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## G = C2×C4×D8order 128 = 27

### Direct product of C2×C4 and D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C4 — C2×C4×D8
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×D4 — C2×C4×D8
 Lower central C1 — C2 — C4 — C2×C4×D8
 Upper central C1 — C22×C4 — C2×C42 — C2×C4×D8
 Jennings C1 — C2 — C2 — C2×C4 — C2×C4×D8

Generators and relations for C2×C4×D8
G = < a,b,c,d | a2=b4=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 604 in 308 conjugacy classes, 156 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C4 [×2], C4 [×6], C4 [×6], C22, C22 [×6], C22 [×32], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×12], C2×C4 [×26], D4 [×8], D4 [×12], C23, C23 [×20], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×2], D8 [×16], C22×C4 [×3], C22×C4 [×18], C2×D4 [×12], C2×D4 [×6], C24 [×2], C4×C8 [×4], D4⋊C4 [×8], C2.D8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C4×D4 [×8], C4×D4 [×4], C22×C8 [×2], C2×D8 [×12], C23×C4 [×2], C22×D4 [×2], C2×C4×C8, C2×D4⋊C4 [×2], C2×C2.D8, C4×D8 [×8], C2×C4×D4 [×2], C22×D8, C2×C4×D8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], D8 [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×D8 [×6], C4○D8 [×2], C23×C4, C22×D4, C2×C4○D4, C4×D8 [×4], C2×C4×D4, C22×D8, C2×C4○D8, C2×C4×D8

Smallest permutation representation of C2×C4×D8
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 55)(10 56)(11 49)(12 50)(13 51)(14 52)(15 53)(16 54)(17 59)(18 60)(19 61)(20 62)(21 63)(22 64)(23 57)(24 58)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(1 34 53 19)(2 35 54 20)(3 36 55 21)(4 37 56 22)(5 38 49 23)(6 39 50 24)(7 40 51 17)(8 33 52 18)(9 63 31 44)(10 64 32 45)(11 57 25 46)(12 58 26 47)(13 59 27 48)(14 60 28 41)(15 61 29 42)(16 62 30 43)
(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 25)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 56)(17 59)(18 58)(19 57)(20 64)(21 63)(22 62)(23 61)(24 60)(33 47)(34 46)(35 45)(36 44)(37 43)(38 42)(39 41)(40 48)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 4A ··· 4H 4I ··· 4P 4Q ··· 4X 8A ··· 8P order 1 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 4 ··· 4 1 ··· 1 2 ··· 2 4 ··· 4 2 ··· 2

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 D4 D4 D8 C4○D4 C4○D8 kernel C2×C4×D8 C2×C4×C8 C2×D4⋊C4 C2×C2.D8 C4×D8 C2×C4×D4 C22×D8 C2×D8 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 1 2 1 8 2 1 16 2 2 8 4 8

Matrix representation of C2×C4×D8 in GL4(𝔽17) generated by

 16 0 0 0 0 16 0 0 0 0 16 0 0 0 0 16
,
 16 0 0 0 0 4 0 0 0 0 4 0 0 0 0 4
,
 1 0 0 0 0 1 0 0 0 0 11 14 0 0 6 0
,
 1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 16
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,0,0,11,6,0,0,14,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,1,16] >;

C2×C4×D8 in GAP, Magma, Sage, TeX

C_2\times C_4\times D_8
% in TeX

G:=Group("C2xC4xD8");
// GroupNames label

G:=SmallGroup(128,1668);
// by ID

G=gap.SmallGroup(128,1668);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2804,1411,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^8=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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