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## G = C4○D4⋊D4order 128 = 27

### 1st semidirect product of C4○D4 and D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4○D4⋊D4
G = < a,b,c,d,e | a4=c2=d4=e2=1, b2=a2, ab=ba, ac=ca, ad=da, eae=a-1, cbc=dbd-1=ebe=a2b, dcd-1=ece=bc, ede=d-1 >

Subgroups: 908 in 392 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×12], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×42], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×19], D4 [×6], D4 [×43], Q8 [×2], Q8 [×3], C23, C23 [×2], C23 [×27], C42, C22⋊C4 [×5], C4⋊C4 [×2], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×12], SD16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×10], C2×D4 [×49], C2×Q8 [×2], C4○D4 [×4], C4○D4 [×22], C24 [×4], C22⋊C8 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×4], C4.4D4, C41D4, C22×C8, C2×M4(2), C2×D8 [×6], C2×D8 [×4], C2×SD16 [×2], C8⋊C22 [×4], C22×D4 [×2], C22×D4 [×2], C22×D4 [×3], C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4, 2+ 1+4 [×8], (C22×C8)⋊C2, C23.24D4, C23.37D4, C22⋊D8 [×4], D4⋊D4 [×4], C22.29C24, C22×D8, C2×C8⋊C22, C2×2+ 1+4, C4○D4⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, D4○D8 [×2], C4○D4⋊D4

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4○D8 kernel C4○D4⋊D4 (C22×C8)⋊C2 C23.24D4 C23.37D4 C22⋊D8 D4⋊D4 C22.29C24 C22×D8 C2×C8⋊C22 C2×2+ 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 1 1 4 4 1 1 1 1 7 1 4 4

Matrix representation of C4○D4⋊D4 in GL6(𝔽17)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 1 0 0 0 0 16 0 0 0
,
 1 0 0 0 0 0 0 1 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 16 0 0 0 0 16 0
,
 0 15 0 0 0 0 9 0 0 0 0 0 0 0 3 3 0 0 0 0 3 14 0 0 0 0 0 0 14 14 0 0 0 0 14 3
,
 0 2 0 0 0 0 9 0 0 0 0 0 0 0 0 0 14 3 0 0 0 0 3 3 0 0 14 3 0 0 0 0 3 3 0 0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,3,3,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,14,3],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,3,3,0,0,14,3,0,0,0,0,3,3,0,0] >;

C4○D4⋊D4 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes D_4
% in TeX

G:=Group("C4oD4:D4");
// GroupNames label

G:=SmallGroup(128,1740);
// by ID

G=gap.SmallGroup(128,1740);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,521,2804,1411,172]);
// Polycyclic

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

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