metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8.10D10, C5⋊12- 1+4, C20.24C23, C10.10C24, D10.5C23, D20.13C22, Dic5.6C23, Dic10.13C22, (C2×Q8)⋊5D5, (Q8×D5)⋊4C2, C4○D20⋊6C2, (Q8×C10)⋊7C2, Q8⋊2D5⋊4C2, (C2×C4).24D10, (C4×D5).5C22, C4.24(C22×D5), C2.11(C23×D5), C5⋊D4.2C22, (C2×C10).68C23, (C2×C20).48C22, (C5×Q8).10C22, C22.7(C22×D5), SmallGroup(160,222)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.10D10
G = < a,b,c,d | a4=1, b2=c10=d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=dbd-1=a2b, dcd-1=c9 >
Subgroups: 392 in 146 conjugacy classes, 85 normal (9 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, Q8, Q8, D5, C10, C10, C2×Q8, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, 2- 1+4, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C4○D20, Q8×D5, Q8⋊2D5, Q8×C10, Q8.10D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2- 1+4, C22×D5, C23×D5, Q8.10D10
►On 80 points
(1 49 11 59)(2 50 12 60)(3 51 13 41)(4 52 14 42)(5 53 15 43)(6 54 16 44)(7 55 17 45)(8 56 18 46)(9 57 19 47)(10 58 20 48)(21 79 31 69)(22 80 32 70)(23 61 33 71)(24 62 34 72)(25 63 35 73)(26 64 36 74)(27 65 37 75)(28 66 38 76)(29 67 39 77)(30 68 40 78)
(1 28 11 38)(2 39 12 29)(3 30 13 40)(4 21 14 31)(5 32 15 22)(6 23 16 33)(7 34 17 24)(8 25 18 35)(9 36 19 26)(10 27 20 37)(41 68 51 78)(42 79 52 69)(43 70 53 80)(44 61 54 71)(45 72 55 62)(46 63 56 73)(47 74 57 64)(48 65 58 75)(49 76 59 66)(50 67 60 77)
(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 65 11 75)(2 74 12 64)(3 63 13 73)(4 72 14 62)(5 61 15 71)(6 70 16 80)(7 79 17 69)(8 68 18 78)(9 77 19 67)(10 66 20 76)(21 45 31 55)(22 54 32 44)(23 43 33 53)(24 52 34 42)(25 41 35 51)(26 50 36 60)(27 59 37 49)(28 48 38 58)(29 57 39 47)(30 46 40 56)
G:=sub<Sym(80)| (1,49,11,59)(2,50,12,60)(3,51,13,41)(4,52,14,42)(5,53,15,43)(6,54,16,44)(7,55,17,45)(8,56,18,46)(9,57,19,47)(10,58,20,48)(21,79,31,69)(22,80,32,70)(23,61,33,71)(24,62,34,72)(25,63,35,73)(26,64,36,74)(27,65,37,75)(28,66,38,76)(29,67,39,77)(30,68,40,78), (1,28,11,38)(2,39,12,29)(3,30,13,40)(4,21,14,31)(5,32,15,22)(6,23,16,33)(7,34,17,24)(8,25,18,35)(9,36,19,26)(10,27,20,37)(41,68,51,78)(42,79,52,69)(43,70,53,80)(44,61,54,71)(45,72,55,62)(46,63,56,73)(47,74,57,64)(48,65,58,75)(49,76,59,66)(50,67,60,77), (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), (1,65,11,75)(2,74,12,64)(3,63,13,73)(4,72,14,62)(5,61,15,71)(6,70,16,80)(7,79,17,69)(8,68,18,78)(9,77,19,67)(10,66,20,76)(21,45,31,55)(22,54,32,44)(23,43,33,53)(24,52,34,42)(25,41,35,51)(26,50,36,60)(27,59,37,49)(28,48,38,58)(29,57,39,47)(30,46,40,56)>;
G:=Group( (1,49,11,59)(2,50,12,60)(3,51,13,41)(4,52,14,42)(5,53,15,43)(6,54,16,44)(7,55,17,45)(8,56,18,46)(9,57,19,47)(10,58,20,48)(21,79,31,69)(22,80,32,70)(23,61,33,71)(24,62,34,72)(25,63,35,73)(26,64,36,74)(27,65,37,75)(28,66,38,76)(29,67,39,77)(30,68,40,78), (1,28,11,38)(2,39,12,29)(3,30,13,40)(4,21,14,31)(5,32,15,22)(6,23,16,33)(7,34,17,24)(8,25,18,35)(9,36,19,26)(10,27,20,37)(41,68,51,78)(42,79,52,69)(43,70,53,80)(44,61,54,71)(45,72,55,62)(46,63,56,73)(47,74,57,64)(48,65,58,75)(49,76,59,66)(50,67,60,77), (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), (1,65,11,75)(2,74,12,64)(3,63,13,73)(4,72,14,62)(5,61,15,71)(6,70,16,80)(7,79,17,69)(8,68,18,78)(9,77,19,67)(10,66,20,76)(21,45,31,55)(22,54,32,44)(23,43,33,53)(24,52,34,42)(25,41,35,51)(26,50,36,60)(27,59,37,49)(28,48,38,58)(29,57,39,47)(30,46,40,56) );
G=PermutationGroup([[(1,49,11,59),(2,50,12,60),(3,51,13,41),(4,52,14,42),(5,53,15,43),(6,54,16,44),(7,55,17,45),(8,56,18,46),(9,57,19,47),(10,58,20,48),(21,79,31,69),(22,80,32,70),(23,61,33,71),(24,62,34,72),(25,63,35,73),(26,64,36,74),(27,65,37,75),(28,66,38,76),(29,67,39,77),(30,68,40,78)], [(1,28,11,38),(2,39,12,29),(3,30,13,40),(4,21,14,31),(5,32,15,22),(6,23,16,33),(7,34,17,24),(8,25,18,35),(9,36,19,26),(10,27,20,37),(41,68,51,78),(42,79,52,69),(43,70,53,80),(44,61,54,71),(45,72,55,62),(46,63,56,73),(47,74,57,64),(48,65,58,75),(49,76,59,66),(50,67,60,77)], [(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)], [(1,65,11,75),(2,74,12,64),(3,63,13,73),(4,72,14,62),(5,61,15,71),(6,70,16,80),(7,79,17,69),(8,68,18,78),(9,77,19,67),(10,66,20,76),(21,45,31,55),(22,54,32,44),(23,43,33,53),(24,52,34,42),(25,41,35,51),(26,50,36,60),(27,59,37,49),(28,48,38,58),(29,57,39,47),(30,46,40,56)]])
Q8.10D10 is a maximal subgroup of
D20.4D4 D20.5D4 D20.14D4 D20.15D4 D20.29D4 D20.30D4 C40.C23 D20.44D4 C10.C25 D5×2- 1+4 D20.39C23 SL2(𝔽3).11D10 D20.38D6 D20.29D6 C30.33C24 Q8.15D30
Q8.10D10 is a maximal quotient of
C10.12- 1+4 C10.82+ 1+4 C10.2- 1+4 C10.2+ 1+4 C10.52- 1+4 C10.62- 1+4 C42.122D10 Q8⋊5Dic10 C42.125D10 C42.126D10 Q8⋊5D20 C42.132D10 C42.133D10 C42.134D10 C10.152- 1+4 C10.162- 1+4 C10.172- 1+4 D20⋊22D4 Dic10⋊22D4 C10.202- 1+4 C10.212- 1+4 C10.222- 1+4 C10.232- 1+4 C10.242- 1+4 C10.582+ 1+4 C10.262- 1+4 C42.147D10 C42.150D10 C42.151D10 C42.154D10 C42.157D10 C42.158D10 Dic10⋊8Q8 C42.171D10 D20⋊12D4 D20⋊8Q8 C42.174D10 C42.176D10 C42.177D10 C42.178D10 C42.180D10 C10.422- 1+4 Q8×C5⋊D4 C10.442- 1+4 C10.452- 1+4 D20.38D6 D20.29D6 C30.33C24 Q8.15D30
37 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
37 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D5 | D10 | D10 | 2- 1+4 | Q8.10D10 |
| kernel | Q8.10D10 | C4○D20 | Q8×D5 | Q8⋊2D5 | Q8×C10 | C2×Q8 | C2×C4 | Q8 | C5 | C1 |
| # reps | 1 | 6 | 4 | 4 | 1 | 2 | 6 | 8 | 1 | 4 |
Matrix representation of Q8.10D10 ►in GL4(𝔽41) generated by
| 1 | 0 | 7 | 2 |
| 0 | 1 | 39 | 34 |
| 17 | 40 | 40 | 0 |
| 1 | 24 | 0 | 40 |
| 28 | 8 | 12 | 0 |
| 33 | 13 | 0 | 12 |
| 39 | 0 | 13 | 33 |
| 0 | 39 | 8 | 28 |
| 3 | 38 | 27 | 27 |
| 3 | 24 | 14 | 2 |
| 7 | 7 | 38 | 3 |
| 34 | 40 | 38 | 17 |
| 26 | 26 | 36 | 5 |
| 8 | 15 | 40 | 5 |
| 6 | 35 | 26 | 26 |
| 34 | 35 | 8 | 15 |
G:=sub<GL(4,GF(41))| [1,0,17,1,0,1,40,24,7,39,40,0,2,34,0,40],[28,33,39,0,8,13,0,39,12,0,13,8,0,12,33,28],[3,3,7,34,38,24,7,40,27,14,38,38,27,2,3,17],[26,8,6,34,26,15,35,35,36,40,26,8,5,5,26,15] >;
Q8.10D10 in GAP, Magma, Sage, TeX
Q_8._{10}D_{10} % in TeX
G:=Group("Q8.10D10"); // GroupNames label
G:=SmallGroup(160,222);
// by ID
G=gap.SmallGroup(160,222);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,188,86,579,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^10=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations