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## G = C2×C4.D8order 128 = 27

### Direct product of C2 and C4.D8

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C4.D8
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C2×C4⋊1D4 — C2×C4.D8
 Lower central C1 — C22 — C2×C4 — C2×C4.D8
 Upper central C1 — C23 — C2×C42 — C2×C4.D8
 Jennings C1 — C22 — C22 — C42 — C2×C4.D8

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

Subgroups: 500 in 180 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, C23, C23, C42, C42, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C4⋊C8, C4⋊C8, C2×C42, C41D4, C41D4, C22×C8, C22×D4, C22×D4, C4.D8, C2×C4⋊C8, C2×C41D4, C2×C4.D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, D8, SD16, C22×C4, C2×D4, C4.D4, D4⋊C4, C2×C22⋊C4, C2×D8, C2×SD16, C4.D8, C2×C4.D4, C2×D4⋊C4, C2×C4.D8

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

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

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

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

38 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 8 8 2 ··· 2 4 4 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 D4 D4 D8 SD16 C4.D4 kernel C2×C4.D8 C4.D8 C2×C4⋊C8 C2×C4⋊1D4 C4⋊1D4 C22×D4 C42 C22×C4 C2×C4 C2×C4 C22 # reps 1 4 2 1 4 4 2 2 8 8 2

Matrix representation of C2×C4.D8 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 16 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 4 0 0 0 0 0 12 5 0 0 0 5 5 0 0 0 0 0 11 11 0 0 0 3 0
,
 13 0 0 0 0 0 12 5 0 0 0 12 12 0 0 0 0 0 11 11 0 0 0 3 6

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,12,5,0,0,0,5,5,0,0,0,0,0,11,3,0,0,0,11,0],[13,0,0,0,0,0,12,12,0,0,0,5,12,0,0,0,0,0,11,3,0,0,0,11,6] >;

C2×C4.D8 in GAP, Magma, Sage, TeX

C_2\times C_4.D_8
% in TeX

G:=Group("C2xC4.D8");
// GroupNames label

G:=SmallGroup(128,270);
// by ID

G=gap.SmallGroup(128,270);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1123,1018,248,1971,242]);
// Polycyclic

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

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