Aliases: Q8.2D30, SL2(𝔽3).8D10, Q8⋊D15⋊1C2, (C2×C10).7S4, (Q8×C10)⋊2S3, (C2×Q8)⋊2D15, (C5×Q8).9D6, C10.21(C2×S4), C22.2(C5⋊S4), Q8.D15⋊1C2, C5⋊3(Q8.D6), (C2×SL2(𝔽3))⋊3D5, (C10×SL2(𝔽3))⋊3C2, (C5×SL2(𝔽3)).8C22, C2.7(C2×C5⋊S4), SmallGroup(480,1029)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2 — Q8 — C5×SL2(𝔽3) — Q8.D30 |
| C5×SL2(𝔽3) — Q8.D30 |
Generators and relations for Q8.D30
G = < a,b,c,d | a4=c30=1, b2=d2=a2, bab-1=a-1, cac-1=b, dad-1=a-1b, cbc-1=ab, dbd-1=a2b, dcd-1=a2c-1 >
Subgroups: 634 in 78 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, SL2(𝔽3), C3⋊D4, D15, C30, C8.C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C5×Q8, CSU2(𝔽3), GL2(𝔽3), C2×SL2(𝔽3), Dic15, D30, C2×C30, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, Q8.D6, C5×SL2(𝔽3), C15⋊7D4, C20.C23, Q8.D15, Q8⋊D15, C10×SL2(𝔽3), Q8.D30
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, Q8.D6, C5⋊S4, C2×C5⋊S4, Q8.D30
►On 80 points
Generators in S80
(1 22 9 39)(2 28 10 45)(3 34 6 36)(4 25 7 42)(5 31 8 48)(11 62 16 77)(12 53 17 68)(13 74 18 59)(14 65 19 80)(15 56 20 71)(21 43 38 26)(23 33 40 50)(24 46 41 29)(27 49 44 32)(30 37 47 35)(51 61 66 76)(52 57 67 72)(54 64 69 79)(55 60 70 75)(58 63 73 78)
(1 27 9 44)(2 33 10 50)(3 24 6 41)(4 30 7 47)(5 21 8 38)(11 52 16 67)(12 73 17 58)(13 64 18 79)(14 55 19 70)(15 76 20 61)(22 32 39 49)(23 45 40 28)(25 35 42 37)(26 48 43 31)(29 36 46 34)(51 56 66 71)(53 63 68 78)(54 59 69 74)(57 62 72 77)(60 65 75 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 16 9 11)(2 20 10 15)(3 14 6 19)(4 18 7 13)(5 12 8 17)(21 58 38 73)(22 72 39 57)(23 56 40 71)(24 70 41 55)(25 54 42 69)(26 68 43 53)(27 52 44 67)(28 66 45 51)(29 80 46 65)(30 64 47 79)(31 78 48 63)(32 62 49 77)(33 76 50 61)(34 60 36 75)(35 74 37 59)
G:=sub<Sym(80)| (1,22,9,39)(2,28,10,45)(3,34,6,36)(4,25,7,42)(5,31,8,48)(11,62,16,77)(12,53,17,68)(13,74,18,59)(14,65,19,80)(15,56,20,71)(21,43,38,26)(23,33,40,50)(24,46,41,29)(27,49,44,32)(30,37,47,35)(51,61,66,76)(52,57,67,72)(54,64,69,79)(55,60,70,75)(58,63,73,78), (1,27,9,44)(2,33,10,50)(3,24,6,41)(4,30,7,47)(5,21,8,38)(11,52,16,67)(12,73,17,58)(13,64,18,79)(14,55,19,70)(15,76,20,61)(22,32,39,49)(23,45,40,28)(25,35,42,37)(26,48,43,31)(29,36,46,34)(51,56,66,71)(53,63,68,78)(54,59,69,74)(57,62,72,77)(60,65,75,80), (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), (1,16,9,11)(2,20,10,15)(3,14,6,19)(4,18,7,13)(5,12,8,17)(21,58,38,73)(22,72,39,57)(23,56,40,71)(24,70,41,55)(25,54,42,69)(26,68,43,53)(27,52,44,67)(28,66,45,51)(29,80,46,65)(30,64,47,79)(31,78,48,63)(32,62,49,77)(33,76,50,61)(34,60,36,75)(35,74,37,59)>;
G:=Group( (1,22,9,39)(2,28,10,45)(3,34,6,36)(4,25,7,42)(5,31,8,48)(11,62,16,77)(12,53,17,68)(13,74,18,59)(14,65,19,80)(15,56,20,71)(21,43,38,26)(23,33,40,50)(24,46,41,29)(27,49,44,32)(30,37,47,35)(51,61,66,76)(52,57,67,72)(54,64,69,79)(55,60,70,75)(58,63,73,78), (1,27,9,44)(2,33,10,50)(3,24,6,41)(4,30,7,47)(5,21,8,38)(11,52,16,67)(12,73,17,58)(13,64,18,79)(14,55,19,70)(15,76,20,61)(22,32,39,49)(23,45,40,28)(25,35,42,37)(26,48,43,31)(29,36,46,34)(51,56,66,71)(53,63,68,78)(54,59,69,74)(57,62,72,77)(60,65,75,80), (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), (1,16,9,11)(2,20,10,15)(3,14,6,19)(4,18,7,13)(5,12,8,17)(21,58,38,73)(22,72,39,57)(23,56,40,71)(24,70,41,55)(25,54,42,69)(26,68,43,53)(27,52,44,67)(28,66,45,51)(29,80,46,65)(30,64,47,79)(31,78,48,63)(32,62,49,77)(33,76,50,61)(34,60,36,75)(35,74,37,59) );
G=PermutationGroup([[(1,22,9,39),(2,28,10,45),(3,34,6,36),(4,25,7,42),(5,31,8,48),(11,62,16,77),(12,53,17,68),(13,74,18,59),(14,65,19,80),(15,56,20,71),(21,43,38,26),(23,33,40,50),(24,46,41,29),(27,49,44,32),(30,37,47,35),(51,61,66,76),(52,57,67,72),(54,64,69,79),(55,60,70,75),(58,63,73,78)], [(1,27,9,44),(2,33,10,50),(3,24,6,41),(4,30,7,47),(5,21,8,38),(11,52,16,67),(12,73,17,58),(13,64,18,79),(14,55,19,70),(15,76,20,61),(22,32,39,49),(23,45,40,28),(25,35,42,37),(26,48,43,31),(29,36,46,34),(51,56,66,71),(53,63,68,78),(54,59,69,74),(57,62,72,77),(60,65,75,80)], [(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)], [(1,16,9,11),(2,20,10,15),(3,14,6,19),(4,18,7,13),(5,12,8,17),(21,58,38,73),(22,72,39,57),(23,56,40,71),(24,70,41,55),(25,54,42,69),(26,68,43,53),(27,52,44,67),(28,66,45,51),(29,80,46,65),(30,64,47,79),(31,78,48,63),(32,62,49,77),(33,76,50,61),(34,60,36,75),(35,74,37,59)]])
41 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | ··· | 10F | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 2 | 60 | 8 | 6 | 6 | 60 | 2 | 2 | 8 | 8 | 8 | 60 | 60 | 2 | ··· | 2 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
41 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | ||
| image | C1 | C2 | C2 | C2 | S3 | D5 | D6 | D10 | D15 | D30 | S4 | C2×S4 | Q8.D6 | Q8.D6 | Q8.D30 | C5⋊S4 | C2×C5⋊S4 |
| kernel | Q8.D30 | Q8.D15 | Q8⋊D15 | C10×SL2(𝔽3) | Q8×C10 | C2×SL2(𝔽3) | C5×Q8 | SL2(𝔽3) | C2×Q8 | Q8 | C2×C10 | C10 | C5 | C5 | C1 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 1 | 2 | 12 | 2 | 2 |
Matrix representation of Q8.D30 ►in GL4(𝔽241) generated by
| 1 | 2 | 0 | 0 |
| 240 | 240 | 0 | 0 |
| 15 | 15 | 225 | 15 |
| 16 | 15 | 15 | 16 |
| 31 | 30 | 0 | 0 |
| 225 | 210 | 0 | 0 |
| 225 | 225 | 0 | 1 |
| 240 | 225 | 240 | 0 |
| 74 | 196 | 0 | 0 |
| 24 | 143 | 0 | 0 |
| 68 | 24 | 0 | 81 |
| 68 | 24 | 91 | 231 |
| 231 | 0 | 0 | 162 |
| 10 | 0 | 150 | 170 |
| 203 | 98 | 0 | 10 |
| 105 | 0 | 0 | 10 |
G:=sub<GL(4,GF(241))| [1,240,15,16,2,240,15,15,0,0,225,15,0,0,15,16],[31,225,225,240,30,210,225,225,0,0,0,240,0,0,1,0],[74,24,68,68,196,143,24,24,0,0,0,91,0,0,81,231],[231,10,203,105,0,0,98,0,0,150,0,0,162,170,10,10] >;
Q8.D30 in GAP, Magma, Sage, TeX
Q_8.D_{30} % in TeX
G:=Group("Q8.D30"); // GroupNames label
G:=SmallGroup(480,1029);
// by ID
G=gap.SmallGroup(480,1029);
# by ID
G:=PCGroup([7,-2,-2,-3,-5,-2,2,-2,3389,170,1347,4204,3168,172,2525,1909,285,124]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^30=1,b^2=d^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,d*a*d^-1=a^-1*b,c*b*c^-1=a*b,d*b*d^-1=a^2*b,d*c*d^-1=a^2*c^-1>;
// generators/relations