D4 [×6], D4×C10, Q8×C10, C23×D5 [×2], C42⋊D5, C4⋊D20, C22⋊D20 [×4], D10⋊D4 [×4], C20⋊D4, C20.23D4, C5×C4.4D4, C2×D4×D5, C2×Q8⋊2D5, C42⋊18D10
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, 2+ (1+4) [×2], C22×D5 [×7], C22.29C24, D4×D5 [×2], C23×D5, C2×D4×D5, D4⋊8D10 [×2], C42⋊18D10
Generators and relations
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 >
►On 80 points
(1 66 8 61)(2 62 9 67)(3 68 10 63)(4 64 6 69)(5 70 7 65)(11 55 19 60)(12 51 20 56)(13 57 16 52)(14 53 17 58)(15 59 18 54)(21 79 46 39)(22 40 47 80)(23 71 48 31)(24 32 49 72)(25 73 50 33)(26 34 41 74)(27 75 42 35)(28 36 43 76)(29 77 44 37)(30 38 45 78)
(1 34 11 79)(2 40 12 75)(3 36 13 71)(4 32 14 77)(5 38 15 73)(6 72 17 37)(7 78 18 33)(8 74 19 39)(9 80 20 35)(10 76 16 31)(21 61 26 60)(22 56 27 67)(23 63 28 52)(24 58 29 69)(25 65 30 54)(41 55 46 66)(42 62 47 51)(43 57 48 68)(44 64 49 53)(45 59 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 10)(2 9)(3 8)(4 7)(5 6)(11 16)(12 20)(13 19)(14 18)(15 17)(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)(52 60)(53 59)(54 58)(55 57)(61 63)(64 70)(65 69)(66 68)(71 74)(72 73)(75 80)(76 79)(77 78)
G:=sub<Sym(80)| (1,66,8,61)(2,62,9,67)(3,68,10,63)(4,64,6,69)(5,70,7,65)(11,55,19,60)(12,51,20,56)(13,57,16,52)(14,53,17,58)(15,59,18,54)(21,79,46,39)(22,40,47,80)(23,71,48,31)(24,32,49,72)(25,73,50,33)(26,34,41,74)(27,75,42,35)(28,36,43,76)(29,77,44,37)(30,38,45,78), (1,34,11,79)(2,40,12,75)(3,36,13,71)(4,32,14,77)(5,38,15,73)(6,72,17,37)(7,78,18,33)(8,74,19,39)(9,80,20,35)(10,76,16,31)(21,61,26,60)(22,56,27,67)(23,63,28,52)(24,58,29,69)(25,65,30,54)(41,55,46,66)(42,62,47,51)(43,57,48,68)(44,64,49,53)(45,59,50,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,16)(12,20)(13,19)(14,18)(15,17)(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)(52,60)(53,59)(54,58)(55,57)(61,63)(64,70)(65,69)(66,68)(71,74)(72,73)(75,80)(76,79)(77,78)>;
G:=Group( (1,66,8,61)(2,62,9,67)(3,68,10,63)(4,64,6,69)(5,70,7,65)(11,55,19,60)(12,51,20,56)(13,57,16,52)(14,53,17,58)(15,59,18,54)(21,79,46,39)(22,40,47,80)(23,71,48,31)(24,32,49,72)(25,73,50,33)(26,34,41,74)(27,75,42,35)(28,36,43,76)(29,77,44,37)(30,38,45,78), (1,34,11,79)(2,40,12,75)(3,36,13,71)(4,32,14,77)(5,38,15,73)(6,72,17,37)(7,78,18,33)(8,74,19,39)(9,80,20,35)(10,76,16,31)(21,61,26,60)(22,56,27,67)(23,63,28,52)(24,58,29,69)(25,65,30,54)(41,55,46,66)(42,62,47,51)(43,57,48,68)(44,64,49,53)(45,59,50,70), (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), (1,10)(2,9)(3,8)(4,7)(5,6)(11,16)(12,20)(13,19)(14,18)(15,17)(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)(52,60)(53,59)(54,58)(55,57)(61,63)(64,70)(65,69)(66,68)(71,74)(72,73)(75,80)(76,79)(77,78) );
G=PermutationGroup([(1,66,8,61),(2,62,9,67),(3,68,10,63),(4,64,6,69),(5,70,7,65),(11,55,19,60),(12,51,20,56),(13,57,16,52),(14,53,17,58),(15,59,18,54),(21,79,46,39),(22,40,47,80),(23,71,48,31),(24,32,49,72),(25,73,50,33),(26,34,41,74),(27,75,42,35),(28,36,43,76),(29,77,44,37),(30,38,45,78)], [(1,34,11,79),(2,40,12,75),(3,36,13,71),(4,32,14,77),(5,38,15,73),(6,72,17,37),(7,78,18,33),(8,74,19,39),(9,80,20,35),(10,76,16,31),(21,61,26,60),(22,56,27,67),(23,63,28,52),(24,58,29,69),(25,65,30,54),(41,55,46,66),(42,62,47,51),(43,57,48,68),(44,64,49,53),(45,59,50,70)], [(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)], [(1,10),(2,9),(3,8),(4,7),(5,6),(11,16),(12,20),(13,19),(14,18),(15,17),(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),(52,60),(53,59),(54,58),(55,57),(61,63),(64,70),(65,69),(66,68),(71,74),(72,73),(75,80),(76,79),(77,78)])
Matrix representation ►G ⊆ 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] >;
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 | C4⋊D20 | 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 |
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