direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C22.32C24, C10.1542+ 1+4, (C4×D4)⋊10C10, (D4×C20)⋊39C2, C4⋊D4⋊8C10, C42⋊7(C2×C10), C22⋊Q8⋊7C10, C22≀C2⋊4C10, (C4×C20)⋊41C22, C4.4D4⋊8C10, C42⋊2C2⋊1C10, (D4×C10)⋊37C22, C24.17(C2×C10), (Q8×C10)⋊28C22, (C2×C20).667C23, (C2×C10).358C24, (C22×C20)⋊49C22, C22.D4⋊4C10, C2.6(C5×2+ 1+4), C22.32(C23×C10), (C23×C10).17C22, C23.11(C22×C10), (C22×C10).93C23, C4⋊C4⋊15(C2×C10), (C2×D4)⋊5(C2×C10), (C2×Q8)⋊3(C2×C10), (C5×C4⋊D4)⋊35C2, (C5×C4⋊C4)⋊71C22, (C22×C4)⋊9(C2×C10), C2.15(C10×C4○D4), (C5×C22⋊Q8)⋊34C2, (C5×C22≀C2)⋊14C2, C22.4(C5×C4○D4), (C2×C22⋊C4)⋊13C10, (C10×C22⋊C4)⋊33C2, C22⋊C4⋊15(C2×C10), C10.234(C2×C4○D4), (C5×C4.4D4)⋊28C2, (C5×C42⋊2C2)⋊12C2, (C2×C10).52(C4○D4), (C5×C22⋊C4)⋊69C22, (C2×C4).25(C22×C10), (C5×C22.D4)⋊23C2, SmallGroup(320,1540)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.32C24
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=f2=g2=1, e2=c, 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: 434 in 250 conjugacy classes, 146 normal (38 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C10, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, C20, C2×C10, C2×C10, C2×C10, C2×C22⋊C4, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C22×C10, C22×C10, C22.32C24, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, C22×C20, D4×C10, D4×C10, Q8×C10, C23×C10, C10×C22⋊C4, D4×C20, C5×C22≀C2, C5×C4⋊D4, C5×C4⋊D4, C5×C22⋊Q8, C5×C22.D4, C5×C4.4D4, C5×C42⋊2C2, C5×C22.32C24
Quotients: C1, C2, C22, C5, C23, C10, C4○D4, C24, C2×C10, C2×C4○D4, 2+ 1+4, C22×C10, C22.32C24, C5×C4○D4, C23×C10, C10×C4○D4, C5×2+ 1+4, C5×C22.32C24
►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 35)(2 31)(3 32)(4 33)(5 34)(6 80)(7 76)(8 77)(9 78)(10 79)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 75)(57 71)(58 72)(59 73)(60 74)(61 67)(62 68)(63 69)(64 70)(65 66)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 54)(7 55)(8 51)(9 52)(10 53)(11 42)(12 43)(13 44)(14 45)(15 41)(16 47)(17 48)(18 49)(19 50)(20 46)(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 41 21 36)(2 42 22 37)(3 43 23 38)(4 44 24 39)(5 45 25 40)(6 60 20 65)(7 56 16 61)(8 57 17 62)(9 58 18 63)(10 59 19 64)(11 68 77 71)(12 69 78 72)(13 70 79 73)(14 66 80 74)(15 67 76 75)(26 54 34 46)(27 55 35 47)(28 51 31 48)(29 52 32 49)(30 53 33 50)
(6 14)(7 15)(8 11)(9 12)(10 13)(16 76)(17 77)(18 78)(19 79)(20 80)(36 55)(37 51)(38 52)(39 53)(40 54)(41 47)(42 48)(43 49)(44 50)(45 46)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 76)(17 77)(18 78)(19 79)(20 80)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 67)(57 68)(58 69)(59 70)(60 66)(61 75)(62 71)(63 72)(64 73)(65 74)
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)(2,31)(3,32)(4,33)(5,34)(6,80)(7,76)(8,77)(9,78)(10,79)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,77)(12,78)(13,79)(14,80)(15,76)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73), (1,56)(2,57)(3,58)(4,59)(5,60)(6,54)(7,55)(8,51)(9,52)(10,53)(11,42)(12,43)(13,44)(14,45)(15,41)(16,47)(17,48)(18,49)(19,50)(20,46)(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), (1,41,21,36)(2,42,22,37)(3,43,23,38)(4,44,24,39)(5,45,25,40)(6,60,20,65)(7,56,16,61)(8,57,17,62)(9,58,18,63)(10,59,19,64)(11,68,77,71)(12,69,78,72)(13,70,79,73)(14,66,80,74)(15,67,76,75)(26,54,34,46)(27,55,35,47)(28,51,31,48)(29,52,32,49)(30,53,33,50), (6,14)(7,15)(8,11)(9,12)(10,13)(16,76)(17,77)(18,78)(19,79)(20,80)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73), (1,21)(2,22)(3,23)(4,24)(5,25)(6,14)(7,15)(8,11)(9,12)(10,13)(16,76)(17,77)(18,78)(19,79)(20,80)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,67)(57,68)(58,69)(59,70)(60,66)(61,75)(62,71)(63,72)(64,73)(65,74)>;
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)(2,31)(3,32)(4,33)(5,34)(6,80)(7,76)(8,77)(9,78)(10,79)(11,17)(12,18)(13,19)(14,20)(15,16)(21,27)(22,28)(23,29)(24,30)(25,26)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,75)(57,71)(58,72)(59,73)(60,74)(61,67)(62,68)(63,69)(64,70)(65,66), (1,21)(2,22)(3,23)(4,24)(5,25)(6,20)(7,16)(8,17)(9,18)(10,19)(11,77)(12,78)(13,79)(14,80)(15,76)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73), (1,56)(2,57)(3,58)(4,59)(5,60)(6,54)(7,55)(8,51)(9,52)(10,53)(11,42)(12,43)(13,44)(14,45)(15,41)(16,47)(17,48)(18,49)(19,50)(20,46)(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), (1,41,21,36)(2,42,22,37)(3,43,23,38)(4,44,24,39)(5,45,25,40)(6,60,20,65)(7,56,16,61)(8,57,17,62)(9,58,18,63)(10,59,19,64)(11,68,77,71)(12,69,78,72)(13,70,79,73)(14,66,80,74)(15,67,76,75)(26,54,34,46)(27,55,35,47)(28,51,31,48)(29,52,32,49)(30,53,33,50), (6,14)(7,15)(8,11)(9,12)(10,13)(16,76)(17,77)(18,78)(19,79)(20,80)(36,55)(37,51)(38,52)(39,53)(40,54)(41,47)(42,48)(43,49)(44,50)(45,46)(56,61)(57,62)(58,63)(59,64)(60,65)(66,74)(67,75)(68,71)(69,72)(70,73), (1,21)(2,22)(3,23)(4,24)(5,25)(6,14)(7,15)(8,11)(9,12)(10,13)(16,76)(17,77)(18,78)(19,79)(20,80)(26,34)(27,35)(28,31)(29,32)(30,33)(36,41)(37,42)(38,43)(39,44)(40,45)(46,54)(47,55)(48,51)(49,52)(50,53)(56,67)(57,68)(58,69)(59,70)(60,66)(61,75)(62,71)(63,72)(64,73)(65,74) );
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),(2,31),(3,32),(4,33),(5,34),(6,80),(7,76),(8,77),(9,78),(10,79),(11,17),(12,18),(13,19),(14,20),(15,16),(21,27),(22,28),(23,29),(24,30),(25,26),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,75),(57,71),(58,72),(59,73),(60,74),(61,67),(62,68),(63,69),(64,70),(65,66)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,20),(7,16),(8,17),(9,18),(10,19),(11,77),(12,78),(13,79),(14,80),(15,76),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,54),(7,55),(8,51),(9,52),(10,53),(11,42),(12,43),(13,44),(14,45),(15,41),(16,47),(17,48),(18,49),(19,50),(20,46),(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)], [(1,41,21,36),(2,42,22,37),(3,43,23,38),(4,44,24,39),(5,45,25,40),(6,60,20,65),(7,56,16,61),(8,57,17,62),(9,58,18,63),(10,59,19,64),(11,68,77,71),(12,69,78,72),(13,70,79,73),(14,66,80,74),(15,67,76,75),(26,54,34,46),(27,55,35,47),(28,51,31,48),(29,52,32,49),(30,53,33,50)], [(6,14),(7,15),(8,11),(9,12),(10,13),(16,76),(17,77),(18,78),(19,79),(20,80),(36,55),(37,51),(38,52),(39,53),(40,54),(41,47),(42,48),(43,49),(44,50),(45,46),(56,61),(57,62),(58,63),(59,64),(60,65),(66,74),(67,75),(68,71),(69,72),(70,73)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,14),(7,15),(8,11),(9,12),(10,13),(16,76),(17,77),(18,78),(19,79),(20,80),(26,34),(27,35),(28,31),(29,32),(30,33),(36,41),(37,42),(38,43),(39,44),(40,45),(46,54),(47,55),(48,51),(49,52),(50,53),(56,67),(57,68),(58,69),(59,70),(60,66),(61,75),(62,71),(63,72),(64,73),(65,74)]])
110 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 10M | ··· | 10T | 10U | ··· | 10AJ | 20A | ··· | 20P | 20Q | ··· | 20AV |
| order | 1 | 2 | 2 | 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 | 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 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C10 | C4○D4 | C5×C4○D4 | 2+ 1+4 | C5×2+ 1+4 |
| kernel | C5×C22.32C24 | C10×C22⋊C4 | D4×C20 | C5×C22≀C2 | C5×C4⋊D4 | C5×C22⋊Q8 | C5×C22.D4 | C5×C4.4D4 | C5×C42⋊2C2 | C22.32C24 | C2×C22⋊C4 | C4×D4 | C22≀C2 | C4⋊D4 | C22⋊Q8 | C22.D4 | C4.4D4 | C42⋊2C2 | C2×C10 | C22 | C10 | C2 |
| # reps | 1 | 1 | 2 | 2 | 3 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 12 | 4 | 8 | 8 | 8 | 4 | 16 | 2 | 8 |
Matrix representation of C5×C22.32C24 ►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 | 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 |
| 2 | 1 | 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 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 40 | 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 1 |
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],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,2,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[40,0,0,0,0,0,40,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C5×C22.32C24 in GAP, Magma, Sage, TeX
C_5\times C_2^2._{32}C_2^4 % in TeX
G:=Group("C5xC2^2.32C2^4"); // GroupNames label
G:=SmallGroup(320,1540);
// by ID
G=gap.SmallGroup(320,1540);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,1128,3446,891,2467]);
// 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=c,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