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G = C23.28C42order 128 = 27

10th non-split extension by C23 of C42 acting via C42/C2×C4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C23.28C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C42 — C23.28C42
 Lower central C1 — C22 — C23.28C42
 Upper central C1 — C22×C4 — C23.28C42
 Jennings C1 — C2 — C2 — C22×C4 — C23.28C42

Generators and relations for C23.28C42
G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, dad-1=ac=ca, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 356 in 240 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×8], C4 [×8], C22 [×3], C22 [×8], C22 [×12], C8 [×8], C2×C4 [×2], C2×C4 [×34], C2×C4 [×24], C23, C23 [×6], C23 [×4], C42 [×8], C2×C8 [×8], C2×C8 [×8], M4(2) [×16], C22×C4 [×2], C22×C4 [×16], C22×C4 [×12], C24, C2×C42 [×4], C2×C42 [×4], C22×C8 [×4], C2×M4(2) [×8], C2×M4(2) [×8], C23×C4, C23×C4 [×2], C22.7C42 [×4], C22×C42, C22×M4(2) [×2], C23.28C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], M4(2) [×8], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×M4(2) [×4], C2×C2.C42, C4×M4(2) [×2], C24.4C4 [×2], C4⋊M4(2) [×2], C23.28C42

Smallest permutation representation of C23.28C42
On 64 points
Generators in S64
```(1 55)(2 52)(3 49)(4 54)(5 51)(6 56)(7 53)(8 50)(9 61)(10 58)(11 63)(12 60)(13 57)(14 62)(15 59)(16 64)(17 25)(18 30)(19 27)(20 32)(21 29)(22 26)(23 31)(24 28)(33 42)(34 47)(35 44)(36 41)(37 46)(38 43)(39 48)(40 45)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 59)(34 60)(35 61)(36 62)(37 63)(38 64)(39 57)(40 58)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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 63 55 11)(2 38 56 47)(3 57 49 13)(4 40 50 41)(5 59 51 15)(6 34 52 43)(7 61 53 9)(8 36 54 45)(10 22 62 30)(12 24 64 32)(14 18 58 26)(16 20 60 28)(17 39 25 48)(19 33 27 42)(21 35 29 44)(23 37 31 46)```

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4AB 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C4 D4 Q8 M4(2) kernel C23.28C42 C22.7C42 C22×C42 C22×M4(2) C2×C42 C2×M4(2) C23×C4 C22×C4 C22×C4 C2×C4 # reps 1 4 1 2 4 16 4 6 2 16

Matrix representation of C23.28C42 in GL5(𝔽17)

 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 4 0 0 0 0 0 8 15 0 0 0 7 9 0 0 0 0 0 0 1 0 0 0 13 0
,
 4 0 0 0 0 0 16 2 0 0 0 16 1 0 0 0 0 0 16 0 0 0 0 0 16

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[4,0,0,0,0,0,8,7,0,0,0,15,9,0,0,0,0,0,0,13,0,0,0,1,0],[4,0,0,0,0,0,16,16,0,0,0,2,1,0,0,0,0,0,16,0,0,0,0,0,16] >;`

C23.28C42 in GAP, Magma, Sage, TeX

`C_2^3._{28}C_4^2`
`% in TeX`

`G:=Group("C2^3.28C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,460);`
`// by ID`

`G=gap.SmallGroup(128,460);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,1430,172]);`
`// Polycyclic`

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

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