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## G = C42.433D4order 128 = 27

### 66th non-split extension by C42 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.433D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4×C8 — C42.433D4
 Lower central C1 — C2 — C2×C4 — C42.433D4
 Upper central C1 — C23 — C2×C42 — C42.433D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.433D4

Generators and relations for C42.433D4
G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=bc-1 >

Subgroups: 372 in 168 conjugacy classes, 64 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×2], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×6], C23, C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42 [×2], C4×C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C22×C8 [×2], C22×D4, C22×Q8, C428C4, C2×C4×C8, C2×D4⋊C4 [×2], C2×Q8⋊C4 [×2], C2×C4.4D4, C42.433D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C4○D8 [×4], C24.3C22, C23.24D4 [×2], C42.78C22 [×2], C8.12D4 [×2], C42.433D4

Smallest permutation representation of C42.433D4
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 52 12 39)(2 49 9 40)(3 50 10 37)(4 51 11 38)(5 42 23 55)(6 43 24 56)(7 44 21 53)(8 41 22 54)(13 29 25 34)(14 30 26 35)(15 31 27 36)(16 32 28 33)(17 46 62 58)(18 47 63 59)(19 48 64 60)(20 45 61 57)
(1 56 34 19)(2 42 35 63)(3 54 36 17)(4 44 33 61)(5 14 47 49)(6 25 48 39)(7 16 45 51)(8 27 46 37)(9 55 30 18)(10 41 31 62)(11 53 32 20)(12 43 29 64)(13 60 52 24)(15 58 50 22)(21 28 57 38)(23 26 59 40)
(1 4 12 11)(2 10 9 3)(5 62 23 17)(6 20 24 61)(7 64 21 19)(8 18 22 63)(13 28 25 16)(14 15 26 27)(29 32 34 33)(30 36 35 31)(37 49 50 40)(38 39 51 52)(41 59 54 47)(42 46 55 58)(43 57 56 45)(44 48 53 60)```

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 4A ··· 4L 4M ··· 4R 8A ··· 8P order 1 2 ··· 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 8 8 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 D4 D4 D4 C4○D4 C4○D8 kernel C42.433D4 C42⋊8C4 C2×C4×C8 C2×D4⋊C4 C2×Q8⋊C4 C2×C4.4D4 C4.4D4 C42 C2×C8 C22×C4 C2×C4 C22 # reps 1 1 1 2 2 1 8 2 4 2 4 16

Matrix representation of C42.433D4 in GL5(𝔽17)

 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 16 0
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 16
,
 13 0 0 0 0 0 3 14 0 0 0 14 14 0 0 0 0 0 0 13 0 0 0 13 0
,
 16 0 0 0 0 0 13 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 1 0

`G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,16,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,3,14,0,0,0,14,14,0,0,0,0,0,0,13,0,0,0,13,0],[16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,1,0] >;`

C42.433D4 in GAP, Magma, Sage, TeX

`C_4^2._{433}D_4`
`% in TeX`

`G:=Group("C4^2.433D4");`
`// GroupNames label`

`G:=SmallGroup(128,690);`
`// by ID`

`G=gap.SmallGroup(128,690);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,436,2019,248]);`
`// Polycyclic`

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

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