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## G = C2×D4.C8order 128 = 27

### Direct product of C2 and D4.C8

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

Series: Derived Chief Lower central Upper central Jennings

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

Generators and relations for C2×D4.C8
G = < a,b,c,d | a2=b4=c2=1, d8=b2, ab=ba, ac=ca, ad=da, cbc=b-1, bd=db, dcd-1=bc >

Subgroups: 172 in 112 conjugacy classes, 60 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×4], C4 [×2], C22 [×3], C22 [×6], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×5], D4 [×2], D4 [×5], Q8 [×2], Q8, C23, C23, C16 [×4], C2×C8 [×6], C2×C8 [×5], M4(2) [×2], M4(2) [×5], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4 [×4], C4○D4 [×2], C2×C16 [×2], C2×C16 [×3], M5(2) [×2], M5(2), C22×C8, C22×C8, C2×M4(2), C2×M4(2), C8○D4 [×4], C8○D4 [×2], C2×C4○D4, D4.C8 [×4], C22×C16, C2×M5(2), C2×C8○D4, C2×D4.C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4 [×2], C22⋊C8 [×4], C2×C22⋊C4, C22×C8, C2×M4(2), D4.C8 [×2], C2×C22⋊C8, C2×D4.C8

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

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 4G 4H 8A ··· 8H 8I 8J 8K 8L 8M 8N 8O 8P 16A ··· 16P 16Q ··· 16X order 1 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 ··· 8 8 8 8 8 8 8 8 8 16 ··· 16 16 ··· 16 size 1 1 1 1 2 2 4 4 1 1 1 1 2 2 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C8 C8 C8 D4 M4(2) M4(2) D4.C8 kernel C2×D4.C8 D4.C8 C22×C16 C2×M5(2) C2×C8○D4 C2×M4(2) C8○D4 C2×C4○D4 C2×D4 C2×Q8 C4○D4 C2×C8 C2×C4 C23 C2 # reps 1 4 1 1 1 2 4 2 4 4 8 4 2 2 16

Matrix representation of C2×D4.C8 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 16 0 0 0 0 7 4 0 0 13 10
,
 16 1 0 0 0 1 0 0 0 0 7 4 0 0 5 10
,
 13 11 0 0 9 4 0 0 0 0 16 8 0 0 9 5
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,7,13,0,0,4,10],[16,0,0,0,1,1,0,0,0,0,7,5,0,0,4,10],[13,9,0,0,11,4,0,0,0,0,16,9,0,0,8,5] >;

C2×D4.C8 in GAP, Magma, Sage, TeX

C_2\times D_4.C_8
% in TeX

G:=Group("C2xD4.C8");
// GroupNames label

G:=SmallGroup(128,848);
// by ID

G=gap.SmallGroup(128,848);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1018,248,1411,102,124]);
// Polycyclic

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

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