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## G = C8×C4○D4order 128 = 27

### Direct product of C8 and C4○D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C8×C4○D4
 Chief series C1 — C2 — C4 — C2×C4 — C42 — C4×C8 — C2×C4×C8 — C8×C4○D4
 Lower central C1 — C2 — C8×C4○D4
 Upper central C1 — C4×C8 — C8×C4○D4
 Jennings C1 — C2 — C2 — C2×C4 — C8×C4○D4

Generators and relations for C8×C4○D4
G = < a,b,c,d | a8=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 276 in 228 conjugacy classes, 180 normal (16 characteristic)
C1, C2 [×3], C2 [×6], C4 [×12], C4 [×6], C22, C22 [×6], C22 [×6], C8 [×4], C8 [×6], C2×C4 [×3], C2×C4 [×21], C2×C4 [×12], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×8], C2×C8 [×12], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8, C4×C8 [×9], C22⋊C8 [×6], C4⋊C8 [×6], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×C4○D4, C2×C4×C8 [×3], C42.12C4 [×3], C8×D4 [×6], C8×Q8 [×2], C4×C4○D4, C8×C4○D4
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C8 [×8], C2×C4 [×28], C23 [×15], C2×C8 [×28], C22×C4 [×14], C4○D4 [×4], C24, C22×C8 [×14], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C4×C4○D4, C23×C8, C2×C8○D4, C8×C4○D4

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

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

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

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

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 1 1 2 2 type + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 C4 C8 C4○D4 C8○D4 kernel C8×C4○D4 C2×C4×C8 C42.12C4 C8×D4 C8×Q8 C4×C4○D4 C42⋊C2 C4×D4 C4×Q8 C2×C4○D4 C4○D4 C8 C4 # reps 1 3 3 6 2 1 6 6 2 2 32 8 8

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

 8 0 0 0 8 0 0 0 8
,
 1 0 0 0 13 0 0 0 13
,
 1 0 0 0 13 0 0 0 4
,
 16 0 0 0 0 13 0 4 0
G:=sub<GL(3,GF(17))| [8,0,0,0,8,0,0,0,8],[1,0,0,0,13,0,0,0,13],[1,0,0,0,13,0,0,0,4],[16,0,0,0,0,4,0,13,0] >;

C8×C4○D4 in GAP, Magma, Sage, TeX

C_8\times C_4\circ D_4
% in TeX

G:=Group("C8xC4oD4");
// GroupNames label

G:=SmallGroup(128,1696);
// by ID

G=gap.SmallGroup(128,1696);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,80,124]);
// Polycyclic

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

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