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## G = Q8.(C2×D4)  order 128 = 27

### 10th non-split extension by Q8 of C2×D4 acting via C2×D4/C23=C2

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2×C4 — Q8.(C2×D4)
 Chief series C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2×2- 1+4 — Q8.(C2×D4)
 Lower central C1 — C2 — C2×C4 — Q8.(C2×D4)
 Upper central C1 — C22 — C2×C4○D4 — Q8.(C2×D4)
 Jennings C1 — C2 — C2 — C2×C4 — Q8.(C2×D4)

Generators and relations for Q8.(C2×D4)
G = < a,b,c,d,e | a4=1, b2=c2=d4=e2=a2, bab-1=eae-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a-1b, ebe-1=ab, cd=dc, ece-1=a2c, ede-1=a2d3 >

Subgroups: 620 in 350 conjugacy classes, 108 normal (28 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4○D4, C22⋊C8, D4⋊C4, Q8⋊C4, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C2×Q16, C8.C22, C22×Q8, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C4○D4, 2- 1+4, (C22×C8)⋊C2, C23.24D4, C23.38D4, C22⋊Q16, D4.7D4, C23.38C23, C22×Q16, C2×C8.C22, C2×2- 1+4, Q8.(C2×D4)
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22≀C2, C22×D4, C2×C22≀C2, Q8○D8, Q8.(C2×D4)

Smallest permutation representation of Q8.(C2×D4)
On 64 points
Generators in S64
(1 57 5 61)(2 58 6 62)(3 59 7 63)(4 60 8 64)(9 31 13 27)(10 32 14 28)(11 25 15 29)(12 26 16 30)(17 55 21 51)(18 56 22 52)(19 49 23 53)(20 50 24 54)(33 43 37 47)(34 44 38 48)(35 45 39 41)(36 46 40 42)
(1 29 5 25)(2 12 6 16)(3 27 7 31)(4 10 8 14)(9 59 13 63)(11 57 15 61)(17 48 21 44)(18 35 22 39)(19 46 23 42)(20 33 24 37)(26 58 30 62)(28 64 32 60)(34 55 38 51)(36 53 40 49)(41 52 45 56)(43 50 47 54)
(1 22 5 18)(2 23 6 19)(3 24 7 20)(4 17 8 21)(9 47 13 43)(10 48 14 44)(11 41 15 45)(12 42 16 46)(25 35 29 39)(26 36 30 40)(27 37 31 33)(28 38 32 34)(49 58 53 62)(50 59 54 63)(51 60 55 64)(52 61 56 57)
(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 26 5 30)(2 25 6 29)(3 32 7 28)(4 31 8 27)(9 64 13 60)(10 63 14 59)(11 62 15 58)(12 61 16 57)(17 37 21 33)(18 36 22 40)(19 35 23 39)(20 34 24 38)(41 53 45 49)(42 52 46 56)(43 51 47 55)(44 50 48 54)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 Q8○D8 kernel Q8.(C2×D4) (C22×C8)⋊C2 C23.24D4 C23.38D4 C22⋊Q16 D4.7D4 C23.38C23 C22×Q16 C2×C8.C22 C2×2- 1+4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 1 1 1 4 4 1 1 1 1 3 5 4 4

Matrix representation of Q8.(C2×D4) in GL6(𝔽17)

 16 0 0 0 0 0 0 16 0 0 0 0 0 0 0 1 0 0 0 0 16 0 0 0 0 0 0 0 0 1 0 0 0 0 16 0
,
 16 0 0 0 0 0 6 1 0 0 0 0 0 0 12 12 0 0 0 0 12 5 0 0 0 0 5 5 5 5 0 0 5 12 5 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 2 0 0 16 0 15 0 0 0 0 0 0 16 0 0 0 0 1 0
,
 11 15 0 0 0 0 9 6 0 0 0 0 0 0 14 14 11 11 0 0 3 14 6 11 0 0 0 0 3 3 0 0 0 0 14 3
,
 6 2 0 0 0 0 8 11 0 0 0 0 0 0 13 0 9 0 0 0 0 4 0 8 0 0 0 0 4 0 0 0 0 0 0 13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,6,0,0,0,0,0,1,0,0,0,0,0,0,12,12,5,5,0,0,12,5,5,12,0,0,0,0,5,5,0,0,0,0,5,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,15,0,1,0,0,2,0,16,0],[11,9,0,0,0,0,15,6,0,0,0,0,0,0,14,3,0,0,0,0,14,14,0,0,0,0,11,6,3,14,0,0,11,11,3,3],[6,8,0,0,0,0,2,11,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,9,0,4,0,0,0,0,8,0,13] >;

Q8.(C2×D4) in GAP, Magma, Sage, TeX

Q_8.(C_2\times D_4)
% in TeX

G:=Group("Q8.(C2xD4)");
// GroupNames label

G:=SmallGroup(128,1743);
// by ID

G=gap.SmallGroup(128,1743);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,521,2804,1411,172]);
// Polycyclic

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

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