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## G = C24.227C23order 128 = 27

### 67th 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 — C24.227C23
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C22⋊Q8 — C24.227C23
 Lower central C1 — C22 — C24.227C23
 Upper central C1 — C23 — C24.227C23
 Jennings C1 — C23 — C24.227C23

Generators and relations for C24.227C23
G = < a,b,c,d,e,f,g | a2=b2=c2=f2=1, d2=c, e2=g2=a, ab=ba, ac=ca, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, gdg-1=abd, fef=abe, fg=gf >

Subgroups: 412 in 232 conjugacy classes, 132 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×40], Q8 [×4], C23, C23 [×2], C23 [×6], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×4], C4⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×16], C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×2], C24, C2.C42 [×2], C2.C42 [×12], C2×C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4, C2×C4⋊C4 [×8], C22⋊Q8 [×8], C23×C4, C22×Q8, C23.7Q8, C23.34D4, C23.63C23 [×4], C24.C22 [×4], C23.65C23 [×2], C23.67C23 [×2], C2×C22⋊Q8, C24.227C23
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], C22.11C24, C23.32C23, C23.33C23, C22.56C24 [×2], C22.57C24 [×2], C24.227C23

Smallest permutation representation of C24.227C23
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)
(1 19 11 47)(2 48 12 20)(3 17 9 45)(4 46 10 18)(5 42 38 14)(6 15 39 43)(7 44 40 16)(8 13 37 41)(21 57 49 29)(22 30 50 58)(23 59 51 31)(24 32 52 60)(25 61 53 35)(26 36 54 62)(27 63 55 33)(28 34 56 64)
(2 52)(4 50)(5 38)(6 63)(7 40)(8 61)(10 22)(12 24)(14 26)(16 28)(17 57)(18 46)(19 59)(20 48)(29 45)(30 58)(31 47)(32 60)(33 39)(34 64)(35 37)(36 62)(42 54)(44 56)
(1 15 11 43)(2 56 12 28)(3 13 9 41)(4 54 10 26)(5 30 38 58)(6 47 39 19)(7 32 40 60)(8 45 37 17)(14 50 42 22)(16 52 44 24)(18 62 46 36)(20 64 48 34)(21 53 49 25)(23 55 51 27)(29 35 57 61)(31 33 59 63)```

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

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

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

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 C24.227C23 C23.7Q8 C23.34D4 C23.63C23 C24.C22 C23.65C23 C23.67C23 C2×C22⋊Q8 C22⋊Q8 C22 C22 # reps 1 1 1 4 4 2 2 1 16 3 3

Matrix representation of C24.227C23 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 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
,
 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 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 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 2 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 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 0 2 0 3 0 0 0 0 0 0 0 3 0 3 0 0 0 0 0 4 0 3 0 0 0 0 0 0 0 4 0 2
,
 1 0 0 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 4 0 0 0 0 0 0 0 0 0 4 0 0 0 0 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 0 4 0 0 0 0 0 0 0 1 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 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 2 0 4 0 0 0 0 0 0 0 3 0 4
,
 1 0 0 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 0 4 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 2

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

C24.227C23 in GAP, Magma, Sage, TeX

`C_2^4._{227}C_2^3`
`% in TeX`

`G:=Group("C2^4.227C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1110);`
`// by ID`

`G=gap.SmallGroup(128,1110);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,555,1571,346,80]);`
`// Polycyclic`

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

׿
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