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## G = (C2×C8).169D4order 128 = 27

### 137th non-split extension by C2×C8 of D4 acting via D4/C2=C22

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22×C4 — (C2×C8).169D4
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C22×D4 — C24.3C22 — (C2×C8).169D4
 Lower central C1 — C2 — C22×C4 — (C2×C8).169D4
 Upper central C1 — C23 — C2×C42 — (C2×C8).169D4
 Jennings C1 — C2 — C2 — C22×C4 — (C2×C8).169D4

Generators and relations for (C2×C8).169D4
G = < a,b,c,d | a2=b8=1, c4=b4, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=ab5, dbd-1=b3, dcd-1=c3 >

Subgroups: 352 in 145 conjugacy classes, 54 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×7], C22 [×3], C22 [×4], C22 [×10], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×6], C23, C23 [×8], C42 [×2], C22⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×5], C24, D4⋊C4 [×4], C4.Q8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22×C8 [×2], C22×D4, C22.7C42, C429C4, C24.3C22, C2×D4⋊C4 [×2], C2×C4.Q8 [×2], (C2×C8).169D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, SD16 [×4], C2×D4 [×3], C2×Q8, C4○D4 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×SD16 [×2], C8⋊C22 [×2], C23.4Q8, D42Q8 [×2], C23.46D4 [×2], C85D4, C83D4, (C2×C8).169D4

Smallest permutation representation of (C2×C8).169D4
On 64 points
Generators in S64
(1 63)(2 64)(3 57)(4 58)(5 59)(6 60)(7 61)(8 62)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 55)(18 56)(19 49)(20 50)(21 51)(22 52)(23 53)(24 54)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(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 20 30 38 5 24 26 34)(2 55 31 47 6 51 27 43)(3 22 32 40 7 18 28 36)(4 49 25 41 8 53 29 45)(9 35 60 21 13 39 64 17)(10 44 61 56 14 48 57 52)(11 37 62 23 15 33 58 19)(12 46 63 50 16 42 59 54)
(1 39 59 47)(2 34 60 42)(3 37 61 45)(4 40 62 48)(5 35 63 43)(6 38 64 46)(7 33 57 41)(8 36 58 44)(9 54 27 20)(10 49 28 23)(11 52 29 18)(12 55 30 21)(13 50 31 24)(14 53 32 19)(15 56 25 22)(16 51 26 17)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + - + image C1 C2 C2 C2 C2 C2 D4 D4 Q8 SD16 C4○D4 C8⋊C22 kernel (C2×C8).169D4 C22.7C42 C42⋊9C4 C24.3C22 C2×D4⋊C4 C2×C4.Q8 C2×C8 C22×C4 C2×D4 C2×C4 C2×C4 C22 # reps 1 1 1 1 2 2 4 2 2 8 6 2

Matrix representation of (C2×C8).169D4 in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16 0 0 0 0 0 0 16
,
 5 5 0 0 0 0 12 5 0 0 0 0 0 0 5 12 0 0 0 0 5 5 0 0 0 0 0 0 0 9 0 0 0 0 2 0
,
 12 12 0 0 0 0 5 12 0 0 0 0 0 0 5 5 0 0 0 0 12 5 0 0 0 0 0 0 0 2 0 0 0 0 9 0
,
 1 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 0 0 0 0 16 0 0 0 0 0 0 0 13 0 0 0 0 0 0 13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[5,12,0,0,0,0,5,5,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,2,0,0,0,0,9,0],[12,5,0,0,0,0,12,12,0,0,0,0,0,0,5,12,0,0,0,0,5,5,0,0,0,0,0,0,0,9,0,0,0,0,2,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13] >;

(C2×C8).169D4 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{169}D_4
% in TeX

G:=Group("(C2xC8).169D4");
// GroupNames label

G:=SmallGroup(128,826);
// by ID

G=gap.SmallGroup(128,826);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,504,141,624,422,387,1018,521,248,1411,718,172]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^4=b^4,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^5,d*b*d^-1=b^3,d*c*d^-1=c^3>;
// generators/relations

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