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G = C42.300C23order 128 = 27

161st non-split extension by C42 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 — C42.300C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.26C24 — C42.300C23
 Lower central C1 — C22 — C42.300C23
 Upper central C1 — C2×C4 — C42.300C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.300C23

Generators and relations for C42.300C23
G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2c, ede=b2d >

Subgroups: 332 in 197 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.6C4 [×2], C89D4 [×8], C22.26C24, C42.300C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×2], C22.11C24, Q8○M4(2) [×2], C42.300C23

Smallest permutation representation of C42.300C23
On 64 points
Generators in S64
```(1 47 55 33)(2 38 56 44)(3 41 49 35)(4 40 50 46)(5 43 51 37)(6 34 52 48)(7 45 53 39)(8 36 54 42)(9 22 64 26)(10 31 57 19)(11 24 58 28)(12 25 59 21)(13 18 60 30)(14 27 61 23)(15 20 62 32)(16 29 63 17)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 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 17)(2 30)(3 19)(4 32)(5 21)(6 26)(7 23)(8 28)(9 48)(10 35)(11 42)(12 37)(13 44)(14 39)(15 46)(16 33)(18 56)(20 50)(22 52)(24 54)(25 51)(27 53)(29 55)(31 49)(34 64)(36 58)(38 60)(40 62)(41 57)(43 59)(45 61)(47 63)
(1 19)(2 32)(3 21)(4 26)(5 23)(6 28)(7 17)(8 30)(9 40)(10 47)(11 34)(12 41)(13 36)(14 43)(15 38)(16 45)(18 54)(20 56)(22 50)(24 52)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 61)(39 63)(42 60)(44 62)(46 64)(48 58)```

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2H 4A 4B 4C 4D 4E ··· 4M 8A ··· 8P order 1 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8 size 1 1 1 1 4 ··· 4 1 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 C4 C4 C4 C4 2+ 1+4 Q8○M4(2) kernel C42.300C23 (C22×C8)⋊C2 C42.6C4 C8⋊9D4 C22.26C24 C4⋊D4 C4.4D4 C4⋊1D4 C4⋊Q8 C4 C2 # reps 1 4 2 8 1 8 4 2 2 2 4

Matrix representation of C42.300C23 in GL8(𝔽17)

 13 0 0 9 0 0 0 0 4 0 13 4 0 0 0 0 13 13 0 13 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 16 16 1 1 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 15 15 0 1
,
 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 0 0 13 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 1
,
 16 0 5 0 0 0 0 0 7 7 6 6 0 0 0 0 16 0 1 0 0 0 0 0 11 11 6 10 0 0 0 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 0 13 13 4 4 0 0 0 0 8 0 9 13
,
 4 8 0 0 0 0 0 0 13 13 0 0 0 0 0 0 4 4 0 4 0 0 0 0 0 13 13 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 16 16 1 1 0 0 0 0 2 2 0 16
,
 16 15 0 0 0 0 0 0 0 1 0 0 0 0 0 0 16 16 0 16 0 0 0 0 1 1 16 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 16 16 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(17))| [13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,9,4,13,4,0,0,0,0,0,0,0,0,16,0,0,15,0,0,0,0,16,0,1,15,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,1],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,7,16,11,0,0,0,0,0,7,0,11,0,0,0,0,5,6,1,6,0,0,0,0,0,6,0,10,0,0,0,0,0,0,0,0,0,4,13,8,0,0,0,0,13,0,13,0,0,0,0,0,0,0,4,9,0,0,0,0,0,0,4,13],[4,13,4,0,0,0,0,0,8,13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,16,2,0,0,0,0,1,0,16,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16],[16,0,16,1,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;`

C42.300C23 in GAP, Magma, Sage, TeX

`C_4^2._{300}C_2^3`
`% in TeX`

`G:=Group("C4^2.300C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1712);`
`// by ID`

`G=gap.SmallGroup(128,1712);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,521,124]);`
`// Polycyclic`

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

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