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 |
Subgroups: 634 in 78 conjugacy classes, 17 normal (all characteristic)
C1, C2, C2 [×2], C3, C4 [×3], C22, C22, C5, S3, C6 [×2], C8 [×2], C2×C4 [×2], D4 [×2], Q8, Q8 [×2], D5, C10, C10, Dic3, D6, C2×C6, C15, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, C20 [×2], D10, C2×C10, SL2(𝔽3), C3⋊D4, D15, C30 [×2], C8.C22, C5⋊2C8 [×2], 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 [×2], C5⋊Q16 [×2], 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 [×3], C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, Q8.D6, C5⋊S4, C2×C5⋊S4, Q8.D30
Generators and relations
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 >
►On 80 points
Generators in S80
(1 40 9 34)(2 46 10 25)(3 37 6 31)(4 43 7 22)(5 49 8 28)(11 77 16 62)(12 68 17 53)(13 59 18 74)(14 80 19 65)(15 71 20 56)(21 47 42 26)(23 33 44 39)(24 50 45 29)(27 38 48 32)(30 41 36 35)(51 61 66 76)(52 57 67 72)(54 64 69 79)(55 60 70 75)(58 63 73 78)
(1 45 9 24)(2 36 10 30)(3 42 6 21)(4 48 7 27)(5 39 8 33)(11 67 16 52)(12 58 17 73)(13 79 18 64)(14 70 19 55)(15 61 20 76)(22 32 43 38)(23 49 44 28)(25 35 46 41)(26 37 47 31)(29 40 50 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 13 9 18)(2 17 10 12)(3 11 6 16)(4 15 7 20)(5 19 8 14)(21 67 42 52)(22 51 43 66)(23 65 44 80)(24 79 45 64)(25 63 46 78)(26 77 47 62)(27 61 48 76)(28 75 49 60)(29 59 50 74)(30 73 36 58)(31 57 37 72)(32 71 38 56)(33 55 39 70)(34 69 40 54)(35 53 41 68)
G:=sub<Sym(80)| (1,40,9,34)(2,46,10,25)(3,37,6,31)(4,43,7,22)(5,49,8,28)(11,77,16,62)(12,68,17,53)(13,59,18,74)(14,80,19,65)(15,71,20,56)(21,47,42,26)(23,33,44,39)(24,50,45,29)(27,38,48,32)(30,41,36,35)(51,61,66,76)(52,57,67,72)(54,64,69,79)(55,60,70,75)(58,63,73,78), (1,45,9,24)(2,36,10,30)(3,42,6,21)(4,48,7,27)(5,39,8,33)(11,67,16,52)(12,58,17,73)(13,79,18,64)(14,70,19,55)(15,61,20,76)(22,32,43,38)(23,49,44,28)(25,35,46,41)(26,37,47,31)(29,40,50,34)(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), (1,13,9,18)(2,17,10,12)(3,11,6,16)(4,15,7,20)(5,19,8,14)(21,67,42,52)(22,51,43,66)(23,65,44,80)(24,79,45,64)(25,63,46,78)(26,77,47,62)(27,61,48,76)(28,75,49,60)(29,59,50,74)(30,73,36,58)(31,57,37,72)(32,71,38,56)(33,55,39,70)(34,69,40,54)(35,53,41,68)>;
G:=Group( (1,40,9,34)(2,46,10,25)(3,37,6,31)(4,43,7,22)(5,49,8,28)(11,77,16,62)(12,68,17,53)(13,59,18,74)(14,80,19,65)(15,71,20,56)(21,47,42,26)(23,33,44,39)(24,50,45,29)(27,38,48,32)(30,41,36,35)(51,61,66,76)(52,57,67,72)(54,64,69,79)(55,60,70,75)(58,63,73,78), (1,45,9,24)(2,36,10,30)(3,42,6,21)(4,48,7,27)(5,39,8,33)(11,67,16,52)(12,58,17,73)(13,79,18,64)(14,70,19,55)(15,61,20,76)(22,32,43,38)(23,49,44,28)(25,35,46,41)(26,37,47,31)(29,40,50,34)(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), (1,13,9,18)(2,17,10,12)(3,11,6,16)(4,15,7,20)(5,19,8,14)(21,67,42,52)(22,51,43,66)(23,65,44,80)(24,79,45,64)(25,63,46,78)(26,77,47,62)(27,61,48,76)(28,75,49,60)(29,59,50,74)(30,73,36,58)(31,57,37,72)(32,71,38,56)(33,55,39,70)(34,69,40,54)(35,53,41,68) );
G=PermutationGroup([(1,40,9,34),(2,46,10,25),(3,37,6,31),(4,43,7,22),(5,49,8,28),(11,77,16,62),(12,68,17,53),(13,59,18,74),(14,80,19,65),(15,71,20,56),(21,47,42,26),(23,33,44,39),(24,50,45,29),(27,38,48,32),(30,41,36,35),(51,61,66,76),(52,57,67,72),(54,64,69,79),(55,60,70,75),(58,63,73,78)], [(1,45,9,24),(2,36,10,30),(3,42,6,21),(4,48,7,27),(5,39,8,33),(11,67,16,52),(12,58,17,73),(13,79,18,64),(14,70,19,55),(15,61,20,76),(22,32,43,38),(23,49,44,28),(25,35,46,41),(26,37,47,31),(29,40,50,34),(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)], [(1,13,9,18),(2,17,10,12),(3,11,6,16),(4,15,7,20),(5,19,8,14),(21,67,42,52),(22,51,43,66),(23,65,44,80),(24,79,45,64),(25,63,46,78),(26,77,47,62),(27,61,48,76),(28,75,49,60),(29,59,50,74),(30,73,36,58),(31,57,37,72),(32,71,38,56),(33,55,39,70),(34,69,40,54),(35,53,41,68)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
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≀