Copied to
clipboard

## G = C4.182+ 1+4order 128 = 27

### 18th 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 — C4.182+ 1+4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C22.31C24 — C4.182+ 1+4
 Lower central C1 — C2 — C2×C4 — C4.182+ 1+4
 Upper central C1 — C22 — C2×C4○D4 — C4.182+ 1+4
 Jennings C1 — C2 — C2 — C2×C4 — C4.182+ 1+4

Generators and relations for C4.182+ 1+4
G = < a,b,c,d,e | a4=1, b4=c2=e2=a2, d2=ab2, dbd-1=ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ab3, be=eb, dcd-1=ece-1=a2c, ede-1=a-1b2d >

Subgroups: 444 in 201 conjugacy classes, 84 normal (26 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C2×Q16, C2×C4○D4, C2×C4○D4, (C22×C8)⋊C2, D4⋊D4, D4.7D4, C87D4, C8.18D4, C8⋊D4, C22.D8, C23.48D4, C22.31C24, C4.182+ 1+4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, C233D4, D4○D8, Q8○D8, C4.182+ 1+4

Character table of C4.182+ 1+4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 8A 8B 8C 8D 8E 8F size 1 1 1 1 4 4 4 8 8 2 2 4 4 4 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 0 0 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 -2 2 0 0 -2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 -2 2 2 0 0 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 2 2 2 2 -2 0 0 -2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 4 -4 4 -4 0 0 0 0 0 4 -4 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 0 0 -4 4 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 orthogonal lifted from D4○D8 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 orthogonal lifted from D4○D8 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 C4.182+ 1+4
On 64 points
Generators in S64
```(1 17 5 21)(2 18 6 22)(3 19 7 23)(4 20 8 24)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(33 41 37 45)(34 42 38 46)(35 43 39 47)(36 44 40 48)(49 62 53 58)(50 63 54 59)(51 64 55 60)(52 57 56 61)
(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 54 5 50)(2 62 6 58)(3 56 7 52)(4 64 8 60)(9 40 13 36)(10 43 14 47)(11 34 15 38)(12 45 16 41)(17 59 21 63)(18 53 22 49)(19 61 23 57)(20 55 24 51)(25 42 29 46)(26 33 30 37)(27 44 31 48)(28 35 32 39)
(1 43 19 33)(2 40 20 42)(3 41 21 39)(4 38 22 48)(5 47 23 37)(6 36 24 46)(7 45 17 35)(8 34 18 44)(9 55 25 62)(10 61 26 54)(11 53 27 60)(12 59 28 52)(13 51 29 58)(14 57 30 50)(15 49 31 64)(16 63 32 56)
(1 37 5 33)(2 38 6 34)(3 39 7 35)(4 40 8 36)(9 60 13 64)(10 61 14 57)(11 62 15 58)(12 63 16 59)(17 45 21 41)(18 46 22 42)(19 47 23 43)(20 48 24 44)(25 53 29 49)(26 54 30 50)(27 55 31 51)(28 56 32 52)```

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

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

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

Matrix representation of C4.182+ 1+4 in GL8(𝔽17)

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

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

C4.182+ 1+4 in GAP, Magma, Sage, TeX

`C_4._{18}2_+^{1+4}`
`% in TeX`

`G:=Group("C4.18ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(128,1935);`
`// by ID`

`G=gap.SmallGroup(128,1935);`
`# by ID`

`G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,1018,248,4037,1027,124]);`
`// Polycyclic`

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

Export

׿
×
𝔽