direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C22⋊D20, C23⋊5D20, C24.54D10, C10⋊1C22≀C2, D10⋊10(C2×D4), (C2×C20)⋊3C23, (D5×C24)⋊1C2, C22⋊3(C2×D20), (C22×C4)⋊7D10, (C22×C10)⋊9D4, C22⋊C4⋊37D10, (C22×D5)⋊13D4, (C22×D20)⋊5C2, C2.8(C22×D20), C10.6(C22×D4), (C2×D20)⋊43C22, (C2×C10).31C24, (C22×C20)⋊7C22, (C2×Dic5)⋊1C23, C22.125(D4×D5), (C22×D5)⋊1C23, (C23×D5)⋊3C22, D10⋊C4⋊44C22, C22.70(C23×D5), (C23×C10).57C22, (C22×Dic5)⋊6C22, C23.146(C22×D5), (C22×C10).123C23, C2.8(C2×D4×D5), C5⋊1(C2×C22≀C2), (C2×C10)⋊4(C2×D4), (C2×C4)⋊3(C22×D5), (C2×C22⋊C4)⋊10D5, (C22×C5⋊D4)⋊3C2, (C10×C22⋊C4)⋊13C2, (C2×C5⋊D4)⋊34C22, (C2×D10⋊C4)⋊16C2, (C5×C22⋊C4)⋊46C22, SmallGroup(320,1159)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 3038 in 662 conjugacy classes, 143 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×14], C4 [×6], C22, C22 [×10], C22 [×86], C5, C2×C4 [×4], C2×C4 [×8], D4 [×24], C23, C23 [×6], C23 [×88], D5 [×10], C10, C10 [×6], C10 [×4], C22⋊C4 [×4], C22⋊C4 [×8], C22×C4 [×2], C22×C4, C2×D4 [×24], C24, C24 [×19], Dic5 [×2], C20 [×4], D10 [×8], D10 [×66], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4 [×2], C22≀C2 [×8], C22×D4 [×3], C25, D20 [×16], C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×4], C2×C20 [×4], C22×D5 [×14], C22×D5 [×70], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22≀C2, D10⋊C4 [×8], C5×C22⋊C4 [×4], C2×D20 [×8], C2×D20 [×8], C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×C20 [×2], C23×D5, C23×D5 [×6], C23×D5 [×12], C23×C10, C22⋊D20 [×8], C2×D10⋊C4 [×2], C10×C22⋊C4, C22×D20 [×2], C22×C5⋊D4, D5×C24, C2×C22⋊D20
Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D5, C2×D4 [×18], C24, D10 [×7], C22≀C2 [×4], C22×D4 [×3], D20 [×4], C22×D5 [×7], C2×C22≀C2, C2×D20 [×6], D4×D5 [×4], C23×D5, C22⋊D20 [×4], C22×D20, C2×D4×D5 [×2], C2×C22⋊D20
Generators and relations
G = < a,b,c,d,e | a2=b2=c2=d20=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=d-1 >
►On 80 points
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(41 76)(42 77)(43 78)(44 79)(45 80)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)(59 74)(60 75)
(1 73)(2 40)(3 75)(4 22)(5 77)(6 24)(7 79)(8 26)(9 61)(10 28)(11 63)(12 30)(13 65)(14 32)(15 67)(16 34)(17 69)(18 36)(19 71)(20 38)(21 60)(23 42)(25 44)(27 46)(29 48)(31 50)(33 52)(35 54)(37 56)(39 58)(41 76)(43 78)(45 80)(47 62)(49 64)(51 66)(53 68)(55 70)(57 72)(59 74)
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 75)(22 76)(23 77)(24 78)(25 79)(26 80)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 73)(40 74)
(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 38)(2 37)(3 36)(4 35)(5 34)(6 33)(7 32)(8 31)(9 30)(10 29)(11 28)(12 27)(13 26)(14 25)(15 24)(16 23)(17 22)(18 21)(19 40)(20 39)(41 69)(42 68)(43 67)(44 66)(45 65)(46 64)(47 63)(48 62)(49 61)(50 80)(51 79)(52 78)(53 77)(54 76)(55 75)(56 74)(57 73)(58 72)(59 71)(60 70)
G:=sub<Sym(80)| (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75), (1,73)(2,40)(3,75)(4,22)(5,77)(6,24)(7,79)(8,26)(9,61)(10,28)(11,63)(12,30)(13,65)(14,32)(15,67)(16,34)(17,69)(18,36)(19,71)(20,38)(21,60)(23,42)(25,44)(27,46)(29,48)(31,50)(33,52)(35,54)(37,56)(39,58)(41,76)(43,78)(45,80)(47,62)(49,64)(51,66)(53,68)(55,70)(57,72)(59,74), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70)>;
G:=Group( (1,39)(2,40)(3,21)(4,22)(5,23)(6,24)(7,25)(8,26)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,33)(16,34)(17,35)(18,36)(19,37)(20,38)(41,76)(42,77)(43,78)(44,79)(45,80)(46,61)(47,62)(48,63)(49,64)(50,65)(51,66)(52,67)(53,68)(54,69)(55,70)(56,71)(57,72)(58,73)(59,74)(60,75), (1,73)(2,40)(3,75)(4,22)(5,77)(6,24)(7,79)(8,26)(9,61)(10,28)(11,63)(12,30)(13,65)(14,32)(15,67)(16,34)(17,69)(18,36)(19,71)(20,38)(21,60)(23,42)(25,44)(27,46)(29,48)(31,50)(33,52)(35,54)(37,56)(39,58)(41,76)(43,78)(45,80)(47,62)(49,64)(51,66)(53,68)(55,70)(57,72)(59,74), (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,75)(22,76)(23,77)(24,78)(25,79)(26,80)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,73)(40,74), (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), (1,38)(2,37)(3,36)(4,35)(5,34)(6,33)(7,32)(8,31)(9,30)(10,29)(11,28)(12,27)(13,26)(14,25)(15,24)(16,23)(17,22)(18,21)(19,40)(20,39)(41,69)(42,68)(43,67)(44,66)(45,65)(46,64)(47,63)(48,62)(49,61)(50,80)(51,79)(52,78)(53,77)(54,76)(55,75)(56,74)(57,73)(58,72)(59,71)(60,70) );
G=PermutationGroup([(1,39),(2,40),(3,21),(4,22),(5,23),(6,24),(7,25),(8,26),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,33),(16,34),(17,35),(18,36),(19,37),(20,38),(41,76),(42,77),(43,78),(44,79),(45,80),(46,61),(47,62),(48,63),(49,64),(50,65),(51,66),(52,67),(53,68),(54,69),(55,70),(56,71),(57,72),(58,73),(59,74),(60,75)], [(1,73),(2,40),(3,75),(4,22),(5,77),(6,24),(7,79),(8,26),(9,61),(10,28),(11,63),(12,30),(13,65),(14,32),(15,67),(16,34),(17,69),(18,36),(19,71),(20,38),(21,60),(23,42),(25,44),(27,46),(29,48),(31,50),(33,52),(35,54),(37,56),(39,58),(41,76),(43,78),(45,80),(47,62),(49,64),(51,66),(53,68),(55,70),(57,72),(59,74)], [(1,58),(2,59),(3,60),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,75),(22,76),(23,77),(24,78),(25,79),(26,80),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,73),(40,74)], [(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)], [(1,38),(2,37),(3,36),(4,35),(5,34),(6,33),(7,32),(8,31),(9,30),(10,29),(11,28),(12,27),(13,26),(14,25),(15,24),(16,23),(17,22),(18,21),(19,40),(20,39),(41,69),(42,68),(43,67),(44,66),(45,65),(46,64),(47,63),(48,62),(49,61),(50,80),(51,79),(52,78),(53,77),(54,76),(55,75),(56,74),(57,73),(58,72),(59,71),(60,70)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 9 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 2 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 4 | 0 | 0 |
| 0 | 0 | 5 | 15 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 9 |
| 0 | 0 | 0 | 0 | 1 | 20 |
| 25 | 39 | 0 | 0 | 0 | 0 |
| 25 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 15 | 37 | 0 | 0 |
| 0 | 0 | 15 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 32 |
| 0 | 0 | 0 | 0 | 17 | 21 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,2,0,0,0,0,39,13,0,0,0,0,0,0,26,5,0,0,0,0,4,15,0,0,0,0,0,0,21,1,0,0,0,0,9,20],[25,25,0,0,0,0,39,16,0,0,0,0,0,0,15,15,0,0,0,0,37,26,0,0,0,0,0,0,20,17,0,0,0,0,32,21] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 2T | 2U | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | D10 | D10 | D10 | D20 | D4×D5 |
| kernel | C2×C22⋊D20 | C22⋊D20 | C2×D10⋊C4 | C10×C22⋊C4 | C22×D20 | C22×C5⋊D4 | D5×C24 | C22×D5 | C22×C10 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C24 | C23 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 8 | 4 | 2 | 8 | 4 | 2 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_2\times C_2^2\rtimes D_{20} % in TeX
G:=Group("C2xC2^2:D20"); // GroupNames label
G:=SmallGroup(320,1159);
// by ID
G=gap.SmallGroup(320,1159);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,675,297,80,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations