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## G = C2×D4.3D4order 128 = 27

### Direct product of C2 and D4.3D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.3D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C2×D4.3D4
 Lower central C1 — C2 — C2×C4 — C2×D4.3D4
 Upper central C1 — C22 — C22×C4 — C2×D4.3D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.3D4

Generators and relations for C2×D4.3D4
G = < a,b,c,d,e | a2=b4=c2=1, d4=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d3 >

Subgroups: 460 in 234 conjugacy classes, 100 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×4], C22 [×3], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×9], D4 [×2], D4 [×11], Q8 [×2], Q8 [×7], C23, C23 [×7], C2×C8 [×2], C2×C8 [×4], C2×C8 [×7], M4(2) [×6], M4(2) [×7], D8 [×4], SD16 [×16], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C4○D4 [×2], C24, C4.D4 [×4], C4.10D4 [×4], C8.C4 [×4], C22×C8, C22×C8, C2×M4(2) [×3], C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×D8, C2×SD16 [×4], C2×SD16 [×6], C2×Q16, C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C22×D4, C22×Q8, C2×C4○D4, C2×C4.D4, C2×C4.10D4, C2×C8.C4, D4.3D4 [×8], C2×C8○D4, C22×SD16, C2×C8⋊C22, C2×C8.C22, C2×D4.3D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, D4.3D4 [×2], C2×C4⋊D4, C2×D4.3D4

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

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 ··· 8 8 8 8 8 size 1 1 1 1 2 2 4 4 8 8 2 2 2 2 4 4 8 8 2 2 2 2 4 ··· 4 8 8 8 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D4 D4.3D4 kernel C2×D4.3D4 C2×C4.D4 C2×C4.10D4 C2×C8.C4 D4.3D4 C2×C8○D4 C22×SD16 C2×C8⋊C22 C2×C8.C22 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 # reps 1 1 1 1 8 1 1 1 1 4 1 1 2 2 2 4

Matrix representation of C2×D4.3D4 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 1 15 0 0 0 0 1 16 0 0 0 0 0 0 16 2 0 0 0 0 16 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 15 0 0 0 0 1 16 0 0 16 2 0 0 0 0 16 1 0 0
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 10 0 0 0 0 12 10 0 0 0 0 0 0 0 10 0 0 0 0 12 10
,
 0 13 0 0 0 0 4 0 0 0 0 0 0 0 0 10 0 0 0 0 5 0 0 0 0 0 0 0 10 7 0 0 0 0 5 7

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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,0,0,0,0,15,16,0,0,0,0,0,0,16,16,0,0,0,0,2,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,12,0,0,0,0,10,10,0,0,0,0,0,0,0,12,0,0,0,0,10,10],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,5,0,0,0,0,10,0,0,0,0,0,0,0,10,5,0,0,0,0,7,7] >;

C2×D4.3D4 in GAP, Magma, Sage, TeX

C_2\times D_4._3D_4
% in TeX

G:=Group("C2xD4.3D4");
// GroupNames label

G:=SmallGroup(128,1796);
// by ID

G=gap.SmallGroup(128,1796);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,2804,172,4037,1027,124]);
// Polycyclic

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

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