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## G = C4×C4○D4order 64 = 26

### Direct product of C4 and C4○D4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×C4○D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C4×C4○D4
 Lower central C1 — C2 — C4×C4○D4
 Upper central C1 — C42 — C4×C4○D4
 Jennings C1 — C22 — C4×C4○D4

Generators and relations for C4×C4○D4
G = < a,b,c,d | a4=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 185 in 155 conjugacy classes, 125 normal (8 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×C4○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C23×C4, C2×C4○D4, C4×C4○D4

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

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 4M ··· 4AD order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 1 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4○D4 kernel C4×C4○D4 C2×C42 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C4 # reps 1 3 3 6 2 1 16 8

Matrix representation of C4×C4○D4 in GL3(𝔽5) generated by

 3 0 0 0 2 0 0 0 2
,
 4 0 0 0 2 0 0 0 2
,
 4 0 0 0 2 0 0 4 3
,
 1 0 0 0 4 1 0 0 1
G:=sub<GL(3,GF(5))| [3,0,0,0,2,0,0,0,2],[4,0,0,0,2,0,0,0,2],[4,0,0,0,2,4,0,0,3],[1,0,0,0,4,0,0,1,1] >;

C4×C4○D4 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_4
% in TeX

G:=Group("C4xC4oD4");
// GroupNames label

G:=SmallGroup(64,198);
// by ID

G=gap.SmallGroup(64,198);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,158,69]);
// Polycyclic

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

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