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G = D4×C4○D4order 128 = 27

Direct product of D4 and C4○D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — D4×C4○D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — D4×C4○D4
 Lower central C1 — C22 — D4×C4○D4
 Upper central C1 — C2×C4 — D4×C4○D4
 Jennings C1 — C22 — D4×C4○D4

Generators and relations for D4×C4○D4
G = < a,b,c,d,e | a4=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1300 in 824 conjugacy classes, 438 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4, C4×D4, C4×Q8, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C41D4, C4⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C22.19C24, C22.26C24, D42, D45D4, D46D4, Q85D4, D4×Q8, Q86D4, C22×C4○D4, D4×C4○D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C25, D4×C23, C22×C4○D4, C2.C25, D4×C4○D4

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

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2M 2N ··· 2S 4A 4B 4C 4D 4E ··· 4R 4S ··· 4AD order 1 2 2 2 2 ··· 2 2 ··· 2 4 4 4 4 4 ··· 4 4 ··· 4 size 1 1 1 1 2 ··· 2 4 ··· 4 1 1 1 1 2 ··· 2 4 ··· 4

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 C4○D4 C2.C25 kernel D4×C4○D4 C2×C4×D4 C4×C4○D4 C22.19C24 C22.26C24 D42 D4⋊5D4 D4⋊6D4 Q8⋊5D4 D4×Q8 Q8⋊6D4 C22×C4○D4 C4○D4 D4 C2 # reps 1 3 1 6 3 3 6 3 2 1 1 2 8 8 2

Matrix representation of D4×C4○D4 in GL4(𝔽5) generated by

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

D4×C4○D4 in GAP, Magma, Sage, TeX

D_4\times C_4\circ D_4
% in TeX

G:=Group("D4xC4oD4");
// GroupNames label

G:=SmallGroup(128,2200);
// by ID

G=gap.SmallGroup(128,2200);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,570,102]);
// Polycyclic

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

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