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## G = C42.22C23order 128 = 27

### 22nd 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 — C42.22C23
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C23.33C23 — C42.22C23
 Lower central C1 — C2 — C2×C4 — C42.22C23
 Upper central C1 — C22 — C42⋊C2 — C42.22C23
 Jennings C1 — C2 — C2 — C2×C4 — C42.22C23

Generators and relations for C42.22C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=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: 332 in 188 conjugacy classes, 100 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×6], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×2], D4 [×5], Q8 [×2], Q8 [×3], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×5], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C4○D4 [×4], C4○D4 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4 [×2], C2×C4⋊C4, C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8, C2×M4(2), C2×C4○D4, C23.24D4, C23.36D4, C42.6C22, C2×C2.D8, M4(2)⋊C4, D4⋊Q8 [×2], C4.Q16 [×2], D4.Q8 [×2], Q8.Q8 [×2], C23.33C23, C23.41C23, C42.22C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×C22⋊Q8, D4○D8, Q8○D8, C42.22C23

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A 8B 8C 8D 8E 8F order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 8 8 8 8 8 size 1 1 1 1 2 2 4 4 2 2 2 2 4 ··· 4 8 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 Q8 C4○D4 D4○D8 Q8○D8 kernel C42.22C23 C23.24D4 C23.36D4 C42.6C22 C2×C2.D8 M4(2)⋊C4 D4⋊Q8 C4.Q16 D4.Q8 Q8.Q8 C23.33C23 C23.41C23 C22⋊C4 C4⋊C4 C4○D4 C2×C4 C2 C2 # reps 1 1 1 1 1 1 2 2 2 2 1 1 2 2 4 4 2 2

Matrix representation of C42.22C23 in GL6(𝔽17)

 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 4 0 0 4 0 0 0 0 0 0 4 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 15 0 0 0 0 1 1 0 0 0 0 0 0 16 15 0 0 0 0 1 1
,
 10 16 0 0 0 0 16 7 0 0 0 0 0 0 0 0 11 11 0 0 0 0 3 6 0 0 6 6 0 0 0 0 14 11 0 0
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16 0 0 0 0 0 0 1 0 0 0 0 0 16 16
,
 0 16 0 0 0 0 1 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13 0 0 0 0 0 0 4 0 0 0 0 0 0 4

`G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,4,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,0,0,0,0,16,1,0,0,0,0,15,1],[10,16,0,0,0,0,16,7,0,0,0,0,0,0,0,0,6,14,0,0,0,0,6,11,0,0,11,3,0,0,0,0,11,6,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,0,16],[0,1,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;`

C42.22C23 in GAP, Magma, Sage, TeX

`C_4^2._{22}C_2^3`
`% in TeX`

`G:=Group("C4^2.22C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,1815);`
`// by ID`

`G=gap.SmallGroup(128,1815);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,120,758,1018,248,4037,1027,124]);`
`// Polycyclic`

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