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## G = C42.409D4order 128 = 27

### 42nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — C42.409D4
 Chief series C1 — C2 — C22 — C2×C4 — C42 — C2×C42 — C22.26C24 — C42.409D4
 Lower central C1 — C22 — C2×C4 — C42.409D4
 Upper central C1 — C2×C4 — C2×C42 — C42.409D4
 Jennings C1 — C22 — C22 — C42 — C42.409D4

Generators and relations for C42.409D4
G = < a,b,c,d | a4=b4=1, c4=a2, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=a2bc3 >

Subgroups: 284 in 128 conjugacy classes, 54 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×5], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×9], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×6], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C4×C8, C22⋊C8, C4⋊C8 [×4], C4⋊C8, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C22×C8, C2×C4○D4, C4.D8 [×2], C4.10D8 [×2], C2×C4⋊C8, C42.12C4, C22.26C24, C42.409D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, C4○D8 [×2], M4(2).8C22, C2×D4⋊C4, C23.24D4, C42.409D4

Smallest permutation representation of C42.409D4
On 64 points
Generators in S64
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 59 13 63)(10 60 14 64)(11 61 15 57)(12 62 16 58)(17 51 21 55)(18 52 22 56)(19 53 23 49)(20 54 24 50)(25 34 29 38)(26 35 30 39)(27 36 31 40)(28 37 32 33)
(1 31 56 57)(2 58 49 32)(3 25 50 59)(4 60 51 26)(5 27 52 61)(6 62 53 28)(7 29 54 63)(8 64 55 30)(9 43 38 24)(10 17 39 44)(11 45 40 18)(12 19 33 46)(13 47 34 20)(14 21 35 48)(15 41 36 22)(16 23 37 42)
(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 64 31 55 56 30 57 8)(2 7 58 29 49 54 32 63)(3 62 25 53 50 28 59 6)(4 5 60 27 51 52 26 61)(9 46 43 12 38 19 24 33)(10 40 17 18 39 11 44 45)(13 42 47 16 34 23 20 37)(14 36 21 22 35 15 48 41)

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

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

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

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 8A ··· 8P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 4 4 4 4 8 ··· 8 size 1 1 1 1 2 2 8 8 1 1 1 1 2 ··· 2 4 4 8 8 4 ··· 4

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 D4 D4 D8 SD16 C4○D8 M4(2).8C22 kernel C42.409D4 C4.D8 C4.10D8 C2×C4⋊C8 C42.12C4 C22.26C24 C4×D4 C4⋊D4 C42 C22×C4 C2×C4 C2×C4 C4 C2 # reps 1 2 2 1 1 1 4 4 2 2 4 4 8 2

Matrix representation of C42.409D4 in GL4(𝔽17) generated by

 4 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 16 0
,
 3 14 0 0 3 3 0 0 0 0 14 14 0 0 14 3
,
 3 14 0 0 14 14 0 0 0 0 14 14 0 0 3 14
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,16,0,0,1,0],[3,3,0,0,14,3,0,0,0,0,14,14,0,0,14,3],[3,14,0,0,14,14,0,0,0,0,14,3,0,0,14,14] >;

C42.409D4 in GAP, Magma, Sage, TeX

C_4^2._{409}D_4
% in TeX

G:=Group("C4^2.409D4");
// GroupNames label

G:=SmallGroup(128,272);
// by ID

G=gap.SmallGroup(128,272);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,184,1123,1018,248,1971,242]);
// Polycyclic

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

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