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## G = C42.9D4order 128 = 27

### 9th non-split extension by C42 of D4 acting faithfully

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — C42.9D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×M4(2) — M4(2).8C22 — C42.9D4
 Lower central C1 — C2 — C22×C4 — C42.9D4
 Upper central C1 — C4 — C22×C4 — C42.9D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.9D4

Generators and relations for C42.9D4
G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b, dad=ab-1, cbc-1=dbd=b-1, dcd=b-1c3 >

Subgroups: 200 in 95 conjugacy classes, 36 normal (34 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×4], C22 [×3], C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×6], D4 [×4], Q8 [×2], C23, C23, C42 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×6], M4(2) [×6], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4.D4, C4.10D4, D4⋊C4, Q8⋊C4, C4≀C2 [×2], C4⋊C8 [×2], C42⋊C2 [×2], C22×C8, C2×M4(2) [×3], C2×C4○D4, C4.9C42, C4.C42, M4(2)⋊4C4, M4(2).8C22, C23.24D4, C42⋊C22, C42.6C22, C42.9D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, C2×D4 [×2], C4○D4 [×5], C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C23.11D4, C42.9D4

Character table of C42.9D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 8A 8B 8C 8D 8E 8F 8G 8H 8I 8J size 1 1 2 2 2 8 1 1 2 2 2 8 8 8 8 8 4 4 4 4 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 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 2 2 -2 2 -2 0 2 2 -2 2 -2 0 -2 0 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ10 2 2 2 -2 -2 2 -2 -2 -2 2 2 0 0 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 -2 2 -2 0 2 2 -2 2 -2 0 2 0 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ12 2 2 2 -2 -2 -2 -2 -2 -2 2 2 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2i 0 2i 0 0 complex lifted from C4○D4 ρ14 2 2 2 2 2 0 -2 -2 -2 -2 -2 0 0 0 0 0 0 0 0 0 0 2i 0 -2i 0 0 complex lifted from C4○D4 ρ15 2 2 -2 2 -2 0 -2 -2 2 -2 2 0 0 0 0 0 2i -2i -2i 2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ16 2 2 2 -2 -2 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 2i 0 0 0 -2i 0 complex lifted from C4○D4 ρ17 2 2 -2 2 -2 0 -2 -2 2 -2 2 0 0 0 0 0 -2i 2i 2i -2i 0 0 0 0 0 0 complex lifted from C4○D4 ρ18 2 2 -2 -2 2 0 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 2i 0 0 -2i complex lifted from C4○D4 ρ19 2 2 -2 -2 2 0 2 2 -2 -2 2 -2i 0 0 0 2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ20 2 2 2 -2 -2 0 2 2 2 -2 -2 0 0 0 0 0 0 0 0 0 -2i 0 0 0 2i 0 complex lifted from C4○D4 ρ21 2 2 -2 -2 2 0 -2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 -2i 0 0 2i complex lifted from C4○D4 ρ22 2 2 -2 -2 2 0 2 2 -2 -2 2 2i 0 0 0 -2i 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ83 2ζ85 2ζ8 2ζ87 0 0 0 0 0 0 complex faithful ρ24 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ8 2ζ87 2ζ83 2ζ85 0 0 0 0 0 0 complex faithful ρ25 4 -4 0 0 0 0 4i -4i 0 0 0 0 0 0 0 0 2ζ87 2ζ8 2ζ85 2ζ83 0 0 0 0 0 0 complex faithful ρ26 4 -4 0 0 0 0 -4i 4i 0 0 0 0 0 0 0 0 2ζ85 2ζ83 2ζ87 2ζ8 0 0 0 0 0 0 complex faithful

Smallest permutation representation of C42.9D4
On 32 points
Generators in S32
(1 8 32 27)(2 30 29 7)(3 6 26 25)(4 28 31 5)(9 10 22 19)(11 16 24 17)(12 20 21 15)(13 14 18 23)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 27 29 31)(26 32 30 28)
(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 13)(2 10)(3 11)(4 16)(5 9)(6 14)(7 15)(8 12)(17 27)(18 28)(19 25)(20 26)(21 31)(22 32)(23 29)(24 30)

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

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

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

Matrix representation of C42.9D4 in GL4(𝔽17) generated by

 0 0 15 6 0 0 0 9 9 7 0 0 0 2 0 0
,
 13 9 0 0 0 4 0 0 0 0 13 9 0 0 0 4
,
 0 0 2 10 0 0 15 15 8 10 0 0 9 9 0 0
,
 2 10 0 0 15 15 0 0 0 0 2 10 0 0 15 15
G:=sub<GL(4,GF(17))| [0,0,9,0,0,0,7,2,15,0,0,0,6,9,0,0],[13,0,0,0,9,4,0,0,0,0,13,0,0,0,9,4],[0,0,8,9,0,0,10,9,2,15,0,0,10,15,0,0],[2,15,0,0,10,15,0,0,0,0,2,15,0,0,10,15] >;

C42.9D4 in GAP, Magma, Sage, TeX

C_4^2._9D_4
% in TeX

G:=Group("C4^2.9D4");
// GroupNames label

G:=SmallGroup(128,812);
// by ID

G=gap.SmallGroup(128,812);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,422,387,58,1018,248,1411,4037,1027]);
// Polycyclic

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

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