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## G = C2×C8.12D4order 128 = 27

### Direct product of C2 and C8.12D4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C2×C8.12D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C42 — C2×C4×C8 — C2×C8.12D4
 Lower central C1 — C2 — C2×C4 — C2×C8.12D4
 Upper central C1 — C23 — C2×C42 — C2×C8.12D4
 Jennings C1 — C2 — C2 — C2×C4 — C2×C8.12D4

Generators and relations for C2×C8.12D4
G = < a,b,c,d | a2=b8=c4=1, d2=b4, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=b4c-1 >

Subgroups: 628 in 296 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×6], C22 [×20], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×12], Q8 [×12], C23, C23 [×16], C42 [×4], C22⋊C4 [×16], C2×C8 [×12], D8 [×8], SD16 [×16], Q16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×4], C2×D4 [×10], C2×Q8 [×4], C2×Q8 [×10], C24 [×2], C4×C8 [×4], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C4.4D4 [×4], C22×C8 [×2], C2×D8 [×4], C2×D8 [×4], C2×SD16 [×8], C2×SD16 [×8], C2×Q16 [×4], C2×Q16 [×4], C22×D4 [×2], C22×Q8 [×2], C2×C4×C8, C8.12D4 [×8], C2×C4.4D4 [×2], C22×D8, C22×SD16 [×2], C22×Q16, C2×C8.12D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C41D4 [×4], C4○D8 [×4], C22×D4 [×3], C8.12D4 [×4], C2×C41D4, C2×C4○D8 [×2], C2×C8.12D4

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

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

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

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

44 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4L 4M 4N 4O 4P 8A ··· 8P order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 ··· 1 8 8 8 8 2 ··· 2 8 8 8 8 2 ··· 2

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D4 C4○D8 kernel C2×C8.12D4 C2×C4×C8 C8.12D4 C2×C4.4D4 C22×D8 C22×SD16 C22×Q16 C42 C2×C8 C22×C4 C22 # reps 1 1 8 2 1 2 1 2 8 2 16

Matrix representation of C2×C8.12D4 in GL5(𝔽17)

 16 0 0 0 0 0 16 0 0 0 0 0 16 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 10 10 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 16 0 0 0 0 0 0 4 0 0 0 0 0 4
,
 16 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 4 0 0 0 0 13 13

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,10,12,0,0,0,10,0],[1,0,0,0,0,0,0,16,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,4,13,0,0,0,0,13] >;

C2×C8.12D4 in GAP, Magma, Sage, TeX

C_2\times C_8._{12}D_4
% in TeX

G:=Group("C2xC8.12D4");
// GroupNames label

G:=SmallGroup(128,1878);
// by ID

G=gap.SmallGroup(128,1878);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,520,2804,172]);
// 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,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=b^4*c^-1>;
// generators/relations

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