metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.44D20, C24.48D10, (C23×D5)⋊8C4, (C22×C4)⋊1D10, C5⋊2(C24⋊3C4), (D5×C24).1C2, C23.51(C4×D5), C10.37C22≀C2, D10⋊4(C22⋊C4), (C22×C20)⋊1C22, C22.43(C2×D20), C22.100(D4×D5), (C22×C10).67D4, C2.2(C23⋊D10), C2.4(C22⋊D20), (C22×D5).124D4, C23.52(C5⋊D4), C22⋊3(D10⋊C4), (C23×C10).38C22, (C22×Dic5)⋊2C22, (C23×D5).99C22, C23.282(C22×D5), (C22×C10).329C23, (C2×C22⋊C4)⋊2D5, (C10×C22⋊C4)⋊2C2, (C2×C23.D5)⋊2C2, C2.28(D5×C22⋊C4), (C2×D10⋊C4)⋊3C2, C22.126(C2×C4×D5), (C2×C10)⋊4(C22⋊C4), (C2×C10).321(C2×D4), C2.9(C2×D10⋊C4), C10.77(C2×C22⋊C4), C22.50(C2×C5⋊D4), (C2×C10).209(C22×C4), (C22×C10).121(C2×C4), (C22×D5).104(C2×C4), SmallGroup(320,582)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.44D20
G = < a,b,c,d,e | a2=b2=c2=d20=1, e2=b, ab=ba, dad-1=eae-1=ac=ca, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 2318 in 506 conjugacy classes, 95 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×12], C4 [×4], C22 [×3], C22 [×8], C22 [×76], C5, C2×C4 [×12], C23, C23 [×6], C23 [×80], D5 [×8], C10, C10 [×6], C10 [×4], C22⋊C4 [×12], C22×C4 [×2], C22×C4 [×2], C24, C24 [×18], Dic5 [×2], C20 [×2], D10 [×8], D10 [×56], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4 [×5], C25, C2×Dic5 [×6], C2×C20 [×6], C22×D5 [×12], C22×D5 [×64], C22×C10, C22×C10 [×6], C22×C10 [×4], C24⋊3C4, D10⋊C4 [×8], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×2], C22×C20 [×2], C23×D5 [×6], C23×D5 [×12], C23×C10, C2×D10⋊C4 [×4], C2×C23.D5, C10×C22⋊C4, D5×C24, C23.44D20
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×12], C23, D5, C22⋊C4 [×12], C22×C4, C2×D4 [×6], D10 [×3], C2×C22⋊C4 [×3], C22≀C2 [×4], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C24⋊3C4, D10⋊C4 [×4], C2×C4×D5, C2×D20, D4×D5 [×4], C2×C5⋊D4, D5×C22⋊C4 [×2], C22⋊D20 [×2], C2×D10⋊C4, C23⋊D10 [×2], C23.44D20
►On 80 points
(1 41)(2 35)(3 43)(4 37)(5 45)(6 39)(7 47)(8 21)(9 49)(10 23)(11 51)(12 25)(13 53)(14 27)(15 55)(16 29)(17 57)(18 31)(19 59)(20 33)(22 74)(24 76)(26 78)(28 80)(30 62)(32 64)(34 66)(36 68)(38 70)(40 72)(42 67)(44 69)(46 71)(48 73)(50 75)(52 77)(54 79)(56 61)(58 63)(60 65)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(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 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 73)(9 74)(10 75)(11 76)(12 77)(13 78)(14 79)(15 80)(16 61)(17 62)(18 63)(19 64)(20 65)(21 48)(22 49)(23 50)(24 51)(25 52)(26 53)(27 54)(28 55)(29 56)(30 57)(31 58)(32 59)(33 60)(34 41)(35 42)(36 43)(37 44)(38 45)(39 46)(40 47)
(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 50 24 65)(2 64 25 49)(3 48 26 63)(4 62 27 47)(5 46 28 61)(6 80 29 45)(7 44 30 79)(8 78 31 43)(9 42 32 77)(10 76 33 41)(11 60 34 75)(12 74 35 59)(13 58 36 73)(14 72 37 57)(15 56 38 71)(16 70 39 55)(17 54 40 69)(18 68 21 53)(19 52 22 67)(20 66 23 51)
G:=sub<Sym(80)| (1,41)(2,35)(3,43)(4,37)(5,45)(6,39)(7,47)(8,21)(9,49)(10,23)(11,51)(12,25)(13,53)(14,27)(15,55)(16,29)(17,57)(18,31)(19,59)(20,33)(22,74)(24,76)(26,78)(28,80)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(42,67)(44,69)(46,71)(48,73)(50,75)(52,77)(54,79)(56,61)(58,63)(60,65), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,50,24,65)(2,64,25,49)(3,48,26,63)(4,62,27,47)(5,46,28,61)(6,80,29,45)(7,44,30,79)(8,78,31,43)(9,42,32,77)(10,76,33,41)(11,60,34,75)(12,74,35,59)(13,58,36,73)(14,72,37,57)(15,56,38,71)(16,70,39,55)(17,54,40,69)(18,68,21,53)(19,52,22,67)(20,66,23,51)>;
G:=Group( (1,41)(2,35)(3,43)(4,37)(5,45)(6,39)(7,47)(8,21)(9,49)(10,23)(11,51)(12,25)(13,53)(14,27)(15,55)(16,29)(17,57)(18,31)(19,59)(20,33)(22,74)(24,76)(26,78)(28,80)(30,62)(32,64)(34,66)(36,68)(38,70)(40,72)(42,67)(44,69)(46,71)(48,73)(50,75)(52,77)(54,79)(56,61)(58,63)(60,65), (1,24)(2,25)(3,26)(4,27)(5,28)(6,29)(7,30)(8,31)(9,32)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,40)(18,21)(19,22)(20,23)(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), (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,73)(9,74)(10,75)(11,76)(12,77)(13,78)(14,79)(15,80)(16,61)(17,62)(18,63)(19,64)(20,65)(21,48)(22,49)(23,50)(24,51)(25,52)(26,53)(27,54)(28,55)(29,56)(30,57)(31,58)(32,59)(33,60)(34,41)(35,42)(36,43)(37,44)(38,45)(39,46)(40,47), (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), (1,50,24,65)(2,64,25,49)(3,48,26,63)(4,62,27,47)(5,46,28,61)(6,80,29,45)(7,44,30,79)(8,78,31,43)(9,42,32,77)(10,76,33,41)(11,60,34,75)(12,74,35,59)(13,58,36,73)(14,72,37,57)(15,56,38,71)(16,70,39,55)(17,54,40,69)(18,68,21,53)(19,52,22,67)(20,66,23,51) );
G=PermutationGroup([(1,41),(2,35),(3,43),(4,37),(5,45),(6,39),(7,47),(8,21),(9,49),(10,23),(11,51),(12,25),(13,53),(14,27),(15,55),(16,29),(17,57),(18,31),(19,59),(20,33),(22,74),(24,76),(26,78),(28,80),(30,62),(32,64),(34,66),(36,68),(38,70),(40,72),(42,67),(44,69),(46,71),(48,73),(50,75),(52,77),(54,79),(56,61),(58,63),(60,65)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,29),(7,30),(8,31),(9,32),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,40),(18,21),(19,22),(20,23),(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)], [(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,73),(9,74),(10,75),(11,76),(12,77),(13,78),(14,79),(15,80),(16,61),(17,62),(18,63),(19,64),(20,65),(21,48),(22,49),(23,50),(24,51),(25,52),(26,53),(27,54),(28,55),(29,56),(30,57),(31,58),(32,59),(33,60),(34,41),(35,42),(36,43),(37,44),(38,45),(39,46),(40,47)], [(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)], [(1,50,24,65),(2,64,25,49),(3,48,26,63),(4,62,27,47),(5,46,28,61),(6,80,29,45),(7,44,30,79),(8,78,31,43),(9,42,32,77),(10,76,33,41),(11,60,34,75),(12,74,35,59),(13,58,36,73),(14,72,37,57),(15,56,38,71),(16,70,39,55),(17,54,40,69),(18,68,21,53),(19,52,22,67),(20,66,23,51)])
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10V | 20A | ··· | 20P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | D5 | D10 | D10 | C4×D5 | D20 | C5⋊D4 | D4×D5 |
| kernel | C23.44D20 | C2×D10⋊C4 | C2×C23.D5 | C10×C22⋊C4 | D5×C24 | C23×D5 | C22×D5 | C22×C10 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C23 | C23 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 8 | 4 | 2 | 4 | 2 | 8 | 8 | 8 | 8 |
Matrix representation of C23.44D20 ►in GL6(𝔽41)
| 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 | 1 |
| 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 |
| 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 |
| 21 | 24 | 0 | 0 | 0 | 0 |
| 38 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 37 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 23 | 24 | 0 | 0 | 0 | 0 |
| 36 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 21 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 0 | 0 | 0 | 40 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[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],[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],[21,38,0,0,0,0,24,18,0,0,0,0,0,0,1,37,0,0,0,0,21,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[23,36,0,0,0,0,24,18,0,0,0,0,0,0,1,0,0,0,0,0,21,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;
C23.44D20 in GAP, Magma, Sage, TeX
C_2^3._{44}D_{20} % in TeX
G:=Group("C2^3.44D20"); // GroupNames label
G:=SmallGroup(320,582);
// by ID
G=gap.SmallGroup(320,582);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,422,387,58,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^20=1,e^2=b,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^-1>;
// generators/relations