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## G = C4×C4.D4order 128 = 27

### Direct product of C4 and C4.D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C4.D4
G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c3 >

Subgroups: 356 in 180 conjugacy classes, 78 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×5], C22 [×3], C22 [×18], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×15], D4 [×8], C23, C23 [×4], C23 [×8], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], M4(2) [×8], M4(2) [×4], C22×C4 [×3], C22×C4 [×2], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C4.D4 [×8], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C2×M4(2) [×4], C23×C4 [×2], C22×D4, C22.C42 [×2], C4×M4(2) [×2], C2×C4.D4 [×2], C2×C4×D4, C4×C4.D4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C4.D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4, C2×C4.D4, M4(2).8C22, C4×C4.D4

Smallest permutation representation of C4×C4.D4
On 32 points
Generators in S32
(1 12 31 17)(2 13 32 18)(3 14 25 19)(4 15 26 20)(5 16 27 21)(6 9 28 22)(7 10 29 23)(8 11 30 24)
(1 29 5 25)(2 26 6 30)(3 31 7 27)(4 28 8 32)(9 24 13 20)(10 21 14 17)(11 18 15 22)(12 23 16 19)
(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 32 29 4 5 28 25 8)(2 7 26 27 6 3 30 31)(9 14 24 17 13 10 20 21)(11 12 18 23 15 16 22 19)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E ··· 4N 4O 4P 4Q 4R 8A ··· 8P order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 4 4 4 4 1 1 1 1 2 ··· 2 4 4 4 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 D4 C4○D4 C4.D4 M4(2).8C22 kernel C4×C4.D4 C22.C42 C4×M4(2) C2×C4.D4 C2×C4×D4 C4.D4 C2×C22⋊C4 C23×C4 C42 C2×C4 C4 C2 # reps 1 2 2 2 1 16 4 4 4 4 2 2

Matrix representation of C4×C4.D4 in GL6(𝔽17)

 4 0 0 0 0 0 0 4 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 0 16 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 0 4 0 0 0 0 13 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 1 0 0 0 0 0 0 16 0 0
,
 0 4 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 16 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0

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

C4×C4.D4 in GAP, Magma, Sage, TeX

C_4\times C_4.D_4
% in TeX

G:=Group("C4xC4.D4");
// GroupNames label

G:=SmallGroup(128,487);
// by ID

G=gap.SmallGroup(128,487);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,100,1018,4037,124]);
// Polycyclic

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

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