Copied to
clipboard

## G = C24.5Q8order 128 = 27

### 4th non-split extension by C24 of Q8 acting via Q8/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.5Q8
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — C22×C22⋊C4 — C24.5Q8
 Lower central C1 — C23 — C24.5Q8
 Upper central C1 — C24 — C24.5Q8
 Jennings C1 — C24 — C24.5Q8

Generators and relations for C24.5Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=be2, ab=ba, faf-1=ac=ca, eae-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 812 in 390 conjugacy classes, 116 normal (34 characteristic)
C1, C2 [×7], C2 [×8], C2 [×4], C4 [×14], C22 [×11], C22 [×24], C22 [×36], C2×C4 [×4], C2×C4 [×62], C23 [×7], C23 [×12], C23 [×52], C22⋊C4 [×24], C4⋊C4 [×8], C22×C4 [×16], C22×C4 [×34], C24, C24 [×6], C24 [×12], C2.C42 [×6], C2×C22⋊C4 [×4], C2×C22⋊C4 [×16], C2×C4⋊C4 [×6], C23×C4 [×2], C23×C4 [×4], C25, C2×C2.C42, C2×C2.C42 [×2], C22×C22⋊C4, C22×C22⋊C4 [×2], C22×C4⋊C4, C24.5Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×14], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×7], C2×Q8, C4○D4 [×6], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×4], C22≀C2 [×3], C4⋊D4 [×8], C22⋊Q8 [×6], C22.D4 [×5], C4.4D4 [×3], C422C2 [×2], C41D4, C23.7Q8, C23.8Q8 [×2], C23.23D4, C24.C22 [×2], C24.3C22, C232D4, C23⋊Q8, C23.10D4 [×2], C23.Q8 [×2], C23.11D4, C23.4Q8, C24.5Q8

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2O 2P 2Q 2R 2S 4A ··· 4X order 1 2 ··· 2 2 2 2 2 4 ··· 4 size 1 1 ··· 1 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C4 D4 D4 Q8 C4○D4 kernel C24.5Q8 C2×C2.C42 C22×C22⋊C4 C22×C4⋊C4 C2×C22⋊C4 C22×C4 C24 C24 C23 # reps 1 3 3 1 8 12 2 2 12

Matrix representation of C24.5Q8 in GL7(𝔽5)

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

`G:=sub<GL(7,GF(5))| [1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,3,3,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,1,0],[3,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,3,0] >;`

C24.5Q8 in GAP, Magma, Sage, TeX

`C_2^4._5Q_8`
`% in TeX`

`G:=Group("C2^4.5Q8");`
`// GroupNames label`

`G:=SmallGroup(128,171);`
`// by ID`

`G=gap.SmallGroup(128,171);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,2,448,141,64,422,387]);`
`// Polycyclic`

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

׿
×
𝔽