metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23⋊C4⋊5D5, C23.5(C4×D5), (C2×Dic10)⋊6C4, C23⋊Dic5⋊3C2, C22.23(D4×D5), (C2×D4).119D10, C22⋊C4.43D10, (C22×Dic5)⋊4C4, (D4×C10).7C22, C23.1(C22×D5), (C2×Dic5).188D4, C23.1D10⋊3C2, (C22×C10).1C23, C23.D5.1C22, C5⋊4(C23.C23), C23.11D10⋊23C2, Dic5.20(C22⋊C4), (C22×Dic5).25C22, (C2×C4×D5)⋊1C4, (C2×C5⋊D4)⋊1C4, (C2×C4).5(C4×D5), (C5×C23⋊C4)⋊3C2, C22.12(C2×C4×D5), (C2×C20).19(C2×C4), (C2×C10).16(C2×D4), C2.11(D5×C22⋊C4), (C2×D4⋊2D5).1C2, C10.51(C2×C22⋊C4), (C22×C10).5(C2×C4), (C2×Dic5).1(C2×C4), (C22×D5).1(C2×C4), (C2×C5⋊D4).1C22, (C2×C10).107(C22×C4), (C5×C22⋊C4).82C22, SmallGroup(320,367)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23⋊C4⋊5D5
G = < a,b,c,d,e,f | a2=b2=c2=d4=e5=f2=1, ab=ba, faf=ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, fdf=bcd, fef=e-1 >
Subgroups: 622 in 158 conjugacy classes, 51 normal (31 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, C23⋊C4, C23⋊C4, C42⋊C2, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C22×D5, C22×C10, C23.C23, C4×Dic5, C10.D4, C23.D5, C5×C22⋊C4, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, C23.1D10, C23⋊Dic5, C5×C23⋊C4, C23.11D10, C2×D4⋊2D5, C23⋊C4⋊5D5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D5, C22⋊C4, C22×C4, C2×D4, D10, C2×C22⋊C4, C4×D5, C22×D5, C23.C23, C2×C4×D5, D4×D5, D5×C22⋊C4, C23⋊C4⋊5D5
►On 80 points
Generators in S80
(1 44)(2 45)(3 41)(4 42)(5 43)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(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 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 66 56 71)(42 67 57 72)(43 68 58 73)(44 69 59 74)(45 70 60 75)(46 61 51 76)(47 62 52 77)(48 63 53 78)(49 64 54 79)(50 65 55 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 33)(2 32)(3 31)(4 35)(5 34)(6 36)(7 40)(8 39)(9 38)(10 37)(11 26)(12 30)(13 29)(14 28)(15 27)(16 21)(17 25)(18 24)(19 23)(20 22)(41 76)(42 80)(43 79)(44 78)(45 77)(46 71)(47 75)(48 74)(49 73)(50 72)(51 61)(52 65)(53 64)(54 63)(55 62)(56 66)(57 70)(58 69)(59 68)(60 67)
G:=sub<Sym(80)| (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(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), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,66,56,71)(42,67,57,72)(43,68,58,73)(44,69,59,74)(45,70,60,75)(46,61,51,76)(47,62,52,77)(48,63,53,78)(49,64,54,79)(50,65,55,80), (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), (1,33)(2,32)(3,31)(4,35)(5,34)(6,36)(7,40)(8,39)(9,38)(10,37)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22)(41,76)(42,80)(43,79)(44,78)(45,77)(46,71)(47,75)(48,74)(49,73)(50,72)(51,61)(52,65)(53,64)(54,63)(55,62)(56,66)(57,70)(58,69)(59,68)(60,67)>;
G:=Group( (1,44)(2,45)(3,41)(4,42)(5,43)(6,46)(7,47)(8,48)(9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,57)(18,58)(19,59)(20,60)(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), (1,14)(2,15)(3,11)(4,12)(5,13)(6,16)(7,17)(8,18)(9,19)(10,20)(21,36)(22,37)(23,38)(24,39)(25,40)(26,31)(27,32)(28,33)(29,34)(30,35)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,9)(2,10)(3,6)(4,7)(5,8)(11,16)(12,17)(13,18)(14,19)(15,20)(21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,46)(42,47)(43,48)(44,49)(45,50)(51,56)(52,57)(53,58)(54,59)(55,60)(61,66)(62,67)(63,68)(64,69)(65,70)(71,76)(72,77)(73,78)(74,79)(75,80), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,66,56,71)(42,67,57,72)(43,68,58,73)(44,69,59,74)(45,70,60,75)(46,61,51,76)(47,62,52,77)(48,63,53,78)(49,64,54,79)(50,65,55,80), (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), (1,33)(2,32)(3,31)(4,35)(5,34)(6,36)(7,40)(8,39)(9,38)(10,37)(11,26)(12,30)(13,29)(14,28)(15,27)(16,21)(17,25)(18,24)(19,23)(20,22)(41,76)(42,80)(43,79)(44,78)(45,77)(46,71)(47,75)(48,74)(49,73)(50,72)(51,61)(52,65)(53,64)(54,63)(55,62)(56,66)(57,70)(58,69)(59,68)(60,67) );
G=PermutationGroup([[(1,44),(2,45),(3,41),(4,42),(5,43),(6,46),(7,47),(8,48),(9,49),(10,50),(11,51),(12,52),(13,53),(14,54),(15,55),(16,56),(17,57),(18,58),(19,59),(20,60),(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)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,16),(7,17),(8,18),(9,19),(10,20),(21,36),(22,37),(23,38),(24,39),(25,40),(26,31),(27,32),(28,33),(29,34),(30,35),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,76),(62,77),(63,78),(64,79),(65,80),(66,71),(67,72),(68,73),(69,74),(70,75)], [(1,9),(2,10),(3,6),(4,7),(5,8),(11,16),(12,17),(13,18),(14,19),(15,20),(21,26),(22,27),(23,28),(24,29),(25,30),(31,36),(32,37),(33,38),(34,39),(35,40),(41,46),(42,47),(43,48),(44,49),(45,50),(51,56),(52,57),(53,58),(54,59),(55,60),(61,66),(62,67),(63,68),(64,69),(65,70),(71,76),(72,77),(73,78),(74,79),(75,80)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,66,56,71),(42,67,57,72),(43,68,58,73),(44,69,59,74),(45,70,60,75),(46,61,51,76),(47,62,52,77),(48,63,53,78),(49,64,54,79),(50,65,55,80)], [(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)], [(1,33),(2,32),(3,31),(4,35),(5,34),(6,36),(7,40),(8,39),(9,38),(10,37),(11,26),(12,30),(13,29),(14,28),(15,27),(16,21),(17,25),(18,24),(19,23),(20,22),(41,76),(42,80),(43,79),(44,78),(45,77),(46,71),(47,75),(48,74),(49,73),(50,72),(51,61),(52,65),(53,64),(54,63),(55,62),(56,66),(57,70),(58,69),(59,68),(60,67)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4E | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4O | 5A | 5B | 10A | 10B | 10C | ··· | 10H | 10I | 10J | 20A | ··· | 20J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | 10 | 10 | ··· | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 20 | 4 | ··· | 4 | 5 | 5 | 10 | 10 | 10 | 20 | ··· | 20 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | ··· | 8 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | D10 | D10 | C4×D5 | C4×D5 | C23.C23 | D4×D5 | C23⋊C4⋊5D5 |
| kernel | C23⋊C4⋊5D5 | C23.1D10 | C23⋊Dic5 | C5×C23⋊C4 | C23.11D10 | C2×D4⋊2D5 | C2×Dic10 | C2×C4×D5 | C22×Dic5 | C2×C5⋊D4 | C2×Dic5 | C23⋊C4 | C22⋊C4 | C2×D4 | C2×C4 | C23 | C5 | C22 | C1 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 2 | 4 | 2 |
Matrix representation of C23⋊C4⋊5D5 ►in GL8(𝔽41)
| 20 | 11 | 33 | 40 | 0 | 0 | 0 | 0 |
| 10 | 18 | 8 | 33 | 0 | 0 | 0 | 0 |
| 8 | 0 | 23 | 30 | 0 | 0 | 0 | 0 |
| 35 | 8 | 31 | 21 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 32 |
| 0 | 0 | 0 | 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 40 |
| 30 | 29 | 28 | 24 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 28 | 0 | 0 | 0 | 0 |
| 15 | 18 | 40 | 12 | 0 | 0 | 0 | 0 |
| 15 | 15 | 0 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 6 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 23 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 18 | 40 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 11 | 6 | 17 | 13 | 0 | 0 | 0 | 0 |
| 0 | 1 | 13 | 28 | 0 | 0 | 0 | 0 |
| 37 | 14 | 29 | 1 | 0 | 0 | 0 | 0 |
| 37 | 14 | 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 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 |
G:=sub<GL(8,GF(41))| [20,10,8,35,0,0,0,0,11,18,0,8,0,0,0,0,33,8,23,31,0,0,0,0,40,33,30,21,0,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[30,0,15,15,0,0,0,0,29,1,18,15,0,0,0,0,28,13,40,0,0,0,0,0,24,28,12,11,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[6,40,23,0,0,0,0,0,1,0,0,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[11,0,37,37,0,0,0,0,6,1,14,14,0,0,0,0,17,13,29,30,0,0,0,0,13,28,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0] >;
C23⋊C4⋊5D5 in GAP, Magma, Sage, TeX
C_2^3\rtimes C_4\rtimes_5D_5
% in TeX
G:=Group("C2^3:C4:5D5"); // GroupNames label
G:=SmallGroup(320,367);
// by ID
G=gap.SmallGroup(320,367);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,58,570,438,12550]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^4=e^5=f^2=1,a*b=b*a,f*a*f=a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=b*c*d,f*e*f=e^-1>;
// generators/relations