metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×D20)⋊25C4, (C2×C4).48D20, (C2×C20).145D4, C42⋊C2⋊5D5, (C2×Dic10)⋊24C4, C22⋊C4.83D10, C22.15(C2×D20), C20.96(C22⋊C4), (C22×C4).114D10, C23.1D10⋊7C2, C4.53(D10⋊C4), C23.73(C22×D5), C5⋊5(C23.C23), C23.D5.78C22, C23.21D10⋊14C2, (C22×C20).155C22, (C22×C10).112C23, (C2×C4×D5)⋊3C4, (C2×C4).47(C4×D5), C22.19(C2×C4×D5), (C2×C4○D20).9C2, (C2×C20).268(C2×C4), (C5×C42⋊C2)⋊5C2, (C2×C10).462(C2×D4), (C2×C4).46(C5⋊D4), C10.89(C2×C22⋊C4), (C2×Dic5).4(C2×C4), (C22×D5).4(C2×C4), C22.28(C2×C5⋊D4), C2.21(C2×D10⋊C4), (C2×C5⋊D4).92C22, (C2×C10).114(C22×C4), (C5×C22⋊C4).94C22, SmallGroup(320,633)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for (C2×D20)⋊25C4
G = < a,b,c,d | a2=b20=c2=d4=1, ab=ba, dcd-1=ac=ca, dad-1=ab10, cbc=b-1, bd=db >
Subgroups: 638 in 158 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2 [×5], C4 [×4], C4 [×6], C22 [×3], C22 [×5], C5, C2×C4 [×6], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×2], C10, C10 [×3], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×4], C20 [×2], D10 [×4], C2×C10 [×3], C2×C10, C23⋊C4 [×4], C42⋊C2, C42⋊C2, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×6], C2×C20 [×2], C22×D5 [×2], C22×C10, C23.C23, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, C23.1D10 [×4], C23.21D10, C5×C42⋊C2, C2×C4○D20, (C2×D20)⋊25C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.C23, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, (C2×D20)⋊25C4
►On 80 points
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 73)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 61)(57 62)(58 63)(59 64)(60 65)
(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 77)(2 76)(3 75)(4 74)(5 73)(6 72)(7 71)(8 70)(9 69)(10 68)(11 67)(12 66)(13 65)(14 64)(15 63)(16 62)(17 61)(18 80)(19 79)(20 78)(21 47)(22 46)(23 45)(24 44)(25 43)(26 42)(27 41)(28 60)(29 59)(30 58)(31 57)(32 56)(33 55)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)
(1 63 26 58)(2 64 27 59)(3 65 28 60)(4 66 29 41)(5 67 30 42)(6 68 31 43)(7 69 32 44)(8 70 33 45)(9 71 34 46)(10 72 35 47)(11 73 36 48)(12 74 37 49)(13 75 38 50)(14 76 39 51)(15 77 40 52)(16 78 21 53)(17 79 22 54)(18 80 23 55)(19 61 24 56)(20 62 25 57)
G:=sub<Sym(80)| (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48), (1,63,26,58)(2,64,27,59)(3,65,28,60)(4,66,29,41)(5,67,30,42)(6,68,31,43)(7,69,32,44)(8,70,33,45)(9,71,34,46)(10,72,35,47)(11,73,36,48)(12,74,37,49)(13,75,38,50)(14,76,39,51)(15,77,40,52)(16,78,21,53)(17,79,22,54)(18,80,23,55)(19,61,24,56)(20,62,25,57)>;
G:=Group( (1,36)(2,37)(3,38)(4,39)(5,40)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31)(17,32)(18,33)(19,34)(20,35)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,73)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,61)(57,62)(58,63)(59,64)(60,65), (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,72)(7,71)(8,70)(9,69)(10,68)(11,67)(12,66)(13,65)(14,64)(15,63)(16,62)(17,61)(18,80)(19,79)(20,78)(21,47)(22,46)(23,45)(24,44)(25,43)(26,42)(27,41)(28,60)(29,59)(30,58)(31,57)(32,56)(33,55)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48), (1,63,26,58)(2,64,27,59)(3,65,28,60)(4,66,29,41)(5,67,30,42)(6,68,31,43)(7,69,32,44)(8,70,33,45)(9,71,34,46)(10,72,35,47)(11,73,36,48)(12,74,37,49)(13,75,38,50)(14,76,39,51)(15,77,40,52)(16,78,21,53)(17,79,22,54)(18,80,23,55)(19,61,24,56)(20,62,25,57) );
G=PermutationGroup([(1,36),(2,37),(3,38),(4,39),(5,40),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31),(17,32),(18,33),(19,34),(20,35),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,73),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,61),(57,62),(58,63),(59,64),(60,65)], [(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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,72),(7,71),(8,70),(9,69),(10,68),(11,67),(12,66),(13,65),(14,64),(15,63),(16,62),(17,61),(18,80),(19,79),(20,78),(21,47),(22,46),(23,45),(24,44),(25,43),(26,42),(27,41),(28,60),(29,59),(30,58),(31,57),(32,56),(33,55),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48)], [(1,63,26,58),(2,64,27,59),(3,65,28,60),(4,66,29,41),(5,67,30,42),(6,68,31,43),(7,69,32,44),(8,70,33,45),(9,71,34,46),(10,72,35,47),(11,73,36,48),(12,74,37,49),(13,75,38,50),(14,76,39,51),(15,77,40,52),(16,78,21,53),(17,79,22,54),(18,80,23,55),(19,61,24,56),(20,62,25,57)])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | ··· | 4O | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | ··· | 20AB |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 20 | 20 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | C23.C23 | (C2×D20)⋊25C4 |
| kernel | (C2×D20)⋊25C4 | C23.1D10 | C23.21D10 | C5×C42⋊C2 | C2×C4○D20 | C2×Dic10 | C2×C4×D5 | C2×D20 | C2×C20 | C42⋊C2 | C22⋊C4 | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C5 | C1 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 2 | 8 |
Matrix representation of (C2×D20)⋊25C4 ►in GL4(𝔽41) generated by
| 24 | 40 | 24 | 40 |
| 1 | 17 | 1 | 17 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 40 | 24 |
| 9 | 30 | 0 | 0 |
| 11 | 14 | 0 | 0 |
| 0 | 0 | 9 | 30 |
| 0 | 0 | 11 | 14 |
| 38 | 38 | 38 | 38 |
| 17 | 3 | 17 | 3 |
| 6 | 6 | 3 | 3 |
| 7 | 35 | 24 | 38 |
| 17 | 1 | 8 | 21 |
| 40 | 24 | 20 | 32 |
| 7 | 39 | 24 | 40 |
| 2 | 34 | 1 | 17 |
G:=sub<GL(4,GF(41))| [24,1,0,0,40,17,0,0,24,1,17,40,40,17,1,24],[9,11,0,0,30,14,0,0,0,0,9,11,0,0,30,14],[38,17,6,7,38,3,6,35,38,17,3,24,38,3,3,38],[17,40,7,2,1,24,39,34,8,20,24,1,21,32,40,17] >;
(C2×D20)⋊25C4 in GAP, Magma, Sage, TeX
(C_2\times D_{20})\rtimes_{25}C_4 % in TeX
G:=Group("(C2xD20):25C4"); // GroupNames label
G:=SmallGroup(320,633);
// by ID
G=gap.SmallGroup(320,633);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1123,438,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^20=c^2=d^4=1,a*b=b*a,d*c*d^-1=a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,b*d=d*b>;
// generators/relations