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## G = C42.19C23order 128 = 27

### 19th non-split extension by C42 of C23 acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.19C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42.19C23
 Lower central C1 — C2 — C2×C4 — C42.19C23
 Upper central C1 — C22 — C42⋊C2 — C42.19C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.19C23

Generators and relations for C42.19C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=a2, ab=ba, cac-1=a-1, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 436 in 236 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×17], D4 [×2], D4 [×9], Q8 [×2], Q8 [×9], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], D8 [×2], SD16 [×8], Q16 [×6], C22×C4, C22×C4 [×6], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×6], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×4], C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C2×SD16 [×2], C2×Q16, C2×Q16 [×2], C4○D8 [×4], C8.C22 [×4], C22×Q8, C2×C4○D4 [×2], C2×Q8⋊C4, C23.36D4, C42.6C22, D4.D4 [×2], C42Q16 [×2], D4.2D4 [×2], Q8.D4 [×2], C23.33C23, C23.38C23, C2×C4○D8, C2×C8.C22, C42.19C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16, Q8○D8, C42.19C23

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 8 2 2 2 2 4 ··· 4 8 8 8 4 4 4 4 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 D4○SD16 Q8○D8 kernel C42.19C23 C2×Q8⋊C4 C23.36D4 C42.6C22 D4.D4 C4⋊2Q16 D4.2D4 Q8.D4 C23.33C23 C23.38C23 C2×C4○D8 C2×C8.C22 C22⋊C4 C4⋊C4 C4○D4 C2×C4 C2 C2 # reps 1 1 1 1 2 2 2 2 1 1 1 1 2 2 4 4 2 2

Matrix representation of C42.19C23 in GL6(𝔽17)

 9 13 0 0 0 0 12 8 0 0 0 0 0 0 5 0 12 0 0 0 0 5 0 12 0 0 12 0 12 0 0 0 0 12 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 1 0 0 0 0 0 13 16 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 16 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 3 3 0 0 0 0 3 14 0 0 0 0 0 0 3 3 0 0 0 0 3 14
,
 9 13 0 0 0 0 12 8 0 0 0 0 0 0 5 0 5 0 0 0 0 5 0 5 0 0 5 0 12 0 0 0 0 5 0 12

`G:=sub<GL(6,GF(17))| [9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,12,0,0,0,0,5,0,12,0,0,12,0,12,0,0,0,0,12,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[1,13,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,3,3,0,0,0,0,3,14],[9,12,0,0,0,0,13,8,0,0,0,0,0,0,5,0,5,0,0,0,0,5,0,5,0,0,5,0,12,0,0,0,0,5,0,12] >;`

C42.19C23 in GAP, Magma, Sage, TeX

`C_4^2._{19}C_2^3`
`% in TeX`

`G:=Group("C4^2.19C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1778);`
`// by ID`

`G=gap.SmallGroup(128,1778);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,248,4037,1027,124]);`
`// Polycyclic`

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

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