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## G = C4.(C4×D4)  order 128 = 27

### 5th non-split extension by C4 of C4×D4 acting via C4×D4/C42=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C4.(C4×D4)
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C4○D4 — Q8○M4(2) — C4.(C4×D4)
 Lower central C1 — C2 — C2×C4 — C4.(C4×D4)
 Upper central C1 — C2 — C22×C4 — C4.(C4×D4)
 Jennings C1 — C2 — C2 — C22×C4 — C4.(C4×D4)

Generators and relations for C4.(C4×D4)
G = < a,b,c,d | a4=c4=1, b4=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a-1b, bd=db, dcd-1=c-1 >

Subgroups: 300 in 156 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×12], D4 [×6], Q8 [×8], C23, C23 [×2], C42 [×2], C42 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C2×C8 [×4], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C23⋊C4 [×2], Q8⋊C4 [×4], C42⋊C2, C42⋊C2 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C22×Q8, C2×C4○D4, C4.9C42 [×2], C23.C23, C23.38D4 [×2], C23.38C23, Q8○M4(2), C4.(C4×D4)
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], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C4.(C4×D4)

Character table of C4.(C4×D4)

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 8A 8B 8C 8D 8E 8F 8G 8H size 1 1 2 2 2 4 4 2 2 2 2 4 4 8 8 8 8 8 8 8 8 4 4 4 4 4 4 4 4 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 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 -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 -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 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 -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 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 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 -1 -1 -1 linear of order 2 ρ9 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 i i 1 -i -i -1 -1 i -i i -i i -i i -i linear of order 4 ρ10 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 i -i -1 -i i 1 -1 i -i i i -i i -i -i linear of order 4 ρ11 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 -i i 1 i -i -1 1 i -i i i -i i -i -i linear of order 4 ρ12 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 -i -i -1 i i 1 1 i -i i -i i -i i -i linear of order 4 ρ13 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 -1 i -i 1 -i i -1 1 -i i -i -i i -i i i linear of order 4 ρ14 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 -1 i i -1 -i -i 1 1 -i i -i i -i i -i i linear of order 4 ρ15 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -i -i 1 i i -1 -1 -i i -i i -i i -i i linear of order 4 ρ16 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -i i -1 i -i 1 -1 -i i -i -i i -i i i linear of order 4 ρ17 2 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 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 -2 orthogonal lifted from D4 ρ19 2 2 2 -2 -2 0 0 2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 2 orthogonal lifted from D4 ρ20 2 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 0 0 orthogonal lifted from D4 ρ21 2 2 -2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 2 0 orthogonal lifted from D4 ρ22 2 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 0 0 orthogonal lifted from D4 ρ23 2 2 -2 2 -2 0 0 2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 orthogonal lifted from D4 ρ24 2 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 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2i -2i -2i 2i 0 complex lifted from C4○D4 ρ26 2 2 -2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 2i -2i -2i 0 0 0 0 2i complex lifted from C4○D4 ρ27 2 2 -2 2 -2 0 0 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 -2i 2i 2i 0 0 0 0 -2i complex lifted from C4○D4 ρ28 2 2 2 -2 -2 0 0 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 -2i 2i 2i -2i 0 complex lifted from C4○D4 ρ29 8 -8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C4.(C4×D4)
On 32 points
Generators in S32
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 27 29 31)(26 28 30 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 18 27 15)(2 17 32 10)(3 24 29 13)(4 23 26 16)(5 22 31 11)(6 21 28 14)(7 20 25 9)(8 19 30 12)
(1 14 5 10)(2 15 6 11)(3 16 7 12)(4 9 8 13)(17 27 21 31)(18 28 22 32)(19 29 23 25)(20 30 24 26)

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

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

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

Matrix representation of C4.(C4×D4) 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 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
,
 16 11 7 11 0 0 0 0 16 1 7 10 0 0 0 0 10 6 16 11 0 0 0 0 10 7 16 1 0 0 0 0 7 14 0 3 13 3 4 3 0 14 1 3 14 4 14 13 0 3 10 3 13 14 13 3 1 3 0 3 3 4 14 4
,
 1 0 7 0 8 6 8 11 0 0 0 0 1 16 7 10 10 0 1 0 8 11 9 11 0 0 0 0 7 10 16 1 7 13 0 13 13 3 4 3 7 14 0 3 13 3 4 3 0 13 10 4 13 14 13 3 0 3 10 3 13 14 13 3
,
 0 0 16 0 15 0 0 0 0 0 0 0 16 1 0 0 16 0 0 0 0 0 15 0 0 0 0 0 0 0 16 1 1 0 0 0 0 0 1 0 1 16 0 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 1 16 1 0 0 0

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

C4.(C4×D4) in GAP, Magma, Sage, TeX

C_4.(C_4\times D_4)
% in TeX

G:=Group("C4.(C4xD4)");
// GroupNames label

G:=SmallGroup(128,641);
// by ID

G=gap.SmallGroup(128,641);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,2019,521,248,2804,1411,1027]);
// Polycyclic

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

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