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## G = C4.172+ 1+4order 128 = 27

### 17th non-split extension by C4 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.172+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.38C23 — C4.172+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.172+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.172+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.172+ 1+4

Generators and relations for C4.172+ 1+4
G = < a,b,c,d,e | a4=c2=1, b4=e2=a2, d2=a-1b2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

Subgroups: 364 in 189 conjugacy classes, 84 normal (16 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×Q16, C22×Q8, C2×C4○D4, (C22×C8)⋊C2, C22⋊Q16, C8.18D4, C8.D4, C23.20D4, C23.38C23, C4.172+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, Q8○D8, C4.172+ 1+4

Character table of C4.172+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 2 2 4 4 4 8 8 8 8 8 8 8 8 4 4 4 4 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ3 1 1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ4 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ5 1 1 1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ7 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 1 -1 1 -1 linear of order 2 ρ9 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 1 1 -1 -1 linear of order 2 ρ10 1 1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ11 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ12 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ13 1 1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 linear of order 2 ρ14 1 1 1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 linear of order 2 ρ16 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ17 2 2 2 2 -2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 -2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 2 -2 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 -2 2 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ22 4 -4 4 -4 0 0 0 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from 2+ 1+4 ρ23 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ24 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ25 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2√2 0 2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2 ρ26 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2√2 0 -2√2 0 0 0 symplectic lifted from Q8○D8, Schur index 2

Smallest permutation representation of C4.172+ 1+4
On 64 points
Generators in S64
```(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 37 13 33)(10 38 14 34)(11 39 15 35)(12 40 16 36)(17 45 21 41)(18 46 22 42)(19 47 23 43)(20 48 24 44)(25 56 29 52)(26 49 30 53)(27 50 31 54)(28 51 32 55)
(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)
(2 60)(4 62)(6 64)(8 58)(9 35)(10 14)(11 37)(12 16)(13 39)(15 33)(17 21)(18 44)(19 23)(20 46)(22 48)(24 42)(25 50)(27 52)(29 54)(31 56)(34 38)(36 40)(41 45)(43 47)
(1 43 63 17)(2 20 64 46)(3 41 57 23)(4 18 58 44)(5 47 59 21)(6 24 60 42)(7 45 61 19)(8 22 62 48)(9 52 35 31)(10 26 36 55)(11 50 37 29)(12 32 38 53)(13 56 39 27)(14 30 40 51)(15 54 33 25)(16 28 34 49)
(1 36 5 40)(2 37 6 33)(3 38 7 34)(4 39 8 35)(9 58 13 62)(10 59 14 63)(11 60 15 64)(12 61 16 57)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(25 46 29 42)(26 47 30 43)(27 48 31 44)(28 41 32 45)```

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

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

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

Matrix representation of C4.172+ 1+4 in GL8(𝔽17)

 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 0 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 3 7 16 2 0 0 0 0 15 5 16 1
,
 9 0 16 6 0 0 0 0 9 0 16 8 0 0 0 0 0 14 8 3 0 0 0 0 9 8 0 0 0 0 0 0 0 0 0 0 14 14 0 0 0 0 0 0 3 14 0 0 0 0 0 0 9 4 11 6 0 0 0 0 11 15 14 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 16 0 0 0 0 0 0 0 0 16 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 16 0 0 0 0 0 14 10 0 16
,
 8 11 0 0 0 0 0 0 8 9 0 0 0 0 0 0 0 14 8 3 0 0 0 0 8 0 1 9 0 0 0 0 0 0 0 0 13 2 6 3 0 0 0 0 14 10 8 15 0 0 0 0 16 7 0 0 0 0 0 0 4 16 14 11
,
 9 10 0 0 0 0 0 0 9 8 0 0 0 0 0 0 9 5 8 5 0 0 0 0 8 0 1 9 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 14 10 1 15 0 0 0 0 16 0 0 0 0 0 0 0 10 8 12 7

`G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,3,15,0,0,0,0,16,0,7,5,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1],[9,9,0,9,0,0,0,0,0,0,14,8,0,0,0,0,16,16,8,0,0,0,0,0,6,8,3,0,0,0,0,0,0,0,0,0,14,3,9,11,0,0,0,0,14,14,4,15,0,0,0,0,0,0,11,14,0,0,0,0,0,0,6,0],[1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,14,0,0,0,0,0,1,0,10,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[8,8,0,8,0,0,0,0,11,9,14,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,3,9,0,0,0,0,0,0,0,0,13,14,16,4,0,0,0,0,2,10,7,16,0,0,0,0,6,8,0,14,0,0,0,0,3,15,0,11],[9,9,9,8,0,0,0,0,10,8,5,0,0,0,0,0,0,0,8,1,0,0,0,0,0,0,5,9,0,0,0,0,0,0,0,0,0,14,16,10,0,0,0,0,0,10,0,8,0,0,0,0,1,1,0,12,0,0,0,0,0,15,0,7] >;`

C4.172+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{17}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.17ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1934);`
`// by ID`

`G=gap.SmallGroup(128,1934);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,219,352,675,1018,4037,1027,124]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^4=c^2=1,b^4=e^2=a^2,d^2=a^-1*b^2,d*b*d^-1=a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a*b^3,b*e=e*b,d*c*d^-1=e*c*e^-1=a^2*c,e*d*e^-1=a^-1*b^2*d>;`
`// generators/relations`

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