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## G = C2×D4.9D4order 128 = 27

### Direct product of C2 and D4.9D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.9D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C2×2+ 1+4 — C2×D4.9D4
 Lower central C1 — C2 — C2×C4 — C2×D4.9D4
 Upper central C1 — C22 — C22×C4 — C2×D4.9D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.9D4

Generators and relations for C2×D4.9D4
G = < a,b,c,d,e | a2=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=b-1c, ece-1=bc, ede-1=b2d-1 >

Subgroups: 804 in 380 conjugacy classes, 108 normal (16 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×4], C4 [×8], C22 [×3], C22 [×32], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×26], D4 [×4], D4 [×34], Q8 [×4], Q8 [×8], C23, C23 [×4], C23 [×22], C42 [×2], C42, C22⋊C4 [×8], C2×C8 [×2], M4(2) [×4], M4(2) [×2], SD16 [×8], Q16 [×8], C22×C4, C22×C4 [×6], C2×D4 [×6], C2×D4 [×42], C2×Q8 [×4], C2×Q8 [×5], C4○D4 [×8], C4○D4 [×20], C24 [×2], C24 [×2], C4.D4 [×4], C4≀C2 [×8], C2×C42, C2×C22⋊C4 [×2], C4.4D4 [×4], C4.4D4 [×2], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×8], C8.C22 [×4], C22×D4, C22×D4 [×4], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], 2+ 1+4 [×4], 2+ 1+4 [×6], C2×C4.D4, C2×C4≀C2 [×2], D4.9D4 [×8], C2×C4.4D4, C2×C8.C22 [×2], C2×2+ 1+4, C2×D4.9D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], D4.9D4 [×2], C2×C22≀C2, C2×D4.9D4

Smallest permutation representation of C2×D4.9D4
On 32 points
Generators in S32
(1 14)(2 15)(3 16)(4 13)(5 28)(6 25)(7 26)(8 27)(9 18)(10 19)(11 20)(12 17)(21 31)(22 32)(23 29)(24 30)
(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 4)(2 3)(6 8)(9 10)(11 12)(13 14)(15 16)(17 20)(18 19)(22 24)(25 27)(30 32)
(1 5 9 32)(2 6 10 29)(3 7 11 30)(4 8 12 31)(13 27 17 21)(14 28 18 22)(15 25 19 23)(16 26 20 24)
(1 30 3 32)(2 29 4 31)(5 9 7 11)(6 12 8 10)(13 21 15 23)(14 24 16 22)(17 27 19 25)(18 26 20 28)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F ··· 2M 4A 4B 4C 4D 4E ··· 4L 4M 4N 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 4 ··· 4 2 2 2 2 4 ··· 4 8 8 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D4.9D4 kernel C2×D4.9D4 C2×C4.D4 C2×C4≀C2 D4.9D4 C2×C4.4D4 C2×C8.C22 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C24 C2 # reps 1 1 2 8 1 2 1 4 2 4 2 4

Matrix representation of C2×D4.9D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 15 0 0 0 0 0 1 1 0 0 1 0 1 0 0 0 16 16 16 0
,
 16 0 0 0 0 0 9 1 0 0 0 0 0 0 16 0 15 0 0 0 1 0 1 16 0 0 0 0 1 0 0 0 16 16 16 0
,
 1 4 0 0 0 0 0 16 0 0 0 0 0 0 13 4 13 4 0 0 0 13 0 0 0 0 4 0 0 13 0 0 13 0 13 0
,
 16 13 0 0 0 0 0 1 0 0 0 0 0 0 13 4 13 4 0 0 0 13 0 0 0 0 0 13 4 0 0 0 0 4 0 4

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

C2×D4.9D4 in GAP, Magma, Sage, TeX

C_2\times D_4._9D_4
% in TeX

G:=Group("C2xD4.9D4");
// GroupNames label

G:=SmallGroup(128,1747);
// by ID

G=gap.SmallGroup(128,1747);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2804,1411,718,172,2028]);
// Polycyclic

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

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