metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (D4×C10)⋊18C4, (C2×D4)⋊4Dic5, C20.203(C2×D4), (C2×C20).189D4, D4.5(C2×Dic5), (C22×D4).2D5, (C2×D4).196D10, D4⋊Dic5⋊37C2, C4⋊Dic5⋊68C22, C4.8(C23.D5), C20.78(C22⋊C4), (C2×C20).470C23, C20.139(C22×C4), (C22×C10).195D4, (C22×C4).147D10, C23.84(C5⋊D4), C5⋊5(C23.37D4), C4.10(C22×Dic5), C10.101(C8⋊C22), C2.5(D4.D10), (D4×C10).238C22, C23.21D10⋊18C2, (C22×C20).195C22, C22.20(C23.D5), (D4×C2×C10).2C2, C4.89(C2×C5⋊D4), (C2×C5⋊2C8)⋊9C22, (C5×D4).36(C2×C4), (C2×C20).289(C2×C4), C2.8(C2×C23.D5), (C2×C10).552(C2×D4), (C2×C4.Dic5)⋊17C2, (C2×C4).23(C2×Dic5), C22.90(C2×C5⋊D4), (C2×C4).196(C5⋊D4), C10.113(C2×C22⋊C4), (C2×C4).557(C22×D5), (C2×C10).175(C22⋊C4), SmallGroup(320,842)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (D4×C10)⋊18C4
G = < a,b,c,d | a10=b4=c2=d4=1, ab=ba, ac=ca, dad-1=a-1b2, cbc=dbd-1=b-1, dcd-1=b-1c >
Subgroups: 526 in 190 conjugacy classes, 71 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×18], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], D4 [×4], D4 [×6], C23, C23 [×10], C10, C10 [×2], C10 [×6], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C2×D4 [×6], C2×D4 [×3], C24, Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×18], D4⋊C4 [×4], C42⋊C2, C2×M4(2), C22×D4, C5⋊2C8 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C5×D4 [×6], C22×C10, C22×C10 [×10], C23.37D4, C2×C5⋊2C8 [×2], C4.Dic5 [×2], C4×Dic5, C4⋊Dic5 [×2], C23.D5, C22×C20, D4×C10 [×6], D4×C10 [×3], C23×C10, D4⋊Dic5 [×4], C2×C4.Dic5, C23.21D10, D4×C2×C10, (D4×C10)⋊18C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], Dic5 [×4], D10 [×3], C2×C22⋊C4, C8⋊C22 [×2], C2×Dic5 [×6], C5⋊D4 [×4], C22×D5, C23.37D4, C23.D5 [×4], C22×Dic5, C2×C5⋊D4 [×2], D4.D10 [×2], C2×C23.D5, (D4×C10)⋊18C4
►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 35 40 9)(2 31 36 10)(3 32 37 6)(4 33 38 7)(5 34 39 8)(11 22 18 29)(12 23 19 30)(13 24 20 26)(14 25 16 27)(15 21 17 28)(41 80 46 75)(42 71 47 76)(43 72 48 77)(44 73 49 78)(45 74 50 79)(51 64 56 69)(52 65 57 70)(53 66 58 61)(54 67 59 62)(55 68 60 63)
(1 29)(2 30)(3 26)(4 27)(5 28)(6 13)(7 14)(8 15)(9 11)(10 12)(16 33)(17 34)(18 35)(19 31)(20 32)(21 39)(22 40)(23 36)(24 37)(25 38)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 71)(69 72)(70 73)
(1 69 22 43)(2 63 23 47)(3 67 24 41)(4 61 25 45)(5 65 21 49)(6 59 13 80)(7 53 14 74)(8 57 15 78)(9 51 11 72)(10 55 12 76)(16 79 33 58)(17 73 34 52)(18 77 35 56)(19 71 31 60)(20 75 32 54)(26 46 37 62)(27 50 38 66)(28 44 39 70)(29 48 40 64)(30 42 36 68)
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), (1,35,40,9)(2,31,36,10)(3,32,37,6)(4,33,38,7)(5,34,39,8)(11,22,18,29)(12,23,19,30)(13,24,20,26)(14,25,16,27)(15,21,17,28)(41,80,46,75)(42,71,47,76)(43,72,48,77)(44,73,49,78)(45,74,50,79)(51,64,56,69)(52,65,57,70)(53,66,58,61)(54,67,59,62)(55,68,60,63), (1,29)(2,30)(3,26)(4,27)(5,28)(6,13)(7,14)(8,15)(9,11)(10,12)(16,33)(17,34)(18,35)(19,31)(20,32)(21,39)(22,40)(23,36)(24,37)(25,38)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (1,69,22,43)(2,63,23,47)(3,67,24,41)(4,61,25,45)(5,65,21,49)(6,59,13,80)(7,53,14,74)(8,57,15,78)(9,51,11,72)(10,55,12,76)(16,79,33,58)(17,73,34,52)(18,77,35,56)(19,71,31,60)(20,75,32,54)(26,46,37,62)(27,50,38,66)(28,44,39,70)(29,48,40,64)(30,42,36,68)>;
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), (1,35,40,9)(2,31,36,10)(3,32,37,6)(4,33,38,7)(5,34,39,8)(11,22,18,29)(12,23,19,30)(13,24,20,26)(14,25,16,27)(15,21,17,28)(41,80,46,75)(42,71,47,76)(43,72,48,77)(44,73,49,78)(45,74,50,79)(51,64,56,69)(52,65,57,70)(53,66,58,61)(54,67,59,62)(55,68,60,63), (1,29)(2,30)(3,26)(4,27)(5,28)(6,13)(7,14)(8,15)(9,11)(10,12)(16,33)(17,34)(18,35)(19,31)(20,32)(21,39)(22,40)(23,36)(24,37)(25,38)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(61,74)(62,75)(63,76)(64,77)(65,78)(66,79)(67,80)(68,71)(69,72)(70,73), (1,69,22,43)(2,63,23,47)(3,67,24,41)(4,61,25,45)(5,65,21,49)(6,59,13,80)(7,53,14,74)(8,57,15,78)(9,51,11,72)(10,55,12,76)(16,79,33,58)(17,73,34,52)(18,77,35,56)(19,71,31,60)(20,75,32,54)(26,46,37,62)(27,50,38,66)(28,44,39,70)(29,48,40,64)(30,42,36,68) );
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)], [(1,35,40,9),(2,31,36,10),(3,32,37,6),(4,33,38,7),(5,34,39,8),(11,22,18,29),(12,23,19,30),(13,24,20,26),(14,25,16,27),(15,21,17,28),(41,80,46,75),(42,71,47,76),(43,72,48,77),(44,73,49,78),(45,74,50,79),(51,64,56,69),(52,65,57,70),(53,66,58,61),(54,67,59,62),(55,68,60,63)], [(1,29),(2,30),(3,26),(4,27),(5,28),(6,13),(7,14),(8,15),(9,11),(10,12),(16,33),(17,34),(18,35),(19,31),(20,32),(21,39),(22,40),(23,36),(24,37),(25,38),(41,59),(42,60),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(61,74),(62,75),(63,76),(64,77),(65,78),(66,79),(67,80),(68,71),(69,72),(70,73)], [(1,69,22,43),(2,63,23,47),(3,67,24,41),(4,61,25,45),(5,65,21,49),(6,59,13,80),(7,53,14,74),(8,57,15,78),(9,51,11,72),(10,55,12,76),(16,79,33,58),(17,73,34,52),(18,77,35,56),(19,71,31,60),(20,75,32,54),(26,46,37,62),(27,50,38,66),(28,44,39,70),(29,48,40,64),(30,42,36,68)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | Dic5 | D10 | C5⋊D4 | C5⋊D4 | C8⋊C22 | D4.D10 |
| kernel | (D4×C10)⋊18C4 | D4⋊Dic5 | C2×C4.Dic5 | C23.21D10 | D4×C2×C10 | D4×C10 | C2×C20 | C22×C10 | C22×D4 | C22×C4 | C2×D4 | C2×D4 | C2×C4 | C23 | C10 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 3 | 1 | 2 | 2 | 8 | 4 | 12 | 4 | 2 | 8 |
Matrix representation of (D4×C10)⋊18C4 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 0 |
| 0 | 0 | 0 | 0 | 0 | 10 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 39 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,2,40],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,39,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
(D4×C10)⋊18C4 in GAP, Magma, Sage, TeX
(D_4\times C_{10})\rtimes_{18}C_4 % in TeX
G:=Group("(D4xC10):18C4"); // GroupNames label
G:=SmallGroup(320,842);
// by ID
G=gap.SmallGroup(320,842);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,1684,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^4=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c=d*b*d^-1=b^-1,d*c*d^-1=b^-1*c>;
// generators/relations