Copied to
clipboard

## G = C24.230C23order 128 = 27

### 70th 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 — C22 — C24.230C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C42⋊2C2 — C24.230C23
 Lower central C1 — C22 — C24.230C23
 Upper central C1 — C23 — C24.230C23
 Jennings C1 — C23 — C24.230C23

Generators and relations for C24.230C23
G = < a,b,c,d,e,f,g | a2=b2=c2=e2=1, d2=c, f2=a, g2=b, ab=ba, ac=ca, ede=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=abd, fef-1=abe, fg=gf >

Subgroups: 396 in 228 conjugacy classes, 132 normal (13 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×20], C22, C22 [×6], C22 [×10], C2×C4 [×12], C2×C4 [×40], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×3], C22⋊C4 [×12], C22⋊C4 [×3], C4⋊C4 [×12], C4⋊C4 [×9], C22×C4 [×2], C22×C4 [×12], C22×C4 [×3], C24, C2.C42 [×12], C2×C42, C2×C42 [×3], C2×C22⋊C4 [×6], C2×C4⋊C4 [×12], C422C2 [×8], C23×C4, C425C4, C429C4, C23.8Q8 [×3], C23.63C23 [×3], C24.C22 [×3], C23.65C23 [×3], C2×C422C2, C24.230C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×3], 2- 1+4 [×3], C23.33C23 [×3], C22.54C24, C22.57C24 [×3], C24.230C23

Smallest permutation representation of C24.230C23
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 62)(6 33)(7 64)(8 35)(13 41)(15 43)(17 57)(18 30)(19 59)(20 32)(22 50)(24 52)(25 53)(27 55)(29 45)(31 47)(34 40)(36 38)(37 61)(39 63)(46 58)(48 60)
(1 19 11 47)(2 32 12 60)(3 17 9 45)(4 30 10 58)(5 26 38 54)(6 15 39 43)(7 28 40 56)(8 13 37 41)(14 62 42 36)(16 64 44 34)(18 22 46 50)(20 24 48 52)(21 57 49 29)(23 59 51 31)(25 61 53 35)(27 63 55 33)
(1 55 51 43)(2 16 52 28)(3 53 49 41)(4 14 50 26)(5 58 36 46)(6 19 33 31)(7 60 34 48)(8 17 35 29)(9 25 21 13)(10 42 22 54)(11 27 23 15)(12 44 24 56)(18 38 30 62)(20 40 32 64)(37 45 61 57)(39 47 63 59)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4AB order 1 2 ··· 2 2 2 4 ··· 4 size 1 1 ··· 1 4 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 4 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 2+ 1+4 2- 1+4 kernel C24.230C23 C42⋊5C4 C42⋊9C4 C23.8Q8 C23.63C23 C24.C22 C23.65C23 C2×C42⋊2C2 C42⋊2C2 C22 C22 # reps 1 1 1 3 3 3 3 1 16 3 3

Matrix representation of C24.230C23 in GL9(𝔽5)

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

`G:=sub<GL(9,GF(5))| [1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,0,1,4,0,1,0,0,0,0,0,2,4,4,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,3,0,0,0,0,0,0,0,3,0,3],[4,0,0,0,0,0,0,0,0,0,1,4,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,0,4,0,1,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,3,1,1,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,2,0,0,0,0,0,1,0,2,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0],[1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,1,1,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,3,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,2] >;`

C24.230C23 in GAP, Magma, Sage, TeX

`C_2^4._{230}C_2^3`
`% in TeX`

`G:=Group("C2^4.230C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1115);`
`// by ID`

`G=gap.SmallGroup(128,1115);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,555,100,1571,346,136]);`
`// Polycyclic`

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

׿
×
𝔽