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## G = C42.7C23order 128 = 27

### 7th 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 — C42 — C42.7C23
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C42.C2 — C22.56C24 — C42.7C23
 Lower central C1 — C22 — C42 — C42.7C23
 Upper central C1 — C22 — C42 — C42.7C23
 Jennings C1 — C22 — C22 — C42 — C42.7C23

Generators and relations for C42.7C23
G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=a2b2, ab=ba, cac-1=dad=a-1, eae-1=a-1b2, cbc-1=ebe-1=b-1, dbd=a2b-1, dcd=ac, ece-1=bc, de=ed >

Subgroups: 264 in 104 conjugacy classes, 30 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×8], C22, C22 [×6], C8 [×5], C2×C4 [×3], C2×C4 [×7], D4 [×4], Q8 [×4], C23 [×2], C42, C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×3], SD16 [×2], Q16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C2×Q8, C8⋊C4 [×3], D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C42.C2, C4⋊Q8, C2×SD16, C2×Q16, C42.C22 [×2], C42.2C22, C42.28C22, C8.2D4, C8⋊Q8, C22.56C24, C42.7C23
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], C22≀C2, D44D4, D4.9D4, D4.10D4, C42.7C23

Character table of C42.7C23

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E 8F size 1 1 1 1 8 8 4 4 4 8 8 8 8 16 8 8 8 8 8 8 ρ1 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 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 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 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 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 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 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 linear of order 2 ρ9 2 2 2 2 0 0 -2 -2 2 0 2 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 2 2 0 -2 2 -2 0 0 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 2 0 2 2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 2 0 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 0 0 -2 -2 2 0 -2 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ14 2 2 2 2 -2 0 -2 2 -2 0 0 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 orthogonal lifted from D4⋊4D4 ρ16 4 -4 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 orthogonal lifted from D4⋊4D4 ρ17 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ18 4 4 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 symplectic lifted from D4.10D4, Schur index 2 ρ19 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 2i -2i 0 0 0 0 complex lifted from D4.9D4 ρ20 4 -4 4 -4 0 0 0 0 0 0 0 0 0 0 -2i 2i 0 0 0 0 complex lifted from D4.9D4

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

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

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

Matrix representation of C42.7C23 in GL8(𝔽17)

 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 1 16 0 0 0 0 0 1 0 0 16 0 0 0 0 2 0 0 16 0 0 0 0 0 2 16 0
,
 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 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 15 0 0 1 0 0 0 0 0 2 16 0
,
 14 4 3 4 0 0 0 0 4 14 4 3 0 0 0 0 3 4 3 13 0 0 0 0 4 3 13 3 0 0 0 0 0 0 0 0 13 4 15 2 0 0 0 0 0 0 2 2 0 0 0 0 0 0 4 0 0 0 0 0 0 8 13 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 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 2 16 0 0 0 0 0 2 0 0 16
,
 1 0 0 7 0 0 0 0 0 1 7 0 0 0 0 0 0 7 16 0 0 0 0 0 7 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 15 1 0 0 0 0 0 2 0 0 16

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

C42.7C23 in GAP, Magma, Sage, TeX

`C_4^2._7C_2^3`
`% in TeX`

`G:=Group("C4^2.7C2^3");`
`// GroupNames label`

`G:=SmallGroup(128,393);`
`// by ID`

`G=gap.SmallGroup(128,393);`
`# by ID`

`G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,352,1123,570,521,136,3924,1411,998,242]);`
`// Polycyclic`

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

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