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## G = C4×C8○D4order 128 = 27

### Direct product of C4 and C8○D4

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C4×C8○D4
G = < a,b,c,d | a4=b8=d2=1, c2=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c >

Subgroups: 316 in 286 conjugacy classes, 256 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4×C8, C4×C8, C8⋊C4, C2×C42, C42⋊C2, C4×D4, C4×Q8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, C2×C4×C8, C4×M4(2), C82M4(2), C4×C4○D4, C2×C8○D4, C4×C8○D4
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, C22×C4, C24, C2×C42, C8○D4, C23×C4, C22×C42, C2×C8○D4, C4×C8○D4

Smallest permutation representation of C4×C8○D4
On 64 points
Generators in S64
(1 33 63 47)(2 34 64 48)(3 35 57 41)(4 36 58 42)(5 37 59 43)(6 38 60 44)(7 39 61 45)(8 40 62 46)(9 18 32 49)(10 19 25 50)(11 20 26 51)(12 21 27 52)(13 22 28 53)(14 23 29 54)(15 24 30 55)(16 17 31 56)
(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 61 5 57)(2 62 6 58)(3 63 7 59)(4 64 8 60)(9 26 13 30)(10 27 14 31)(11 28 15 32)(12 29 16 25)(17 50 21 54)(18 51 22 55)(19 52 23 56)(20 53 24 49)(33 45 37 41)(34 46 38 42)(35 47 39 43)(36 48 40 44)
(1 20)(2 21)(3 22)(4 23)(5 24)(6 17)(7 18)(8 19)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 61)(50 62)(51 63)(52 64)(53 57)(54 58)(55 59)(56 60)

G:=sub<Sym(64)| (1,33,63,47)(2,34,64,48)(3,35,57,41)(4,36,58,42)(5,37,59,43)(6,38,60,44)(7,39,61,45)(8,40,62,46)(9,18,32,49)(10,19,25,50)(11,20,26,51)(12,21,27,52)(13,22,28,53)(14,23,29,54)(15,24,30,55)(16,17,31,56), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(17,50,21,54)(18,51,22,55)(19,52,23,56)(20,53,24,49)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60)>;

G:=Group( (1,33,63,47)(2,34,64,48)(3,35,57,41)(4,36,58,42)(5,37,59,43)(6,38,60,44)(7,39,61,45)(8,40,62,46)(9,18,32,49)(10,19,25,50)(11,20,26,51)(12,21,27,52)(13,22,28,53)(14,23,29,54)(15,24,30,55)(16,17,31,56), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,61,5,57)(2,62,6,58)(3,63,7,59)(4,64,8,60)(9,26,13,30)(10,27,14,31)(11,28,15,32)(12,29,16,25)(17,50,21,54)(18,51,22,55)(19,52,23,56)(20,53,24,49)(33,45,37,41)(34,46,38,42)(35,47,39,43)(36,48,40,44), (1,20)(2,21)(3,22)(4,23)(5,24)(6,17)(7,18)(8,19)(9,45)(10,46)(11,47)(12,48)(13,41)(14,42)(15,43)(16,44)(25,40)(26,33)(27,34)(28,35)(29,36)(30,37)(31,38)(32,39)(49,61)(50,62)(51,63)(52,64)(53,57)(54,58)(55,59)(56,60) );

G=PermutationGroup([[(1,33,63,47),(2,34,64,48),(3,35,57,41),(4,36,58,42),(5,37,59,43),(6,38,60,44),(7,39,61,45),(8,40,62,46),(9,18,32,49),(10,19,25,50),(11,20,26,51),(12,21,27,52),(13,22,28,53),(14,23,29,54),(15,24,30,55),(16,17,31,56)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,61,5,57),(2,62,6,58),(3,63,7,59),(4,64,8,60),(9,26,13,30),(10,27,14,31),(11,28,15,32),(12,29,16,25),(17,50,21,54),(18,51,22,55),(19,52,23,56),(20,53,24,49),(33,45,37,41),(34,46,38,42),(35,47,39,43),(36,48,40,44)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,17),(7,18),(8,19),(9,45),(10,46),(11,47),(12,48),(13,41),(14,42),(15,43),(16,44),(25,40),(26,33),(27,34),(28,35),(29,36),(30,37),(31,38),(32,39),(49,61),(50,62),(51,63),(52,64),(53,57),(54,58),(55,59),(56,60)]])

80 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 4M ··· 4AD 8A ··· 8P 8Q ··· 8AN order 1 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8○D4 kernel C4×C8○D4 C2×C4×C8 C4×M4(2) C8○2M4(2) C4×C4○D4 C2×C8○D4 C4×D4 C4×Q8 C8○D4 C4 # reps 1 3 3 6 1 2 12 4 32 16

Matrix representation of C4×C8○D4 in GL3(𝔽17) generated by

 13 0 0 0 1 0 0 0 1
,
 1 0 0 0 2 0 0 0 2
,
 1 0 0 0 13 0 0 0 4
,
 1 0 0 0 0 4 0 13 0
G:=sub<GL(3,GF(17))| [13,0,0,0,1,0,0,0,1],[1,0,0,0,2,0,0,0,2],[1,0,0,0,13,0,0,0,4],[1,0,0,0,0,13,0,4,0] >;

C4×C8○D4 in GAP, Magma, Sage, TeX

C_4\times C_8\circ D_4
% in TeX

G:=Group("C4xC8oD4");
// GroupNames label

G:=SmallGroup(128,1606);
// by ID

G=gap.SmallGroup(128,1606);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,232,521,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^8=d^2=1,c^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c>;
// generators/relations

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