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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 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×37], D4 [×2], D4 [×19], Q8 [×6], Q8 [×23], C23, C23 [×2], C23, C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×4], Q16 [×12], C22×C4, C22×C4 [×2], C22×C4 [×7], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C2×Q8 [×8], C2×Q8 [×26], C4○D4 [×4], C4○D4 [×38], C22⋊C8 [×4], D4⋊C4 [×2], Q8⋊C4 [×6], C42⋊C2, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×6], C2×Q16 [×4], C8.C22 [×4], C22×Q8 [×2], C22×Q8 [×2], C22×Q8, C2×C4○D4 [×2], C2×C4○D4 [×2], C2×C4○D4 [×3], 2- 1+4 [×8], (C22×C8)⋊C2, C23.24D4, C23.38D4, C22⋊Q16 [×4], D4.7D4 [×4], C23.38C23, C22×Q16, C2×C8.C22, C2×2- 1+4, Q8.(C2×D4)
Quotients: C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, Q8○D8 [×2], 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 40 13 36)(10 33 14 37)(11 34 15 38)(12 35 16 39)(17 54 21 50)(18 55 22 51)(19 56 23 52)(20 49 24 53)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)
(1 38 5 34)(2 12 6 16)(3 36 7 40)(4 10 8 14)(9 59 13 63)(11 57 15 61)(17 30 21 26)(18 48 22 44)(19 28 23 32)(20 46 24 42)(25 49 29 53)(27 55 31 51)(33 60 37 64)(35 58 39 62)(41 52 45 56)(43 50 47 54)
(1 23 5 19)(2 24 6 20)(3 17 7 21)(4 18 8 22)(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 35 5 39)(2 34 6 38)(3 33 7 37)(4 40 8 36)(9 64 13 60)(10 63 14 59)(11 62 15 58)(12 61 16 57)(17 31 21 27)(18 30 22 26)(19 29 23 25)(20 28 24 32)(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)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41), (1,38,5,34)(2,12,6,16)(3,36,7,40)(4,10,8,14)(9,59,13,63)(11,57,15,61)(17,30,21,26)(18,48,22,44)(19,28,23,32)(20,46,24,42)(25,49,29,53)(27,55,31,51)(33,60,37,64)(35,58,39,62)(41,52,45,56)(43,50,47,54), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,64,13,60)(10,63,14,59)(11,62,15,58)(12,61,16,57)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(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)(9,40,13,36)(10,33,14,37)(11,34,15,38)(12,35,16,39)(17,54,21,50)(18,55,22,51)(19,56,23,52)(20,49,24,53)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41), (1,38,5,34)(2,12,6,16)(3,36,7,40)(4,10,8,14)(9,59,13,63)(11,57,15,61)(17,30,21,26)(18,48,22,44)(19,28,23,32)(20,46,24,42)(25,49,29,53)(27,55,31,51)(33,60,37,64)(35,58,39,62)(41,52,45,56)(43,50,47,54), (1,23,5,19)(2,24,6,20)(3,17,7,21)(4,18,8,22)(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), (1,35,5,39)(2,34,6,38)(3,33,7,37)(4,40,8,36)(9,64,13,60)(10,63,14,59)(11,62,15,58)(12,61,16,57)(17,31,21,27)(18,30,22,26)(19,29,23,25)(20,28,24,32)(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),(9,40,13,36),(10,33,14,37),(11,34,15,38),(12,35,16,39),(17,54,21,50),(18,55,22,51),(19,56,23,52),(20,49,24,53),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41)], [(1,38,5,34),(2,12,6,16),(3,36,7,40),(4,10,8,14),(9,59,13,63),(11,57,15,61),(17,30,21,26),(18,48,22,44),(19,28,23,32),(20,46,24,42),(25,49,29,53),(27,55,31,51),(33,60,37,64),(35,58,39,62),(41,52,45,56),(43,50,47,54)], [(1,23,5,19),(2,24,6,20),(3,17,7,21),(4,18,8,22),(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)], [(1,35,5,39),(2,34,6,38),(3,33,7,37),(4,40,8,36),(9,64,13,60),(10,63,14,59),(11,62,15,58),(12,61,16,57),(17,31,21,27),(18,30,22,26),(19,29,23,25),(20,28,24,32),(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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