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## G = Q8.D30order 480 = 25·3·5

### 2nd non-split extension by Q8 of D30 acting via D30/C10=S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C5×SL2(𝔽3) — Q8.D30
 Chief series C1 — C2 — Q8 — C5×Q8 — C5×SL2(𝔽3) — Q8⋊D15 — Q8.D30
 Lower central C5×SL2(𝔽3) — Q8.D30
 Upper central C1 — C2 — C22

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, C52C8, 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), C157D4, 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

Smallest permutation representation of 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`

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