Copied to
clipboard

## G = C24.308C23order 128 = 27

### 148th non-split extension by C24 of C23 acting via C23/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C24.308C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C22⋊C4 — C24.308C23
 Lower central C1 — C23 — C24.308C23
 Upper central C1 — C23 — C24.308C23
 Jennings C1 — C23 — C24.308C23

Generators and relations for C24.308C23
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=b, g2=a, ab=ba, ac=ca, ede=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Subgroups: 420 in 242 conjugacy classes, 104 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×44], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×8], C22⋊C4 [×12], C4⋊C4 [×18], C22×C4 [×2], C22×C4 [×16], C22×C4 [×4], C2×Q8 [×6], C24, C2.C42 [×14], C2×C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×3], C2×C4⋊C4 [×6], C42⋊C2 [×4], C22⋊Q8 [×4], C23×C4, C22×Q8, C4×C22⋊C4, C428C4, C429C4, C23.63C23 [×4], C23.67C23 [×2], C23.11D4 [×4], C2×C42⋊C2, C2×C22⋊Q8, C24.308C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22.D4 [×4], C22×D4, C2×C4○D4 [×4], 2- 1+4 [×2], C2×C22.D4, C22.26C24, C23.38C23, C22.46C24 [×2], C22.50C24 [×2], C24.308C23

Smallest permutation representation of C24.308C23
On 64 points
Generators in S64
```(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(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)
(2 12)(4 10)(5 40)(6 8)(7 38)(14 42)(16 44)(17 19)(18 48)(20 46)(22 50)(24 52)(26 54)(28 56)(29 31)(30 60)(32 58)(33 35)(34 62)(36 64)(37 39)(45 47)(57 59)(61 63)
(1 59 51 47)(2 48 52 60)(3 57 49 45)(4 46 50 58)(5 26 36 14)(6 15 33 27)(7 28 34 16)(8 13 35 25)(9 29 21 17)(10 18 22 30)(11 31 23 19)(12 20 24 32)(37 41 61 53)(38 54 62 42)(39 43 63 55)(40 56 64 44)
(1 15 11 43)(2 44 12 16)(3 13 9 41)(4 42 10 14)(5 46 38 18)(6 19 39 47)(7 48 40 20)(8 17 37 45)(21 53 49 25)(22 26 50 54)(23 55 51 27)(24 28 52 56)(29 61 57 35)(30 36 58 62)(31 63 59 33)(32 34 60 64)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4H 4I ··· 4X 4Y 4Z 4AA 4AB order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 4 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 2- 1+4 kernel C24.308C23 C4×C22⋊C4 C42⋊8C4 C42⋊9C4 C23.63C23 C23.67C23 C23.11D4 C2×C42⋊C2 C2×C22⋊Q8 C22×C4 C2×C4 C22 # reps 1 1 1 1 4 2 4 1 1 4 16 2

Matrix representation of C24.308C23 in GL6(𝔽5)

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

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

C24.308C23 in GAP, Magma, Sage, TeX

`C_2^4._{308}C_2^3`
`% in TeX`

`G:=Group("C2^4.308C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1231);`
`// by ID`

`G=gap.SmallGroup(128,1231);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,723,184,675,80]);`
`// Polycyclic`

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

׿
×
𝔽