metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊21D10, (C4×D5)⋊12D4, (C2×D4)⋊22D10, C4⋊D4⋊26D5, C4.182(D4×D5), D10⋊3(C4○D4), C23⋊D10⋊8C2, C20⋊2D4⋊17C2, C22⋊C4⋊26D10, (D4×Dic5)⋊20C2, D10.74(C2×D4), C20.226(C2×D4), D10⋊2Q8⋊20C2, Dic5⋊4D4⋊8C2, (D4×C10)⋊11C22, C4⋊Dic5⋊30C22, C10.65(C22×D4), C20.48D4⋊33C2, C22⋊2(D4⋊2D5), (C2×C20).594C23, (C2×C10).150C24, Dic5.118(C2×D4), C5⋊4(C22.19C24), (C4×Dic5)⋊20C22, (C22×C4).368D10, D10.12D4⋊18C2, C23.D5⋊22C22, C23.15(C22×D5), (C2×Dic10)⋊24C22, C10.D4⋊28C22, (C22×C10).19C23, (C2×Dic5).71C23, C22.171(C23×D5), D10⋊C4.14C22, (C22×C20).240C22, (C22×Dic5)⋊19C22, (C22×D5).195C23, (C23×D5).119C22, C2.38(C2×D4×D5), (D5×C22×C4)⋊4C2, (C5×C4⋊C4)⋊9C22, C2.38(D5×C4○D4), C4⋊C4⋊7D5⋊19C2, (C2×C10)⋊5(C4○D4), (C5×C4⋊D4)⋊12C2, (C2×D4⋊2D5)⋊12C2, C10.151(C2×C4○D4), C2.36(C2×D4⋊2D5), (C2×C4×D5).317C22, (C5×C22⋊C4)⋊11C22, (C2×C4).294(C22×D5), (C2×C5⋊D4).27C22, SmallGroup(320,1278)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊21D10
G = < a,b,c,d | a4=b4=c10=d2=1, bab-1=a-1, ac=ca, ad=da, cbc-1=b-1, dbd=a2b, dcd=c-1 >
Subgroups: 1198 in 330 conjugacy classes, 109 normal (43 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, C2×C10, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, Dic10, C4×D5, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22.19C24, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×C4×D5, D4⋊2D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, C22×C20, D4×C10, D4×C10, C23×D5, Dic5⋊4D4, D10.12D4, C4⋊C4⋊7D5, D10⋊2Q8, C20.48D4, D4×Dic5, C23⋊D10, C20⋊2D4, C5×C4⋊D4, D5×C22×C4, C2×D4⋊2D5, C4⋊C4⋊21D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, C24, D10, C22×D4, C2×C4○D4, C22×D5, C22.19C24, D4×D5, D4⋊2D5, C23×D5, C2×D4×D5, C2×D4⋊2D5, D5×C4○D4, C4⋊C4⋊21D10
►On 80 points
(1 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(21 66 26 61)(22 67 27 62)(23 68 28 63)(24 69 29 64)(25 70 30 65)(31 76 36 71)(32 77 37 72)(33 78 38 73)(34 79 39 74)(35 80 40 75)
(1 78 18 63)(2 64 19 79)(3 80 20 65)(4 66 11 71)(5 72 12 67)(6 68 13 73)(7 74 14 69)(8 70 15 75)(9 76 16 61)(10 62 17 77)(21 56 36 41)(22 42 37 57)(23 58 38 43)(24 44 39 59)(25 60 40 45)(26 46 31 51)(27 52 32 47)(28 48 33 53)(29 54 34 49)(30 50 35 55)
(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 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 20)(9 19)(10 18)(21 29)(22 28)(23 27)(24 26)(31 39)(32 38)(33 37)(34 36)(41 59)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(50 60)(61 69)(62 68)(63 67)(64 66)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(80)| (1,48,13,58)(2,49,14,59)(3,50,15,60)(4,41,16,51)(5,42,17,52)(6,43,18,53)(7,44,19,54)(8,45,20,55)(9,46,11,56)(10,47,12,57)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,78,18,63)(2,64,19,79)(3,80,20,65)(4,66,11,71)(5,72,12,67)(6,68,13,73)(7,74,14,69)(8,70,15,75)(9,76,16,61)(10,62,17,77)(21,56,36,41)(22,42,37,57)(23,58,38,43)(24,44,39,59)(25,60,40,45)(26,46,31,51)(27,52,32,47)(28,48,33,53)(29,54,34,49)(30,50,35,55), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,20)(9,19)(10,18)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,48,13,58)(2,49,14,59)(3,50,15,60)(4,41,16,51)(5,42,17,52)(6,43,18,53)(7,44,19,54)(8,45,20,55)(9,46,11,56)(10,47,12,57)(21,66,26,61)(22,67,27,62)(23,68,28,63)(24,69,29,64)(25,70,30,65)(31,76,36,71)(32,77,37,72)(33,78,38,73)(34,79,39,74)(35,80,40,75), (1,78,18,63)(2,64,19,79)(3,80,20,65)(4,66,11,71)(5,72,12,67)(6,68,13,73)(7,74,14,69)(8,70,15,75)(9,76,16,61)(10,62,17,77)(21,56,36,41)(22,42,37,57)(23,58,38,43)(24,44,39,59)(25,60,40,45)(26,46,31,51)(27,52,32,47)(28,48,33,53)(29,54,34,49)(30,50,35,55), (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), (1,17)(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,20)(9,19)(10,18)(21,29)(22,28)(23,27)(24,26)(31,39)(32,38)(33,37)(34,36)(41,59)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(50,60)(61,69)(62,68)(63,67)(64,66)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,48,13,58),(2,49,14,59),(3,50,15,60),(4,41,16,51),(5,42,17,52),(6,43,18,53),(7,44,19,54),(8,45,20,55),(9,46,11,56),(10,47,12,57),(21,66,26,61),(22,67,27,62),(23,68,28,63),(24,69,29,64),(25,70,30,65),(31,76,36,71),(32,77,37,72),(33,78,38,73),(34,79,39,74),(35,80,40,75)], [(1,78,18,63),(2,64,19,79),(3,80,20,65),(4,66,11,71),(5,72,12,67),(6,68,13,73),(7,74,14,69),(8,70,15,75),(9,76,16,61),(10,62,17,77),(21,56,36,41),(22,42,37,57),(23,58,38,43),(24,44,39,59),(25,60,40,45),(26,46,31,51),(27,52,32,47),(28,48,33,53),(29,54,34,49),(30,50,35,55)], [(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)], [(1,17),(2,16),(3,15),(4,14),(5,13),(6,12),(7,11),(8,20),(9,19),(10,18),(21,29),(22,28),(23,27),(24,26),(31,39),(32,38),(33,37),(34,36),(41,59),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(50,60),(61,69),(62,68),(63,67),(64,66),(71,79),(72,78),(73,77),(74,76)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | C4○D4 | D10 | D10 | D10 | D10 | D4×D5 | D4⋊2D5 | D5×C4○D4 |
| kernel | C4⋊C4⋊21D10 | Dic5⋊4D4 | D10.12D4 | C4⋊C4⋊7D5 | D10⋊2Q8 | C20.48D4 | D4×Dic5 | C23⋊D10 | C20⋊2D4 | C5×C4⋊D4 | D5×C22×C4 | C2×D4⋊2D5 | C4×D5 | C4⋊D4 | D10 | C2×C10 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 6 | 4 | 4 | 4 |
Matrix representation of C4⋊C4⋊21D10 ►in GL6(𝔽41)
| 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 | 32 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 9 |
| 0 | 0 | 0 | 0 | 32 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 35 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,40,0,0,0,0,35,6,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;
C4⋊C4⋊21D10 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_{21}D_{10} % in TeX
G:=Group("C4:C4:21D10"); // GroupNames label
G:=SmallGroup(320,1278);
// by ID
G=gap.SmallGroup(320,1278);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,100,1123,794,297,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations