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## G = C43⋊C2order 128 = 27

### 7th semidirect product of C43 and C2 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C43⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C42 — C43 — C43⋊C2
 Lower central C1 — C2 — C2×C4 — C43⋊C2
 Upper central C1 — C2×C4 — C2×C42 — C43⋊C2
 Jennings C1 — C2 — C2 — C22×C4 — C43⋊C2

Generators and relations for C43⋊C2
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, dad=a-1c-1, bc=cb, dbd=b-1, cd=dc >

Subgroups: 356 in 180 conjugacy classes, 60 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×6], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×22], D4 [×12], Q8 [×4], C23, C23 [×2], C42 [×2], C42 [×6], C42 [×10], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×8], C4≀C2 [×8], C4⋊C8 [×2], C2×C42, C2×C42 [×2], C2×C42 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×C4○D4 [×2], C43, C2×C4≀C2 [×4], C4⋊M4(2), C22.26C24, C43⋊C2
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], C4≀C2 [×4], C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C24.3C22, C2×C4≀C2 [×2], C43⋊C2

Smallest permutation representation of C43⋊C2
On 32 points
Generators in S32
(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)
(1 12 14 4)(2 11 13 3)(5 8 15 10)(6 7 16 9)(17 26 24 32)(18 27 21 29)(19 28 22 30)(20 25 23 31)
(1 5 13 16)(2 6 14 15)(3 9 12 8)(4 10 11 7)(17 21 19 23)(18 22 20 24)(25 32 27 30)(26 29 28 31)
(1 26)(2 30)(3 19)(4 24)(5 29)(6 25)(7 20)(8 21)(9 23)(10 18)(11 22)(12 17)(13 28)(14 32)(15 27)(16 31)

G:=sub<Sym(32)| (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), (1,12,14,4)(2,11,13,3)(5,8,15,10)(6,7,16,9)(17,26,24,32)(18,27,21,29)(19,28,22,30)(20,25,23,31), (1,5,13,16)(2,6,14,15)(3,9,12,8)(4,10,11,7)(17,21,19,23)(18,22,20,24)(25,32,27,30)(26,29,28,31), (1,26)(2,30)(3,19)(4,24)(5,29)(6,25)(7,20)(8,21)(9,23)(10,18)(11,22)(12,17)(13,28)(14,32)(15,27)(16,31)>;

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), (1,12,14,4)(2,11,13,3)(5,8,15,10)(6,7,16,9)(17,26,24,32)(18,27,21,29)(19,28,22,30)(20,25,23,31), (1,5,13,16)(2,6,14,15)(3,9,12,8)(4,10,11,7)(17,21,19,23)(18,22,20,24)(25,32,27,30)(26,29,28,31), (1,26)(2,30)(3,19)(4,24)(5,29)(6,25)(7,20)(8,21)(9,23)(10,18)(11,22)(12,17)(13,28)(14,32)(15,27)(16,31) );

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)], [(1,12,14,4),(2,11,13,3),(5,8,15,10),(6,7,16,9),(17,26,24,32),(18,27,21,29),(19,28,22,30),(20,25,23,31)], [(1,5,13,16),(2,6,14,15),(3,9,12,8),(4,10,11,7),(17,21,19,23),(18,22,20,24),(25,32,27,30),(26,29,28,31)], [(1,26),(2,30),(3,19),(4,24),(5,29),(6,25),(7,20),(8,21),(9,23),(10,18),(11,22),(12,17),(13,28),(14,32),(15,27),(16,31)])

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4AD 4AE 4AF 8A 8B 8C 8D order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 8 8 8 8 size 1 1 1 1 2 2 8 8 1 1 1 1 2 ··· 2 8 8 8 8 8 8

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 C4○D4 C4≀C2 kernel C43⋊C2 C43 C2×C4≀C2 C4⋊M4(2) C22.26C24 C4.4D4 C4⋊1D4 C4⋊Q8 C42 C22×C4 C2×C4 C4 # reps 1 1 4 1 1 4 2 2 6 2 4 16

Matrix representation of C43⋊C2 in GL4(𝔽17) generated by

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

C43⋊C2 in GAP, Magma, Sage, TeX

C_4^3\rtimes C_2
% in TeX

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

G:=SmallGroup(128,694);
// by ID

G=gap.SmallGroup(128,694);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,100,2019,248,124]);
// Polycyclic

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

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