direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.29C24, C10.1522+ 1+4, (C2×C20)⋊26D4, C4⋊D4⋊6C10, C4⋊1D4⋊5C10, C42⋊6(C2×C10), C4.16(D4×C10), C22≀C2⋊3C10, (C4×C20)⋊40C22, C4.4D4⋊6C10, C20.323(C2×D4), (C22×D4)⋊7C10, (D4×C10)⋊36C22, C24.16(C2×C10), (Q8×C10)⋊51C22, C22.21(D4×C10), C42⋊C2⋊10C10, (C2×C10).355C24, (C2×C20).664C23, (C22×C20)⋊48C22, C10.190(C22×D4), C23.9(C22×C10), C2.4(C5×2+ 1+4), (C23×C10).16C22, (C22×C10).91C23, C22.29(C23×C10), (C2×C4)⋊4(C5×D4), (D4×C2×C10)⋊22C2, C4⋊C4⋊14(C2×C10), C2.14(D4×C2×C10), (C2×C4○D4)⋊4C10, (C2×D4)⋊4(C2×C10), (C10×C4○D4)⋊20C2, (C5×C4⋊D4)⋊33C2, (C5×C4⋊1D4)⋊16C2, (C5×C4⋊C4)⋊70C22, C22⋊C4⋊4(C2×C10), (C22×C4)⋊8(C2×C10), (C2×Q8)⋊11(C2×C10), (C5×C22≀C2)⋊13C2, (C5×C4.4D4)⋊26C2, (C2×C10).417(C2×D4), (C5×C42⋊C2)⋊31C2, (C5×C22⋊C4)⋊39C22, (C2×C4).22(C22×C10), SmallGroup(320,1537)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.29C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=g2=1, e2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=gdg=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 610 in 334 conjugacy classes, 162 normal (26 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C20, C20, C2×C10, C2×C10, C2×C10, C42⋊C2, C22≀C2, C4⋊D4, C4.4D4, C4⋊1D4, C22×D4, C2×C4○D4, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C22.29C24, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, D4×C10, Q8×C10, C5×C4○D4, C23×C10, C5×C42⋊C2, C5×C22≀C2, C5×C4⋊D4, C5×C4.4D4, C5×C4⋊1D4, D4×C2×C10, C10×C4○D4, C5×C22.29C24
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C24, C2×C10, C22×D4, 2+ 1+4, C5×D4, C22×C10, C22.29C24, D4×C10, C23×C10, D4×C2×C10, C5×2+ 1+4, C5×C22.29C24
►On 80 points
Generators in S80
(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 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 71 16 61)(7 72 17 62)(8 73 18 63)(9 74 19 64)(10 75 20 65)(11 56 76 66)(12 57 77 67)(13 58 78 68)(14 59 79 69)(15 60 80 70)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)
(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 71)(57 72)(58 73)(59 74)(60 75)(61 66)(62 67)(63 68)(64 69)(65 70)
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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,71,16,61)(7,72,17,62)(8,73,18,63)(9,74,19,64)(10,75,20,65)(11,56,76,66)(12,57,77,67)(13,58,78,68)(14,59,79,69)(15,60,80,70)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70)>;
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), (1,26)(2,27)(3,28)(4,29)(5,30)(6,16)(7,17)(8,18)(9,19)(10,20)(11,76)(12,77)(13,78)(14,79)(15,80)(21,31)(22,32)(23,33)(24,34)(25,35)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,66)(57,67)(58,68)(59,69)(60,70)(61,71)(62,72)(63,73)(64,74)(65,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,56)(2,57)(3,58)(4,59)(5,60)(6,41)(7,42)(8,43)(9,44)(10,45)(11,46)(12,47)(13,48)(14,49)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,61)(22,62)(23,63)(24,64)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,73)(34,74)(35,75)(36,76)(37,77)(38,78)(39,79)(40,80), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,71,16,61)(7,72,17,62)(8,73,18,63)(9,74,19,64)(10,75,20,65)(11,56,76,66)(12,57,77,67)(13,58,78,68)(14,59,79,69)(15,60,80,70)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(36,46)(37,47)(38,48)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,11)(7,12)(8,13)(9,14)(10,15)(16,76)(17,77)(18,78)(19,79)(20,80)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,71)(57,72)(58,73)(59,74)(60,75)(61,66)(62,67)(63,68)(64,69)(65,70) );
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)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,16),(7,17),(8,18),(9,19),(10,20),(11,76),(12,77),(13,78),(14,79),(15,80),(21,31),(22,32),(23,33),(24,34),(25,35),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,66),(57,67),(58,68),(59,69),(60,70),(61,71),(62,72),(63,73),(64,74),(65,75)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,76),(7,77),(8,78),(9,79),(10,80),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,41),(7,42),(8,43),(9,44),(10,45),(11,46),(12,47),(13,48),(14,49),(15,50),(16,51),(17,52),(18,53),(19,54),(20,55),(21,61),(22,62),(23,63),(24,64),(25,65),(26,66),(27,67),(28,68),(29,69),(30,70),(31,71),(32,72),(33,73),(34,74),(35,75),(36,76),(37,77),(38,78),(39,79),(40,80)], [(1,46,26,36),(2,47,27,37),(3,48,28,38),(4,49,29,39),(5,50,30,40),(6,71,16,61),(7,72,17,62),(8,73,18,63),(9,74,19,64),(10,75,20,65),(11,56,76,66),(12,57,77,67),(13,58,78,68),(14,59,79,69),(15,60,80,70),(21,51,31,41),(22,52,32,42),(23,53,33,43),(24,54,34,44),(25,55,35,45)], [(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(36,46),(37,47),(38,48),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(56,61),(57,62),(58,63),(59,64),(60,65),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,11),(7,12),(8,13),(9,14),(10,15),(16,76),(17,77),(18,78),(19,79),(20,80),(26,31),(27,32),(28,33),(29,34),(30,35),(36,41),(37,42),(38,43),(39,44),(40,45),(46,51),(47,52),(48,53),(49,54),(50,55),(56,71),(57,72),(58,73),(59,74),(60,75),(61,66),(62,67),(63,68),(64,69),(65,70)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | ··· | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AR | 20A | ··· | 20P | 20Q | ··· | 20AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
110 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | D4 | C5×D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×C22.29C24 | C5×C42⋊C2 | C5×C22≀C2 | C5×C4⋊D4 | C5×C4.4D4 | C5×C4⋊1D4 | D4×C2×C10 | C10×C4○D4 | C22.29C24 | C42⋊C2 | C22≀C2 | C4⋊D4 | C4.4D4 | C4⋊1D4 | C22×D4 | C2×C4○D4 | C2×C20 | C2×C4 | C10 | C2 |
| # reps | 1 | 1 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 16 | 16 | 8 | 8 | 4 | 4 | 4 | 16 | 2 | 8 |
Matrix representation of C5×C22.29C24 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 |
| 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 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 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 | 1 |
| 9 | 1 | 0 | 0 | 0 | 0 |
| 2 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 7 | 7 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 34 | 34 | 40 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 32 | 32 | 0 |
| 0 | 0 | 1 | 7 | 7 | 1 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 32 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 14 | 0 | 2 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 14 | 2 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,2,0,0,0,0,1,32,0,0,0,0,0,0,7,0,40,0,0,0,7,0,34,1,0,0,7,0,34,0,0,0,0,1,40,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,34,1,0,0,0,0,32,7,0,0,0,0,32,7,0,1,0,0,0,1,40,0],[1,0,0,0,0,0,32,40,0,0,0,0,0,0,1,0,0,0,0,0,14,40,0,0,0,0,0,0,1,0,0,0,2,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,14,0,40,0,0,0,2,0,0,40] >;
C5×C22.29C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{29}C_2^4 % in TeX
G:=Group("C5xC2^2.29C2^4"); // GroupNames label
G:=SmallGroup(320,1537);
// by ID
G=gap.SmallGroup(320,1537);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,568,3446,891,2467]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=f^2=g^2=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e^-1=g*d*g=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations