metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.10M4(2), (C2×C8).2F5, (C2×C40).7C4, C5⋊2C8.19D4, C5⋊1(C23.C8), C20.C8⋊5C2, C4.7(C4.F5), (C2×Dic5).2C8, (C22×D5).2C8, C10.4(C22⋊C8), C22.2(D5⋊C8), C4.36(C22⋊F5), C2.5(D10⋊C8), C20.34(C22⋊C4), (C2×C4×D5).7C4, (C2×C10).7(C2×C8), (C2×C8⋊D5).6C2, (C2×C4).123(C2×F5), (C2×C20).140(C2×C4), (C2×C5⋊2C8).213C22, SmallGroup(320,229)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.10M4(2)
G = < a,b,c | a20=c2=1, b8=a10, bab-1=a7, cac=a9, cbc=a5b5 >
Subgroups: 226 in 58 conjugacy classes, 24 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C10, C16, C2×C8, C2×C8, M4(2), C22×C4, Dic5, C20, D10, C2×C10, M5(2), C2×M4(2), C5⋊2C8, C40, C4×D5, C2×Dic5, C2×C20, C22×D5, C23.C8, C5⋊C16, C8⋊D5, C2×C5⋊2C8, C2×C40, C2×C4×D5, C20.C8, C2×C8⋊D5, C20.10M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C2×F5, C23.C8, D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, C20.10M4(2)
►On 80 points
(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 70 27 55 6 65 32 50 11 80 37 45 16 75 22 60)(2 73 36 42 7 68 21 57 12 63 26 52 17 78 31 47)(3 76 25 49 8 71 30 44 13 66 35 59 18 61 40 54)(4 79 34 56 9 74 39 51 14 69 24 46 19 64 29 41)(5 62 23 43 10 77 28 58 15 72 33 53 20 67 38 48)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 23)(22 32)(24 30)(25 39)(26 28)(27 37)(29 35)(31 33)(34 40)(36 38)(41 49)(42 58)(43 47)(44 56)(46 54)(48 52)(51 59)(53 57)(61 79)(62 68)(63 77)(64 66)(65 75)(67 73)(69 71)(70 80)(72 78)(74 76)
G:=sub<Sym(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), (1,70,27,55,6,65,32,50,11,80,37,45,16,75,22,60)(2,73,36,42,7,68,21,57,12,63,26,52,17,78,31,47)(3,76,25,49,8,71,30,44,13,66,35,59,18,61,40,54)(4,79,34,56,9,74,39,51,14,69,24,46,19,64,29,41)(5,62,23,43,10,77,28,58,15,72,33,53,20,67,38,48), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,49)(42,58)(43,47)(44,56)(46,54)(48,52)(51,59)(53,57)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76)>;
G:=Group( (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), (1,70,27,55,6,65,32,50,11,80,37,45,16,75,22,60)(2,73,36,42,7,68,21,57,12,63,26,52,17,78,31,47)(3,76,25,49,8,71,30,44,13,66,35,59,18,61,40,54)(4,79,34,56,9,74,39,51,14,69,24,46,19,64,29,41)(5,62,23,43,10,77,28,58,15,72,33,53,20,67,38,48), (2,10)(3,19)(4,8)(5,17)(7,15)(9,13)(12,20)(14,18)(21,23)(22,32)(24,30)(25,39)(26,28)(27,37)(29,35)(31,33)(34,40)(36,38)(41,49)(42,58)(43,47)(44,56)(46,54)(48,52)(51,59)(53,57)(61,79)(62,68)(63,77)(64,66)(65,75)(67,73)(69,71)(70,80)(72,78)(74,76) );
G=PermutationGroup([[(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)], [(1,70,27,55,6,65,32,50,11,80,37,45,16,75,22,60),(2,73,36,42,7,68,21,57,12,63,26,52,17,78,31,47),(3,76,25,49,8,71,30,44,13,66,35,59,18,61,40,54),(4,79,34,56,9,74,39,51,14,69,24,46,19,64,29,41),(5,62,23,43,10,77,28,58,15,72,33,53,20,67,38,48)], [(2,10),(3,19),(4,8),(5,17),(7,15),(9,13),(12,20),(14,18),(21,23),(22,32),(24,30),(25,39),(26,28),(27,37),(29,35),(31,33),(34,40),(36,38),(41,49),(42,58),(43,47),(44,56),(46,54),(48,52),(51,59),(53,57),(61,79),(62,68),(63,77),(64,66),(65,75),(67,73),(69,71),(70,80),(72,78),(74,76)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 10A | 10B | 10C | 16A | ··· | 16H | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 2 | 20 | 1 | 1 | 2 | 20 | 4 | 4 | 4 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C8 | D4 | M4(2) | F5 | C2×F5 | C23.C8 | C4.F5 | C22⋊F5 | D5⋊C8 | C20.10M4(2) |
| kernel | C20.10M4(2) | C20.C8 | C2×C8⋊D5 | C2×C40 | C2×C4×D5 | C2×Dic5 | C22×D5 | C5⋊2C8 | C20 | C2×C8 | C2×C4 | C5 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 8 |
Matrix representation of C20.10M4(2) ►in GL4(𝔽241) generated by
| 46 | 64 | 0 | 0 |
| 223 | 64 | 0 | 0 |
| 0 | 0 | 0 | 110 |
| 0 | 0 | 195 | 46 |
| 0 | 0 | 189 | 1 |
| 0 | 0 | 189 | 52 |
| 25 | 90 | 0 | 0 |
| 50 | 216 | 0 | 0 |
| 189 | 1 | 0 | 0 |
| 189 | 52 | 0 | 0 |
| 0 | 0 | 189 | 1 |
| 0 | 0 | 189 | 52 |
G:=sub<GL(4,GF(241))| [46,223,0,0,64,64,0,0,0,0,0,195,0,0,110,46],[0,0,25,50,0,0,90,216,189,189,0,0,1,52,0,0],[189,189,0,0,1,52,0,0,0,0,189,189,0,0,1,52] >;
C20.10M4(2) in GAP, Magma, Sage, TeX
C_{20}._{10}M_4(2) % in TeX
G:=Group("C20.10M4(2)"); // GroupNames label
G:=SmallGroup(320,229);
// by ID
G=gap.SmallGroup(320,229);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,102,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^20=c^2=1,b^8=a^10,b*a*b^-1=a^7,c*a*c=a^9,c*b*c=a^5*b^5>;
// generators/relations