metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.6Dic10, M4(2).30D10, C20.68(C4⋊C4), (C2×C20).28Q8, C20.441(C2×D4), (C2×C20).483D4, (C2×C4).16Dic10, C4.Dic5.10C4, C20.53D4⋊13C2, (C22×C10).16Q8, C5⋊3(M4(2).C4), C20.127(C22×C4), (C2×C20).415C23, (C22×C4).134D10, (C2×M4(2)).16D5, C22.4(C2×Dic10), C4.21(C10.D4), (C10×M4(2)).27C2, C4.Dic5.41C22, (C22×C20).183C22, (C5×M4(2)).33C22, C22.17(C10.D4), C4.90(C2×C4×D5), C10.75(C2×C4⋊C4), C5⋊2C8.5(C2×C4), (C2×C4).49(C4×D5), C4.131(C2×C5⋊D4), (C2×C10).11(C2×Q8), (C2×C10).82(C4⋊C4), (C2×C20).276(C2×C4), (C2×C4).194(C5⋊D4), C2.19(C2×C10.D4), (C2×C4).511(C22×D5), (C2×C4.Dic5).24C2, (C2×C5⋊2C8).143C22, SmallGroup(320,751)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.Dic10
G = < a,b,c,d,e | a2=b2=c2=1, d20=c, e2=bcd10, ab=ba, eae-1=ac=ca, ad=da, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd19 >
Subgroups: 238 in 102 conjugacy classes, 59 normal (39 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C10, C10, C2×C8, M4(2), M4(2), C22×C4, C20, C2×C10, C2×C10, C8.C4, C2×M4(2), C2×M4(2), C5⋊2C8, C5⋊2C8, C40, C2×C20, C22×C10, M4(2).C4, C2×C5⋊2C8, C2×C5⋊2C8, C4.Dic5, C4.Dic5, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C20.53D4, C2×C4.Dic5, C10×M4(2), C23.Dic10
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C4⋊C4, C22×C4, C2×D4, C2×Q8, D10, C2×C4⋊C4, Dic10, C4×D5, C5⋊D4, C22×D5, M4(2).C4, C10.D4, C2×Dic10, C2×C4×D5, C2×C5⋊D4, C2×C10.D4, C23.Dic10
►On 80 points
(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)
(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)(42 62)(44 64)(46 66)(48 68)(50 70)(52 72)(54 74)(56 76)(58 78)(60 80)
(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 71 31 61 21 51 11 41)(2 50 12 60 22 70 32 80)(3 49 33 79 23 69 13 59)(4 68 14 78 24 48 34 58)(5 67 35 57 25 47 15 77)(6 46 16 56 26 66 36 76)(7 45 37 75 27 65 17 55)(8 64 18 74 28 44 38 54)(9 63 39 53 29 43 19 73)(10 42 20 52 30 62 40 72)
G:=sub<Sym(80)| (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), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (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), (1,71,31,61,21,51,11,41)(2,50,12,60,22,70,32,80)(3,49,33,79,23,69,13,59)(4,68,14,78,24,48,34,58)(5,67,35,57,25,47,15,77)(6,46,16,56,26,66,36,76)(7,45,37,75,27,65,17,55)(8,64,18,74,28,44,38,54)(9,63,39,53,29,43,19,73)(10,42,20,52,30,62,40,72)>;
G:=Group( (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), (2,22)(4,24)(6,26)(8,28)(10,30)(12,32)(14,34)(16,36)(18,38)(20,40)(42,62)(44,64)(46,66)(48,68)(50,70)(52,72)(54,74)(56,76)(58,78)(60,80), (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), (1,71,31,61,21,51,11,41)(2,50,12,60,22,70,32,80)(3,49,33,79,23,69,13,59)(4,68,14,78,24,48,34,58)(5,67,35,57,25,47,15,77)(6,46,16,56,26,66,36,76)(7,45,37,75,27,65,17,55)(8,64,18,74,28,44,38,54)(9,63,39,53,29,43,19,73)(10,42,20,52,30,62,40,72) );
G=PermutationGroup([[(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)], [(2,22),(4,24),(6,26),(8,28),(10,30),(12,32),(14,34),(16,36),(18,38),(20,40),(42,62),(44,64),(46,66),(48,68),(50,70),(52,72),(54,74),(56,76),(58,78),(60,80)], [(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)], [(1,71,31,61,21,51,11,41),(2,50,12,60,22,70,32,80),(3,49,33,79,23,69,13,59),(4,68,14,78,24,48,34,58),(5,67,35,57,25,47,15,77),(6,46,16,56,26,66,36,76),(7,45,37,75,27,65,17,55),(8,64,18,74,28,44,38,54),(9,63,39,53,29,43,19,73),(10,42,20,52,30,62,40,72)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 8A | 8B | 8C | 8D | 8E | ··· | 8L | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | ··· | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | - | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | D4 | Q8 | Q8 | D5 | D10 | D10 | Dic10 | C4×D5 | C5⋊D4 | Dic10 | M4(2).C4 | C23.Dic10 |
| kernel | C23.Dic10 | C20.53D4 | C2×C4.Dic5 | C10×M4(2) | C4.Dic5 | C2×C20 | C2×C20 | C22×C10 | C2×M4(2) | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C5 | C1 |
| # reps | 1 | 4 | 2 | 1 | 8 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 4 | 2 | 8 |
Matrix representation of C23.Dic10 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 10 | 0 | 0 |
| 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 36 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,8,0,0,10,0,0,0,0,0,0,4,0,0,36,0],[0,0,32,0,0,0,0,9,1,0,0,0,0,1,0,0] >;
C23.Dic10 in GAP, Magma, Sage, TeX
C_2^3.{\rm Dic}_{10} % in TeX
G:=Group("C2^3.Dic10"); // GroupNames label
G:=SmallGroup(320,751);
// by ID
G=gap.SmallGroup(320,751);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,477,422,58,136,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=b*c*d^10,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d^19>;
// generators/relations