metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.4D20, C4.D4.D5, (C2×D4).6D10, (C2×C20).14D4, C23.D5⋊4C4, C23.4(C4×D5), (C22×Dic5)⋊2C4, (C22×C10).13D4, C5⋊4(C23.D4), C23⋊Dic5.3C2, C10.33(C23⋊C4), (D4×C10).171C22, C23.18D10.4C2, C22.13(D10⋊C4), C2.13(C23.1D10), (C2×C4).2(C5⋊D4), (C22×C10).4(C2×C4), (C5×C4.D4).1C2, (C2×C10).70(C22⋊C4), SmallGroup(320,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.4D20
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=ca=ac, ab=ba, dad-1=abc, ae=ea, dbd-1=ebe-1=bc=cb, cd=dc, ce=ec, ede-1=acd19 >
Subgroups: 318 in 68 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C10, C10, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, Dic5, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22.D4, C40, C2×Dic5, C2×C20, C5×D4, C22×C10, C23.D4, C10.D4, C23.D5, C23.D5, C5×M4(2), C22×Dic5, D4×C10, C23⋊Dic5, C5×C4.D4, C23.18D10, C23.4D20
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C23.D4, D10⋊C4, C23.1D10, C23.4D20
►On 80 points
Generators in S80
(1 79)(3 61)(4 24)(5 43)(7 65)(8 28)(9 47)(11 69)(12 32)(13 51)(15 73)(16 36)(17 55)(19 77)(20 40)(21 59)(23 41)(25 63)(27 45)(29 67)(31 49)(33 71)(35 53)(37 75)(39 57)(42 62)(46 66)(50 70)(54 74)(58 78)
(1 59)(2 80)(3 61)(4 42)(5 63)(6 44)(7 65)(8 46)(9 67)(10 48)(11 69)(12 50)(13 71)(14 52)(15 73)(16 54)(17 75)(18 56)(19 77)(20 58)(21 79)(22 60)(23 41)(24 62)(25 43)(26 64)(27 45)(28 66)(29 47)(30 68)(31 49)(32 70)(33 51)(34 72)(35 53)(36 74)(37 55)(38 76)(39 57)(40 78)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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 73 59 35)(2 72 22 52)(3 33 41 51)(4 12)(5 69 63 31)(6 68 26 48)(7 29 45 47)(9 65 67 27)(10 64 30 44)(11 25 49 43)(13 61 71 23)(14 60 34 80)(15 21 53 79)(16 40)(17 57 75 19)(18 56 38 76)(20 36)(24 32)(37 77 55 39)(42 70)(46 66)(50 62)(54 58)(74 78)
G:=sub<Sym(80)| (1,79)(3,61)(4,24)(5,43)(7,65)(8,28)(9,47)(11,69)(12,32)(13,51)(15,73)(16,36)(17,55)(19,77)(20,40)(21,59)(23,41)(25,63)(27,45)(29,67)(31,49)(33,71)(35,53)(37,75)(39,57)(42,62)(46,66)(50,70)(54,74)(58,78), (1,59)(2,80)(3,61)(4,42)(5,63)(6,44)(7,65)(8,46)(9,67)(10,48)(11,69)(12,50)(13,71)(14,52)(15,73)(16,54)(17,75)(18,56)(19,77)(20,58)(21,79)(22,60)(23,41)(24,62)(25,43)(26,64)(27,45)(28,66)(29,47)(30,68)(31,49)(32,70)(33,51)(34,72)(35,53)(36,74)(37,55)(38,76)(39,57)(40,78), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,73,59,35)(2,72,22,52)(3,33,41,51)(4,12)(5,69,63,31)(6,68,26,48)(7,29,45,47)(9,65,67,27)(10,64,30,44)(11,25,49,43)(13,61,71,23)(14,60,34,80)(15,21,53,79)(16,40)(17,57,75,19)(18,56,38,76)(20,36)(24,32)(37,77,55,39)(42,70)(46,66)(50,62)(54,58)(74,78)>;
G:=Group( (1,79)(3,61)(4,24)(5,43)(7,65)(8,28)(9,47)(11,69)(12,32)(13,51)(15,73)(16,36)(17,55)(19,77)(20,40)(21,59)(23,41)(25,63)(27,45)(29,67)(31,49)(33,71)(35,53)(37,75)(39,57)(42,62)(46,66)(50,70)(54,74)(58,78), (1,59)(2,80)(3,61)(4,42)(5,63)(6,44)(7,65)(8,46)(9,67)(10,48)(11,69)(12,50)(13,71)(14,52)(15,73)(16,54)(17,75)(18,56)(19,77)(20,58)(21,79)(22,60)(23,41)(24,62)(25,43)(26,64)(27,45)(28,66)(29,47)(30,68)(31,49)(32,70)(33,51)(34,72)(35,53)(36,74)(37,55)(38,76)(39,57)(40,78), (1,21)(2,22)(3,23)(4,24)(5,25)(6,26)(7,27)(8,28)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80), (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), (1,73,59,35)(2,72,22,52)(3,33,41,51)(4,12)(5,69,63,31)(6,68,26,48)(7,29,45,47)(9,65,67,27)(10,64,30,44)(11,25,49,43)(13,61,71,23)(14,60,34,80)(15,21,53,79)(16,40)(17,57,75,19)(18,56,38,76)(20,36)(24,32)(37,77,55,39)(42,70)(46,66)(50,62)(54,58)(74,78) );
G=PermutationGroup([[(1,79),(3,61),(4,24),(5,43),(7,65),(8,28),(9,47),(11,69),(12,32),(13,51),(15,73),(16,36),(17,55),(19,77),(20,40),(21,59),(23,41),(25,63),(27,45),(29,67),(31,49),(33,71),(35,53),(37,75),(39,57),(42,62),(46,66),(50,70),(54,74),(58,78)], [(1,59),(2,80),(3,61),(4,42),(5,63),(6,44),(7,65),(8,46),(9,67),(10,48),(11,69),(12,50),(13,71),(14,52),(15,73),(16,54),(17,75),(18,56),(19,77),(20,58),(21,79),(22,60),(23,41),(24,62),(25,43),(26,64),(27,45),(28,66),(29,47),(30,68),(31,49),(32,70),(33,51),(34,72),(35,53),(36,74),(37,55),(38,76),(39,57),(40,78)], [(1,21),(2,22),(3,23),(4,24),(5,25),(6,26),(7,27),(8,28),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(41,61),(42,62),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80)], [(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)], [(1,73,59,35),(2,72,22,52),(3,33,41,51),(4,12),(5,69,63,31),(6,68,26,48),(7,29,45,47),(9,65,67,27),(10,64,30,44),(11,25,49,43),(13,61,71,23),(14,60,34,80),(15,21,53,79),(16,40),(17,57,75,19),(18,56,38,76),(20,36),(24,32),(37,77,55,39),(42,70),(46,66),(50,62),(54,58),(74,78)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 20 | 20 | 40 | 40 | 40 | 2 | 2 | 8 | 8 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | ··· | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D10 | C5⋊D4 | C4×D5 | D20 | C23⋊C4 | C23.D4 | C23.1D10 | C23.4D20 |
| kernel | C23.4D20 | C23⋊Dic5 | C5×C4.D4 | C23.18D10 | C23.D5 | C22×Dic5 | C2×C20 | C22×C10 | C4.D4 | C2×D4 | C2×C4 | C23 | C23 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 4 | 2 |
Matrix representation of C23.4D20 ►in GL6(𝔽41)
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 32 | 0 | 0 |
| 0 | 0 | 1 | 25 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 0 | 40 | 16 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 25 | 9 | 0 | 0 |
| 0 | 0 | 40 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 0 | 0 | 0 | 0 | 40 | 16 |
| 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 |
| 19 | 9 | 0 | 0 | 0 | 0 |
| 32 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 21 | 26 | 20 |
| 0 | 0 | 10 | 35 | 40 | 15 |
| 0 | 0 | 26 | 20 | 35 | 20 |
| 0 | 0 | 40 | 15 | 31 | 6 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 16 | 32 |
| 0 | 0 | 0 | 0 | 24 | 25 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 40 | 0 | 0 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,16,1,0,0,0,0,32,25,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,40,0,0,0,0,9,16,0,0,0,0,0,0,25,40,0,0,0,0,9,16],[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],[19,32,0,0,0,0,9,0,0,0,0,0,0,0,6,10,26,40,0,0,21,35,20,15,0,0,26,40,35,31,0,0,20,15,20,6],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,16,24,0,0,0,0,32,25,0,0] >;
C23.4D20 in GAP, Magma, Sage, TeX
C_2^3._4D_{20} % in TeX
G:=Group("C2^3.4D20"); // GroupNames label
G:=SmallGroup(320,34);
// by ID
G=gap.SmallGroup(320,34);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,346,297,851,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=c*a=a*c,a*b=b*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^19>;
// generators/relations