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## G = C2×C8.18D4order 128 = 27

### Direct product of C2 and C8.18D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8.18D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C23×C4 — C23×C8 — C2×C8.18D4
 Lower central C1 — C2 — C2×C4 — C2×C8.18D4
 Upper central C1 — C23 — C23×C4 — C2×C8.18D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8.18D4

Generators and relations for C2×C8.18D4
G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b4c-1 >

Subgroups: 436 in 256 conjugacy classes, 116 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C22 [×12], C8 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×6], C2×C4 [×26], Q8 [×12], C23, C23 [×6], C23 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×8], C2×C8 [×10], Q16 [×8], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×4], C2×Q8 [×10], C24, Q8⋊C4 [×8], C2.D8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22⋊Q8 [×4], C22×C8 [×2], C22×C8 [×4], C22×C8 [×4], C2×Q16 [×4], C2×Q16 [×4], C23×C4, C22×Q8 [×2], C2×Q8⋊C4 [×2], C2×C2.D8, C8.18D4 [×8], C2×C22⋊Q8 [×2], C23×C8, C22×Q16, C2×C8.18D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], Q16 [×4], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C2×Q16 [×6], C4○D8 [×2], C22×D4 [×2], C2×C4○D4, C8.18D4 [×4], C2×C4⋊D4, C22×Q16, C2×C4○D8, C2×C8.18D4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D4 Q16 C4○D8 kernel C2×C8.18D4 C2×Q8⋊C4 C2×C2.D8 C8.18D4 C2×C22⋊Q8 C23×C8 C22×Q16 C2×C8 C22×C4 C24 C2×C4 C23 C22 # reps 1 2 1 8 2 1 1 4 3 1 4 8 8

Matrix representation of C2×C8.18D4 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
,
 1 0 0 0 0 0 3 14 0 0 0 3 3 0 0 0 0 0 2 0 0 0 0 9 9
,
 16 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 12 15 0 0 0 12 5
,
 1 0 0 0 0 0 0 4 0 0 0 4 0 0 0 0 0 0 12 15 0 0 0 13 5

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],[1,0,0,0,0,0,3,3,0,0,0,14,3,0,0,0,0,0,2,9,0,0,0,0,9],[16,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,12,0,0,0,15,5],[1,0,0,0,0,0,0,4,0,0,0,4,0,0,0,0,0,0,12,13,0,0,0,15,5] >;

C2×C8.18D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{18}D_4
% in TeX

G:=Group("C2xC8.18D4");
// GroupNames label

G:=SmallGroup(128,1781);
// by ID

G=gap.SmallGroup(128,1781);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,2804,172]);
// Polycyclic

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

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