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## G = D4×C2×C8order 128 = 27

### Direct product of C2×C8 and D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4×C2×C8
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C23×C8 — D4×C2×C8
 Lower central C1 — C2 — D4×C2×C8
 Upper central C1 — C22×C8 — D4×C2×C8
 Jennings C1 — C2 — C2 — C2×C4 — D4×C2×C8

Generators and relations for D4×C2×C8
G = < a,b,c,d | a2=b8=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 420 in 308 conjugacy classes, 196 normal (22 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×14], C22 [×24], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×16], C2×C4 [×22], D4 [×16], C23, C23 [×12], C23 [×8], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×22], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×12], C24 [×2], C4×C8 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×8], C22×C8 [×2], C22×C8 [×10], C22×C8 [×8], C23×C4 [×2], C22×D4, C2×C4×C8, C2×C22⋊C8 [×2], C2×C4⋊C8, C8×D4 [×8], C2×C4×D4, C23×C8 [×2], D4×C2×C8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], D4 [×4], C23 [×15], C2×C8 [×28], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C22×C8 [×14], C8○D4 [×2], C23×C4, C22×D4, C2×C4○D4, C8×D4 [×4], C2×C4×D4, C23×C8, C2×C8○D4, D4×C2×C8

Smallest permutation representation of D4×C2×C8
On 64 points
Generators in S64
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)
(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 31 41 54)(2 32 42 55)(3 25 43 56)(4 26 44 49)(5 27 45 50)(6 28 46 51)(7 29 47 52)(8 30 48 53)(9 33 22 62)(10 34 23 63)(11 35 24 64)(12 36 17 57)(13 37 18 58)(14 38 19 59)(15 39 20 60)(16 40 21 61)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 44)(10 45)(11 46)(12 47)(13 48)(14 41)(15 42)(16 43)(25 40)(26 33)(27 34)(28 35)(29 36)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 57)(53 58)(54 59)(55 60)(56 61)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4X 8A ··· 8P 8Q ··· 8AN order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 8 ··· 8 8 ··· 8 size 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 1 1 1 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 D4 C4○D4 C8○D4 kernel D4×C2×C8 C2×C4×C8 C2×C22⋊C8 C2×C4⋊C8 C8×D4 C2×C4×D4 C23×C8 C2×C22⋊C4 C2×C4⋊C4 C4×D4 C22×D4 C2×D4 C2×C8 C2×C4 C22 # reps 1 1 2 1 8 1 2 4 2 8 2 32 4 4 8

Matrix representation of D4×C2×C8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 15 0 0 0 0 16 0 0 0 0 15 0 0 0 0 15
,
 16 0 0 0 0 16 0 0 0 0 16 1 0 0 15 1
,
 1 0 0 0 0 1 0 0 0 0 16 1 0 0 0 1
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[15,0,0,0,0,16,0,0,0,0,15,0,0,0,0,15],[16,0,0,0,0,16,0,0,0,0,16,15,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,1,1] >;

D4×C2×C8 in GAP, Magma, Sage, TeX

D_4\times C_2\times C_8
% in TeX

G:=Group("D4xC2xC8");
// GroupNames label

G:=SmallGroup(128,1658);
// by ID

G=gap.SmallGroup(128,1658);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,124]);
// Polycyclic

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

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