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## G = C42.173D4order 128 = 27

### 155th non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C23 — C42.173D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C2×C4×D4 — C42.173D4
 Lower central C1 — C23 — C42.173D4
 Upper central C1 — C23 — C42.173D4
 Jennings C1 — C23 — C42.173D4

Generators and relations for C42.173D4
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c-1 >

Subgroups: 548 in 282 conjugacy classes, 116 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×8], C4 [×12], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×12], C4⋊C4 [×14], C22×C4 [×3], C22×C4 [×10], C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×8], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4, C2×C4⋊C4 [×6], C4×D4 [×4], C4⋊Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8 [×2], C4×C4⋊C4, C23.7Q8 [×4], C428C4 [×2], C24.3C22 [×2], C23⋊Q8 [×4], C2×C4×D4, C2×C4⋊Q8, C42.173D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C4.4D4 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ 1+4 [×2], C2×C22⋊Q8, C2×C4.4D4, C22.29C24, D43Q8 [×2], C22.49C24 [×2], C42.173D4

Smallest permutation representation of C42.173D4
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 15 11 20)(2 16 12 17)(3 13 9 18)(4 14 10 19)(5 29 28 63)(6 30 25 64)(7 31 26 61)(8 32 27 62)(21 44 45 52)(22 41 46 49)(23 42 47 50)(24 43 48 51)(33 38 55 59)(34 39 56 60)(35 40 53 57)(36 37 54 58)
(1 5 42 33)(2 25 43 56)(3 7 44 35)(4 27 41 54)(6 51 34 12)(8 49 36 10)(9 26 52 53)(11 28 50 55)(13 61 45 57)(14 32 46 37)(15 63 47 59)(16 30 48 39)(17 64 24 60)(18 31 21 40)(19 62 22 58)(20 29 23 38)
(1 37 3 39)(2 40 4 38)(5 48 7 46)(6 47 8 45)(9 60 11 58)(10 59 12 57)(13 34 15 36)(14 33 16 35)(17 53 19 55)(18 56 20 54)(21 25 23 27)(22 28 24 26)(29 43 31 41)(30 42 32 44)(49 63 51 61)(50 62 52 64)```

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I ··· 4V 4W 4X 4Y 4Z order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 ··· 4 4 4 4 4 size 1 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4 8 8 8 8

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 Q8 C4○D4 2+ 1+4 kernel C42.173D4 C4×C4⋊C4 C23.7Q8 C42⋊8C4 C24.3C22 C23⋊Q8 C2×C4×D4 C2×C4⋊Q8 C42 C2×D4 C2×C4 C22 # reps 1 1 4 2 2 4 1 1 4 4 12 2

Matrix representation of C42.173D4 in GL6(𝔽5)

 3 0 0 0 0 0 0 2 0 0 0 0 0 0 4 2 0 0 0 0 4 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3
,
 3 0 0 0 0 0 0 2 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 3
,
 0 1 0 0 0 0 4 0 0 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 4 0 0 0 0 1 0 0 0 0 0 0 0 2 1 0 0 0 0 0 3 0 0 0 0 0 0 0 4 0 0 0 0 1 0

`G:=sub<GL(6,GF(5))| [3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,4,0,0,0,0,2,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,2,0,0,0,0,0,1,3,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;`

C42.173D4 in GAP, Magma, Sage, TeX

`C_4^2._{173}D_4`
`% in TeX`

`G:=Group("C4^2.173D4");`
`// GroupNames label`

`G:=SmallGroup(128,1295);`
`// by ID`

`G=gap.SmallGroup(128,1295);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,2,784,253,568,758,723,675,80]);`
`// Polycyclic`

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

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