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## G = C2×D4.4D4order 128 = 27

### Direct product of C2 and D4.4D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×D4.4D4
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C2×C8○D4 — C2×D4.4D4
 Lower central C1 — C2 — C2×C4 — C2×D4.4D4
 Upper central C1 — C22 — C22×C4 — C2×D4.4D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×D4.4D4

Generators and relations for C2×D4.4D4
G = < a,b,c,d,e | a2=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ece=bc, ede=b2d3 >

Subgroups: 540 in 244 conjugacy classes, 100 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C2×C8, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C24, C4.D4, C8.C4, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C2×M4(2), C8○D4, C8○D4, C2×D8, C2×D8, C2×SD16, C8⋊C22, C8⋊C22, C22×D4, C2×C4○D4, C2×C4.D4, C2×C8.C4, D4.4D4, C2×C8○D4, C22×D8, C2×C8⋊C22, C2×D4.4D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4⋊D4, C22×D4, C2×C4○D4, D4.4D4, C2×C4⋊D4, C2×D4.4D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 C4○D4 C4○D4 D4.4D4 kernel C2×D4.4D4 C2×C4.D4 C2×C8.C4 D4.4D4 C2×C8○D4 C22×D8 C2×C8⋊C22 C2×C8 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C2 # reps 1 2 1 8 1 1 2 4 1 1 2 2 2 4

Matrix representation of C2×D4.4D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 1 1 1 2 0 0 0 16 16 16
,
 1 15 0 0 0 0 0 16 0 0 0 0 0 0 1 1 1 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 16 16 16
,
 13 8 0 0 0 0 0 4 0 0 0 0 0 0 14 14 0 0 0 0 3 14 0 0 0 0 3 3 0 6 0 0 14 0 14 11
,
 16 0 0 0 0 0 16 1 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 0 16 16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1,0,0,0,1,0,1,16,0,0,0,0,1,16,0,0,0,0,2,16],[1,0,0,0,0,0,15,16,0,0,0,0,0,0,1,0,0,0,0,0,1,0,1,16,0,0,1,1,0,16,0,0,2,0,0,16],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,14,3,3,14,0,0,14,14,3,0,0,0,0,0,0,14,0,0,0,0,6,11],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,16,0,0,0,0,0,16] >;

C2×D4.4D4 in GAP, Magma, Sage, TeX

C_2\times D_4._4D_4
% in TeX

G:=Group("C2xD4.4D4");
// GroupNames label

G:=SmallGroup(128,1797);
// by ID

G=gap.SmallGroup(128,1797);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,2804,172,4037,1027,124]);
// Polycyclic

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

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