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## G = C42.7D4order 128 = 27

### 7th 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 — C2×C4 — C42.7D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×Q8 — C23.32C23 — C42.7D4
 Lower central C1 — C2 — C2×C4 — C42.7D4
 Upper central C1 — C2 — C22×C4 — C42.7D4
 Jennings C1 — C2 — C2 — C22×C4 — C42.7D4

Generators and relations for C42.7D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 300 in 157 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×11], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×4], Q8 [×10], C23, C23, C42 [×6], C42 [×4], C22⋊C4 [×7], C4⋊C4 [×11], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×4], C4.10D4 [×2], C4≀C2 [×4], C42⋊C2, C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4, C4.9C42 [×2], C2×C4.10D4, C42⋊C22 [×2], C23.32C23, C23.38C23, C42.7D4
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, C42.7D4

Character table of C42.7D4

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 4Q 4R 4S 8A 8B 8C 8D size 1 1 2 2 2 8 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 8 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 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 i i -1 1 -1 -i i -i -i i -i 1 -1 1 1 i i -i -i linear of order 4 ρ10 1 1 -1 1 -1 1 1 1 -1 -1 -i -i -1 1 -1 i -i i i -i i 1 1 -1 -1 i i -i -i linear of order 4 ρ11 1 1 -1 1 -1 -1 1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i -1 1 1 -1 -i i i -i linear of order 4 ρ12 1 1 -1 1 -1 1 1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i -1 -1 -1 1 -i i i -i linear of order 4 ρ13 1 1 -1 1 -1 -1 1 1 -1 -1 i -i 1 -1 1 -i i -i i -i i -1 1 1 -1 i -i -i i linear of order 4 ρ14 1 1 -1 1 -1 1 1 1 -1 -1 -i i 1 -1 1 i -i i -i i -i -1 -1 -1 1 i -i -i i linear of order 4 ρ15 1 1 -1 1 -1 -1 1 1 -1 -1 -i -i -1 1 -1 i -i i i -i i 1 -1 1 1 -i -i i i linear of order 4 ρ16 1 1 -1 1 -1 1 1 1 -1 -1 i i -1 1 -1 -i i -i -i i -i 1 1 -1 -1 -i -i i i linear of order 4 ρ17 2 2 -2 -2 2 0 2 -2 -2 2 0 2 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 2 2 0 -2 -2 -2 -2 0 0 -2 -2 2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 2 2 2 2 0 -2 -2 -2 -2 0 0 2 2 -2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 2 -2 2 -2 0 -2 -2 2 2 0 0 2 -2 -2 0 0 0 0 0 0 2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ21 2 2 -2 2 -2 0 -2 -2 2 2 0 0 -2 2 2 0 0 0 0 0 0 -2 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 0 -2 2 2 -2 -2 0 0 0 0 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ23 2 2 -2 -2 2 0 2 -2 -2 2 0 -2 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 -2 2 0 -2 2 2 -2 2 0 0 0 0 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 2 -2 -2 0 -2 2 -2 2 -2i 0 0 0 0 2i 2i -2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ26 2 2 2 -2 -2 0 2 -2 2 -2 0 -2i 0 0 0 0 0 0 -2i 2i 2i 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ27 2 2 2 -2 -2 0 -2 2 -2 2 2i 0 0 0 0 -2i -2i 2i 0 0 0 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ28 2 2 2 -2 -2 0 2 -2 2 -2 0 2i 0 0 0 0 0 0 2i -2i -2i 0 0 0 0 0 0 0 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 C42.7D4
On 32 points
Generators in S32
(1 28 31 2)(3 26 25 8)(4 7 30 29)(5 32 27 6)(9 19 22 10)(11 17 24 16)(12 15 21 20)(13 23 18 14)
(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 31 29 27)(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 12 7 14 5 16 3 10)(2 13 4 11 6 9 8 15)(17 29 19 27 21 25 23 31)(18 26 24 28 22 30 20 32)

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

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

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

Matrix representation of C42.7D4 in GL8(𝔽17)

 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 4 13 4 13 1 16 1 15 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 1 0 0 0 0 4
,
 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 1 0 0 0 0 0 0 16 0 0 0 4 13 4 13 1 16 1 15 4 0 4 0 1 0 1 16
,
 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 4 13 4 13 1 16 1 15 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 13
,
 0 0 0 0 10 0 16 0 13 4 13 4 16 11 16 2 0 0 0 0 16 0 7 0 11 6 11 6 7 9 7 3 0 7 0 1 0 0 0 0 10 0 16 0 0 0 0 0 0 1 0 10 0 0 0 0 7 1 0 11 6 11 0 7

G:=sub<GL(8,GF(17))| [0,0,0,4,0,0,0,0,0,0,0,13,0,0,16,16,0,0,0,4,0,16,0,1,0,0,0,13,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,0,0,4],[0,16,0,0,0,0,4,4,1,0,0,0,0,0,13,0,0,0,0,16,0,0,4,4,0,0,1,0,0,0,13,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,4,0,1,0,0,0,0,0,13,0,0,16,0,1,0,0,4,0,0,0,1,0,0,0,13,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,0,0,13],[0,13,0,11,0,10,0,7,0,4,0,6,7,0,1,1,0,13,0,11,0,16,0,0,0,4,0,6,1,0,10,11,10,16,16,7,0,0,0,6,0,11,0,9,0,0,0,11,16,16,7,7,0,0,0,0,0,2,0,3,0,0,0,7] >;

C42.7D4 in GAP, Magma, Sage, TeX

C_4^2._7D_4
% in TeX

G:=Group("C4^2.7D4");
// GroupNames label

G:=SmallGroup(128,644);
// by ID

G=gap.SmallGroup(128,644);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,2804,1411,1027,124]);
// Polycyclic

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

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