Copied to
clipboard

## G = C24.203C23order 128 = 27

### 43rd 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 — C24.203C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4 — C24.203C23
 Lower central C1 — C22 — C24.203C23
 Upper central C1 — C23 — C24.203C23
 Jennings C1 — C23 — C24.203C23

Generators and relations for C24.203C23
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=e2=c, f2=a, g2=b, ab=ba, ac=ca, ede-1=gdg-1=ad=da, fef-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, eg=ge, fg=gf >

Subgroups: 396 in 238 conjugacy classes, 136 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×22], C22 [×7], C22 [×10], C2×C4 [×16], C2×C4 [×38], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×6], C22⋊C4 [×12], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×10], C22×C4 [×14], C22×C4 [×4], C24, C2.C42 [×12], C2×C42 [×6], C2×C22⋊C4 [×6], C2×C4⋊C4 [×10], C422C2 [×8], C23×C4, C4×C22⋊C4, C4×C4⋊C4 [×2], C428C4, C23.8Q8 [×2], C23.63C23 [×3], C24.C22 [×3], C23.65C23 [×2], C2×C422C2, C24.203C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C23×C4, C2×C4○D4 [×2], 2+ 1+4 [×2], 2- 1+4 [×2], C4×C4○D4, C23.33C23 [×2], C22.32C24, C22.33C24, C22.35C24, C22.36C24, C24.203C23

Smallest permutation representation of C24.203C23
On 64 points
Generators in S64
```(1 30)(2 31)(3 32)(4 29)(5 54)(6 55)(7 56)(8 53)(9 52)(10 49)(11 50)(12 51)(13 48)(14 45)(15 46)(16 47)(17 40)(18 37)(19 38)(20 39)(21 42)(22 43)(23 44)(24 41)(25 34)(26 35)(27 36)(28 33)(57 64)(58 61)(59 62)(60 63)
(1 61)(2 62)(3 63)(4 64)(5 35)(6 36)(7 33)(8 34)(9 18)(10 19)(11 20)(12 17)(13 41)(14 42)(15 43)(16 44)(21 45)(22 46)(23 47)(24 48)(25 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(37 52)(38 49)(39 50)(40 51)
(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)
(1 24 3 22)(2 42 4 44)(5 19 7 17)(6 39 8 37)(9 27 11 25)(10 33 12 35)(13 60 15 58)(14 64 16 62)(18 55 20 53)(21 29 23 31)(26 49 28 51)(30 41 32 43)(34 52 36 50)(38 56 40 54)(45 57 47 59)(46 61 48 63)
(1 10 30 49)(2 20 31 39)(3 12 32 51)(4 18 29 37)(5 46 54 15)(6 23 55 44)(7 48 56 13)(8 21 53 42)(9 57 52 64)(11 59 50 62)(14 34 45 25)(16 36 47 27)(17 60 40 63)(19 58 38 61)(22 26 43 35)(24 28 41 33)
(1 34 61 8)(2 26 62 54)(3 36 63 6)(4 28 64 56)(5 31 35 59)(7 29 33 57)(9 13 18 41)(10 45 19 21)(11 15 20 43)(12 47 17 23)(14 38 42 49)(16 40 44 51)(22 50 46 39)(24 52 48 37)(25 58 53 30)(27 60 55 32)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 4M ··· 4AH order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 4 4 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 4 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 C4○D4 2+ 1+4 2- 1+4 kernel C24.203C23 C4×C22⋊C4 C4×C4⋊C4 C42⋊8C4 C23.8Q8 C23.63C23 C24.C22 C23.65C23 C2×C42⋊2C2 C42⋊2C2 C2×C4 C22 C22 # reps 1 1 2 1 2 3 3 2 1 16 8 2 2

Matrix representation of C24.203C23 in GL8(𝔽5)

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

`G:=sub<GL(8,GF(5))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0] >;`

C24.203C23 in GAP, Magma, Sage, TeX

`C_2^4._{203}C_2^3`
`% in TeX`

`G:=Group("C2^4.203C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1066);`
`// by ID`

`G=gap.SmallGroup(128,1066);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758,219,100,675,136]);`
`// Polycyclic`

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

׿
×
𝔽