metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.196D10, M4(2).23D10, C4≀C2⋊7D5, D4⋊D5⋊8C4, C5⋊5(C8○D8), Q8⋊D5⋊8C4, D4.D5⋊8C4, D4.4(C4×D5), C5⋊Q16⋊8C4, Q8.4(C4×D5), C4.203(D4×D5), C10.69(C4×D4), C5⋊2C8.55D4, D20⋊4C4⋊7C2, C4○D4.21D10, D20.21(C2×C4), C20.362(C2×D4), D4.Dic5⋊2C2, D20.2C4⋊9C2, C20.53D4⋊6C2, C20.55(C22×C4), (C4×C20).51C22, (C2×C20).264C23, Dic10.22(C2×C4), D4.8D10.2C2, C4○D20.13C22, C4.Dic5.9C22, C22.9(D4⋊2D5), C2.23(Dic5⋊4D4), (C5×M4(2)).17C22, (C5×C4≀C2)⋊8C2, (C4×C5⋊2C8)⋊3C2, C4.20(C2×C4×D5), C5⋊2C8.24(C2×C4), (C5×D4).21(C2×C4), (C5×Q8).22(C2×C4), (C5×C4○D4).5C22, (C2×C10).35(C4○D4), (C2×C4).370(C22×D5), (C2×C5⋊2C8).230C22, SmallGroup(320,451)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.196D10
G = < a,b,c,d | a4=b4=c10=1, d2=cbc-1=b-1, ab=ba, cac-1=ab-1, ad=da, bd=db, dcd-1=b-1c-1 >
Subgroups: 326 in 106 conjugacy classes, 45 normal (all characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C5, C8 [×6], C2×C4, C2×C4 [×3], D4, D4 [×3], Q8, Q8, D5, C10, C10 [×2], C42, C2×C8 [×4], M4(2), M4(2) [×3], D8, SD16 [×2], Q16, C4○D4, C4○D4, Dic5, C20 [×2], C20 [×3], D10, C2×C10, C2×C10, C4×C8, C4≀C2, C4≀C2, C8.C4, C8○D4 [×2], C4○D8, C5⋊2C8 [×4], C5⋊2C8, C40, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C8○D8, C8×D5, C8⋊D5, C2×C5⋊2C8 [×2], C2×C5⋊2C8, C4.Dic5, C4.Dic5, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4×C20, C5×M4(2), C4○D20, C5×C4○D4, C4×C5⋊2C8, D20⋊4C4, C20.53D4, C5×C4≀C2, D20.2C4, D4.Dic5, D4.8D10, C42.196D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C22×D5, C8○D8, C2×C4×D5, D4×D5, D4⋊2D5, Dic5⋊4D4, C42.196D10
►On 80 points
(1 19)(2 36 20 52)(3 11)(4 38 12 54)(5 13)(6 40 14 56)(7 15)(8 32 16 58)(9 17)(10 34 18 60)(21 73 68 47)(22 69)(23 75 70 49)(24 61)(25 77 62 41)(26 63)(27 79 64 43)(28 65)(29 71 66 45)(30 67)(31 57)(33 59)(35 51)(37 53)(39 55)(42 78)(44 80)(46 72)(48 74)(50 76)
(1 35 19 51)(2 52 20 36)(3 37 11 53)(4 54 12 38)(5 39 13 55)(6 56 14 40)(7 31 15 57)(8 58 16 32)(9 33 17 59)(10 60 18 34)(21 47 68 73)(22 74 69 48)(23 49 70 75)(24 76 61 50)(25 41 62 77)(26 78 63 42)(27 43 64 79)(28 80 65 44)(29 45 66 71)(30 72 67 46)
(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 24 51 50 19 61 35 76)(2 75 36 70 20 49 52 23)(3 22 53 48 11 69 37 74)(4 73 38 68 12 47 54 21)(5 30 55 46 13 67 39 72)(6 71 40 66 14 45 56 29)(7 28 57 44 15 65 31 80)(8 79 32 64 16 43 58 27)(9 26 59 42 17 63 33 78)(10 77 34 62 18 41 60 25)
G:=sub<Sym(80)| (1,19)(2,36,20,52)(3,11)(4,38,12,54)(5,13)(6,40,14,56)(7,15)(8,32,16,58)(9,17)(10,34,18,60)(21,73,68,47)(22,69)(23,75,70,49)(24,61)(25,77,62,41)(26,63)(27,79,64,43)(28,65)(29,71,66,45)(30,67)(31,57)(33,59)(35,51)(37,53)(39,55)(42,78)(44,80)(46,72)(48,74)(50,76), (1,35,19,51)(2,52,20,36)(3,37,11,53)(4,54,12,38)(5,39,13,55)(6,56,14,40)(7,31,15,57)(8,58,16,32)(9,33,17,59)(10,60,18,34)(21,47,68,73)(22,74,69,48)(23,49,70,75)(24,76,61,50)(25,41,62,77)(26,78,63,42)(27,43,64,79)(28,80,65,44)(29,45,66,71)(30,72,67,46), (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), (1,24,51,50,19,61,35,76)(2,75,36,70,20,49,52,23)(3,22,53,48,11,69,37,74)(4,73,38,68,12,47,54,21)(5,30,55,46,13,67,39,72)(6,71,40,66,14,45,56,29)(7,28,57,44,15,65,31,80)(8,79,32,64,16,43,58,27)(9,26,59,42,17,63,33,78)(10,77,34,62,18,41,60,25)>;
G:=Group( (1,19)(2,36,20,52)(3,11)(4,38,12,54)(5,13)(6,40,14,56)(7,15)(8,32,16,58)(9,17)(10,34,18,60)(21,73,68,47)(22,69)(23,75,70,49)(24,61)(25,77,62,41)(26,63)(27,79,64,43)(28,65)(29,71,66,45)(30,67)(31,57)(33,59)(35,51)(37,53)(39,55)(42,78)(44,80)(46,72)(48,74)(50,76), (1,35,19,51)(2,52,20,36)(3,37,11,53)(4,54,12,38)(5,39,13,55)(6,56,14,40)(7,31,15,57)(8,58,16,32)(9,33,17,59)(10,60,18,34)(21,47,68,73)(22,74,69,48)(23,49,70,75)(24,76,61,50)(25,41,62,77)(26,78,63,42)(27,43,64,79)(28,80,65,44)(29,45,66,71)(30,72,67,46), (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), (1,24,51,50,19,61,35,76)(2,75,36,70,20,49,52,23)(3,22,53,48,11,69,37,74)(4,73,38,68,12,47,54,21)(5,30,55,46,13,67,39,72)(6,71,40,66,14,45,56,29)(7,28,57,44,15,65,31,80)(8,79,32,64,16,43,58,27)(9,26,59,42,17,63,33,78)(10,77,34,62,18,41,60,25) );
G=PermutationGroup([(1,19),(2,36,20,52),(3,11),(4,38,12,54),(5,13),(6,40,14,56),(7,15),(8,32,16,58),(9,17),(10,34,18,60),(21,73,68,47),(22,69),(23,75,70,49),(24,61),(25,77,62,41),(26,63),(27,79,64,43),(28,65),(29,71,66,45),(30,67),(31,57),(33,59),(35,51),(37,53),(39,55),(42,78),(44,80),(46,72),(48,74),(50,76)], [(1,35,19,51),(2,52,20,36),(3,37,11,53),(4,54,12,38),(5,39,13,55),(6,56,14,40),(7,31,15,57),(8,58,16,32),(9,33,17,59),(10,60,18,34),(21,47,68,73),(22,74,69,48),(23,49,70,75),(24,76,61,50),(25,41,62,77),(26,78,63,42),(27,43,64,79),(28,80,65,44),(29,45,66,71),(30,72,67,46)], [(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)], [(1,24,51,50,19,61,35,76),(2,75,36,70,20,49,52,23),(3,22,53,48,11,69,37,74),(4,73,38,68,12,47,54,21),(5,30,55,46,13,67,39,72),(6,71,40,66,14,45,56,29),(7,28,57,44,15,65,31,80),(8,79,32,64,16,43,58,27),(9,26,59,42,17,63,33,78),(10,77,34,62,18,41,60,25)])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | ··· | 8L | 8M | 8N | 10A | 10B | 10C | 10D | 10E | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20N | 20O | 20P | 40A | 40B | 40C | 40D |
| order | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 20 | 20 | 40 | 40 | 40 | 40 |
| size | 1 | 1 | 2 | 4 | 20 | 1 | 1 | 2 | ··· | 2 | 4 | 20 | 2 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | 20 | 2 | 2 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 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 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4×D5 | C4×D5 | C8○D8 | D4×D5 | D4⋊2D5 | C42.196D10 |
| kernel | C42.196D10 | C4×C5⋊2C8 | D20⋊4C4 | C20.53D4 | C5×C4≀C2 | D20.2C4 | D4.Dic5 | D4.8D10 | D4⋊D5 | D4.D5 | Q8⋊D5 | C5⋊Q16 | C5⋊2C8 | C4≀C2 | C2×C10 | C42 | M4(2) | C4○D4 | D4 | Q8 | C5 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 8 |
Matrix representation of C42.196D10 ►in GL4(𝔽41) generated by
| 40 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 |
| 40 | 0 | 0 | 0 |
| 0 | 0 | 6 | 6 |
| 0 | 0 | 35 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 38 | 0 | 0 |
| 0 | 0 | 2 | 16 |
| 0 | 0 | 28 | 39 |
G:=sub<GL(4,GF(41))| [40,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,32,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,6,35,0,0,6,1],[27,0,0,0,0,38,0,0,0,0,2,28,0,0,16,39] >;
C42.196D10 in GAP, Magma, Sage, TeX
C_4^2._{196}D_{10} % in TeX
G:=Group("C4^2.196D10"); // GroupNames label
G:=SmallGroup(320,451);
// by ID
G=gap.SmallGroup(320,451);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,555,58,136,1684,851,438,102,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c*b*c^-1=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations