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## G = C2×C8.4Q8order 128 = 27

### Direct product of C2 and C8.4Q8

direct product, p-group, metabelian, nilpotent (class 4), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C8 — C2×C8.4Q8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C8 — C22×C8 — C22×C16 — C2×C8.4Q8
 Lower central C1 — C2 — C4 — C8 — C2×C8.4Q8
 Upper central C1 — C2×C4 — C22×C4 — C22×C8 — C2×C8.4Q8
 Jennings C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — C2×C8.4Q8

Generators and relations for C2×C8.4Q8
G = < a,b,c,d | a2=b8=1, c4=b2, d2=bc2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b6c3 >

Subgroups: 108 in 72 conjugacy classes, 52 normal (32 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C8, C2×C4, C23, C16, C2×C8, C2×C8, M4(2), C22×C4, C8.C4, C8.C4, C2×C16, C2×C16, C22×C8, C2×M4(2), C8.4Q8, C2×C8.C4, C22×C16, C2×C8.4Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, D8, Q16, C22×C4, C2×D4, C2×Q8, C2.D8, C2×C4⋊C4, C2×D8, C2×Q16, C8.4Q8, C2×C2.D8, C2×C8.4Q8

Smallest permutation representation of C2×C8.4Q8
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 41)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 33)
(1 11 5 15 9 3 13 7)(2 12 6 16 10 4 14 8)(17 19 21 23 25 27 29 31)(18 20 22 24 26 28 30 32)(33 35 37 39 41 43 45 47)(34 36 38 40 42 44 46 48)(49 59 53 63 57 51 61 55)(50 60 54 64 58 52 62 56)
(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 24 13 28 9 32 5 20)(2 23 14 27 10 31 6 19)(3 22 15 26 11 30 7 18)(4 21 16 25 12 29 8 17)(33 53 37 49 41 61 45 57)(34 52 38 64 42 60 46 56)(35 51 39 63 43 59 47 55)(36 50 40 62 44 58 48 54)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 8A ··· 8H 8I ··· 8P 16A ··· 16P order 1 2 2 2 2 2 4 4 4 4 4 4 8 ··· 8 8 ··· 8 16 ··· 16 size 1 1 1 1 2 2 1 1 1 1 2 2 2 ··· 2 8 ··· 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + - + + - - image C1 C2 C2 C2 C4 D4 Q8 D4 D8 Q16 Q16 C8.4Q8 kernel C2×C8.4Q8 C8.4Q8 C2×C8.C4 C22×C16 C2×C16 C2×C8 C2×C8 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 4 2 1 8 1 2 1 4 2 2 16

Matrix representation of C2×C8.4Q8 in GL3(𝔽17) generated by

 16 0 0 0 1 0 0 0 1
,
 16 0 0 0 9 0 0 4 15
,
 1 0 0 0 5 0 0 7 7
,
 13 0 0 0 7 2 0 3 10
G:=sub<GL(3,GF(17))| [16,0,0,0,1,0,0,0,1],[16,0,0,0,9,4,0,0,15],[1,0,0,0,5,7,0,0,7],[13,0,0,0,7,3,0,2,10] >;

C2×C8.4Q8 in GAP, Magma, Sage, TeX

C_2\times C_8._4Q_8
% in TeX

G:=Group("C2xC8.4Q8");
// GroupNames label

G:=SmallGroup(128,892);
// by ID

G=gap.SmallGroup(128,892);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,288,1123,360,172,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^2,d^2=b*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^6*c^3>;
// generators/relations

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