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## G = C23.36C42order 128 = 27

### 18th 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 — C23.36C42
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C23.36C42
 Lower central C1 — C22 — C23.36C42
 Upper central C1 — C22×C4 — C23.36C42
 Jennings C1 — C2 — C2 — C22×C4 — C23.36C42

Generators and relations for C23.36C42
G = < a,b,c,d,e | a2=b2=c2=d4=1, e4=c, ab=ba, ac=ca, dad-1=abc, ae=ea, bc=cb, ede-1=bd=db, be=eb, cd=dc, ce=ec >

Subgroups: 276 in 180 conjugacy classes, 92 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×6], C2×C4 [×22], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×4], C2×C8 [×12], C2×C8 [×14], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C8⋊C4 [×4], C22⋊C8 [×4], C2×C42 [×2], C2×C22⋊C4 [×2], C22×C8 [×2], C22×C8 [×6], C22×C8 [×4], C23×C4, C22.7C42 [×2], C4×C22⋊C4, C2×C8⋊C4 [×2], C2×C22⋊C8, C23×C8, C23.36C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], M4(2) [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C8⋊C4 [×4], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C2×M4(2) [×2], C8○D4 [×2], C4×C22⋊C4, C2×C8⋊C4, C82M4(2), C89D4 [×4], C23.36C42

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

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 4M ··· 4T 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 D4 C4○D4 M4(2) C8○D4 kernel C23.36C42 C22.7C42 C4×C22⋊C4 C2×C8⋊C4 C2×C22⋊C8 C23×C8 C2.C42 C22⋊C8 C2×C22⋊C4 C22×C8 C2×C8 C2×C4 C23 C22 # reps 1 2 1 2 1 1 4 8 4 8 4 4 8 8

Matrix representation of C23.36C42 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 16 16
,
 1 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 16 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
,
 4 0 0 0 0 0 12 16 0 0 0 9 5 0 0 0 0 0 16 15 0 0 0 0 1
,
 1 0 0 0 0 0 4 15 0 0 0 10 13 0 0 0 0 0 4 0 0 0 0 13 13

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,16,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[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],[4,0,0,0,0,0,12,9,0,0,0,16,5,0,0,0,0,0,16,0,0,0,0,15,1],[1,0,0,0,0,0,4,10,0,0,0,15,13,0,0,0,0,0,4,13,0,0,0,0,13] >;`

C23.36C42 in GAP, Magma, Sage, TeX

`C_2^3._{36}C_4^2`
`% in TeX`

`G:=Group("C2^3.36C4^2");`
`// GroupNames label`

`G:=SmallGroup(128,484);`
`// by ID`

`G=gap.SmallGroup(128,484);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,172]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^4=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,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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