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## G = C24.17Q8order 128 = 27

### 1st non-split extension by C24 of Q8 acting via Q8/C4=C2

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C24.17Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C24×C4 — C24.17Q8
 Lower central C1 — C22 — C24.17Q8
 Upper central C1 — C24 — C24.17Q8
 Jennings C1 — C24 — C24.17Q8

Generators and relations for C24.17Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=de2, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be-1 >

Subgroups: 964 in 544 conjugacy classes, 172 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C23×C4, C25, C2×C2.C42, C22×C22⋊C4, C24×C4, C24.17Q8
Quotients:

Smallest permutation representation of C24.17Q8
On 64 points
Generators in S64
```(1 25)(2 26)(3 27)(4 28)(5 42)(6 43)(7 44)(8 41)(9 13)(10 14)(11 15)(12 16)(17 45)(18 46)(19 47)(20 48)(21 29)(22 30)(23 31)(24 32)(33 58)(34 59)(35 60)(36 57)(37 55)(38 56)(39 53)(40 54)(49 62)(50 63)(51 64)(52 61)
(1 23)(2 24)(3 21)(4 22)(5 12)(6 9)(7 10)(8 11)(13 43)(14 44)(15 41)(16 42)(17 37)(18 38)(19 39)(20 40)(25 31)(26 32)(27 29)(28 30)(33 51)(34 52)(35 49)(36 50)(45 55)(46 56)(47 53)(48 54)(57 63)(58 64)(59 61)(60 62)
(1 9)(2 10)(3 11)(4 12)(5 22)(6 23)(7 24)(8 21)(13 25)(14 26)(15 27)(16 28)(17 63)(18 64)(19 61)(20 62)(29 41)(30 42)(31 43)(32 44)(33 56)(34 53)(35 54)(36 55)(37 57)(38 58)(39 59)(40 60)(45 50)(46 51)(47 52)(48 49)
(1 25)(2 26)(3 27)(4 28)(5 42)(6 43)(7 44)(8 41)(9 13)(10 14)(11 15)(12 16)(17 50)(18 51)(19 52)(20 49)(21 29)(22 30)(23 31)(24 32)(33 38)(34 39)(35 40)(36 37)(45 63)(46 64)(47 61)(48 62)(53 59)(54 60)(55 57)(56 58)
(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)
(1 63 27 47)(2 60 28 56)(3 61 25 45)(4 58 26 54)(5 18 44 49)(6 37 41 34)(7 20 42 51)(8 39 43 36)(9 17 15 52)(10 40 16 33)(11 19 13 50)(12 38 14 35)(21 59 31 55)(22 64 32 48)(23 57 29 53)(24 62 30 46)```

`G:=sub<Sym(64)| (1,25)(2,26)(3,27)(4,28)(5,42)(6,43)(7,44)(8,41)(9,13)(10,14)(11,15)(12,16)(17,45)(18,46)(19,47)(20,48)(21,29)(22,30)(23,31)(24,32)(33,58)(34,59)(35,60)(36,57)(37,55)(38,56)(39,53)(40,54)(49,62)(50,63)(51,64)(52,61), (1,23)(2,24)(3,21)(4,22)(5,12)(6,9)(7,10)(8,11)(13,43)(14,44)(15,41)(16,42)(17,37)(18,38)(19,39)(20,40)(25,31)(26,32)(27,29)(28,30)(33,51)(34,52)(35,49)(36,50)(45,55)(46,56)(47,53)(48,54)(57,63)(58,64)(59,61)(60,62), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,63)(18,64)(19,61)(20,62)(29,41)(30,42)(31,43)(32,44)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (1,25)(2,26)(3,27)(4,28)(5,42)(6,43)(7,44)(8,41)(9,13)(10,14)(11,15)(12,16)(17,50)(18,51)(19,52)(20,49)(21,29)(22,30)(23,31)(24,32)(33,38)(34,39)(35,40)(36,37)(45,63)(46,64)(47,61)(48,62)(53,59)(54,60)(55,57)(56,58), (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), (1,63,27,47)(2,60,28,56)(3,61,25,45)(4,58,26,54)(5,18,44,49)(6,37,41,34)(7,20,42,51)(8,39,43,36)(9,17,15,52)(10,40,16,33)(11,19,13,50)(12,38,14,35)(21,59,31,55)(22,64,32,48)(23,57,29,53)(24,62,30,46)>;`

`G:=Group( (1,25)(2,26)(3,27)(4,28)(5,42)(6,43)(7,44)(8,41)(9,13)(10,14)(11,15)(12,16)(17,45)(18,46)(19,47)(20,48)(21,29)(22,30)(23,31)(24,32)(33,58)(34,59)(35,60)(36,57)(37,55)(38,56)(39,53)(40,54)(49,62)(50,63)(51,64)(52,61), (1,23)(2,24)(3,21)(4,22)(5,12)(6,9)(7,10)(8,11)(13,43)(14,44)(15,41)(16,42)(17,37)(18,38)(19,39)(20,40)(25,31)(26,32)(27,29)(28,30)(33,51)(34,52)(35,49)(36,50)(45,55)(46,56)(47,53)(48,54)(57,63)(58,64)(59,61)(60,62), (1,9)(2,10)(3,11)(4,12)(5,22)(6,23)(7,24)(8,21)(13,25)(14,26)(15,27)(16,28)(17,63)(18,64)(19,61)(20,62)(29,41)(30,42)(31,43)(32,44)(33,56)(34,53)(35,54)(36,55)(37,57)(38,58)(39,59)(40,60)(45,50)(46,51)(47,52)(48,49), (1,25)(2,26)(3,27)(4,28)(5,42)(6,43)(7,44)(8,41)(9,13)(10,14)(11,15)(12,16)(17,50)(18,51)(19,52)(20,49)(21,29)(22,30)(23,31)(24,32)(33,38)(34,39)(35,40)(36,37)(45,63)(46,64)(47,61)(48,62)(53,59)(54,60)(55,57)(56,58), (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), (1,63,27,47)(2,60,28,56)(3,61,25,45)(4,58,26,54)(5,18,44,49)(6,37,41,34)(7,20,42,51)(8,39,43,36)(9,17,15,52)(10,40,16,33)(11,19,13,50)(12,38,14,35)(21,59,31,55)(22,64,32,48)(23,57,29,53)(24,62,30,46) );`

`G=PermutationGroup([[(1,25),(2,26),(3,27),(4,28),(5,42),(6,43),(7,44),(8,41),(9,13),(10,14),(11,15),(12,16),(17,45),(18,46),(19,47),(20,48),(21,29),(22,30),(23,31),(24,32),(33,58),(34,59),(35,60),(36,57),(37,55),(38,56),(39,53),(40,54),(49,62),(50,63),(51,64),(52,61)], [(1,23),(2,24),(3,21),(4,22),(5,12),(6,9),(7,10),(8,11),(13,43),(14,44),(15,41),(16,42),(17,37),(18,38),(19,39),(20,40),(25,31),(26,32),(27,29),(28,30),(33,51),(34,52),(35,49),(36,50),(45,55),(46,56),(47,53),(48,54),(57,63),(58,64),(59,61),(60,62)], [(1,9),(2,10),(3,11),(4,12),(5,22),(6,23),(7,24),(8,21),(13,25),(14,26),(15,27),(16,28),(17,63),(18,64),(19,61),(20,62),(29,41),(30,42),(31,43),(32,44),(33,56),(34,53),(35,54),(36,55),(37,57),(38,58),(39,59),(40,60),(45,50),(46,51),(47,52),(48,49)], [(1,25),(2,26),(3,27),(4,28),(5,42),(6,43),(7,44),(8,41),(9,13),(10,14),(11,15),(12,16),(17,50),(18,51),(19,52),(20,49),(21,29),(22,30),(23,31),(24,32),(33,38),(34,39),(35,40),(36,37),(45,63),(46,64),(47,61),(48,62),(53,59),(54,60),(55,57),(56,58)], [(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)], [(1,63,27,47),(2,60,28,56),(3,61,25,45),(4,58,26,54),(5,18,44,49),(6,37,41,34),(7,20,42,51),(8,39,43,36),(9,17,15,52),(10,40,16,33),(11,19,13,50),(12,38,14,35),(21,59,31,55),(22,64,32,48),(23,57,29,53),(24,62,30,46)]])`

56 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 4A ··· 4P 4Q ··· 4AF order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C4 C4 D4 D4 Q8 C4○D4 kernel C24.17Q8 C2×C2.C42 C22×C22⋊C4 C24×C4 C2×C22⋊C4 C23×C4 C22×C4 C24 C24 C23 # reps 1 4 2 1 16 8 8 6 2 8

Matrix representation of C24.17Q8 in GL6(𝔽5)

 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 4 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 0 0 0 0 4
,
 3 0 0 0 0 0 0 4 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 2
,
 3 0 0 0 0 0 0 3 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,2],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C24.17Q8 in GAP, Magma, Sage, TeX

`C_2^4._{17}Q_8`
`% in TeX`

`G:=Group("C2^4.17Q8");`
`// GroupNames label`

`G:=SmallGroup(128,165);`
`// by ID`

`G=gap.SmallGroup(128,165);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,2,224,141,456,422]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=d*e^2,a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*e^-1>;`
`// generators/relations`

׿
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