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G = Q8.(C2xD4)  order 128 = 27

10th non-split extension by Q8 of C2xD4 acting via C2xD4/C23=C2

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

Aliases: C4oD4.14D4, D4.46(C2xD4), Q8.46(C2xD4), C2.6(Q8oD8), C4.42C22wrC2, (C2xD4).295D4, C4:C4.13C23, (C2xQ8).123D4, (C22xQ16):7C2, C4.48(C22xD4), D4.7D4:16C2, (C2xC8).130C23, (C2xC4).230C24, C22:Q16:13C2, C23.234(C2xD4), (C2xQ8).23C23, C22.22C22wrC2, (C2xD4).383C23, C23.24D4:7C2, C23.38D4:5C2, C22:C8.12C22, (C2xSD16).1C22, C22:Q8.17C22, (C22xC4).278C23, (C22xC8).135C22, Q8:C4.20C22, (C2xQ16).113C22, C22.490(C22xD4), (C2x2- 1+4).6C2, D4:C4.153C22, C42:C2.98C22, C23.38C23:4C2, (C2xM4(2)).42C22, (C22xQ8).262C22, (C2xC4).457(C2xD4), (C2xC8.C22):9C2, (C22xC8):C2:6C2, C2.48(C2xC22wrC2), (C2xC4oD4).102C22, SmallGroup(128,1743)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — Q8.(C2xD4)
C1C2C22C2xC4C22xC4C2xC4oD4C2x2- 1+4 — Q8.(C2xD4)
C1C2C2xC4 — Q8.(C2xD4)
C1C22C2xC4oD4 — Q8.(C2xD4)
C1C2C2C2xC4 — Q8.(C2xD4)

Generators and relations for Q8.(C2xD4)
 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, C2xC4, C2xC4, C2xC4, D4, D4, Q8, Q8, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), SD16, Q16, C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C2xQ8, C4oD4, C4oD4, C22:C8, D4:C4, Q8:C4, C42:C2, C22:Q8, C22.D4, C4.4D4, C4:Q8, C22xC8, C2xM4(2), C2xSD16, C2xQ16, C2xQ16, C8.C22, C22xQ8, C22xQ8, C22xQ8, C2xC4oD4, C2xC4oD4, C2xC4oD4, 2- 1+4, (C22xC8):C2, C23.24D4, C23.38D4, C22:Q16, D4.7D4, C23.38C23, C22xQ16, C2xC8.C22, C2x2- 1+4, Q8.(C2xD4)
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C22wrC2, C22xD4, C2xC22wrC2, Q8oD8, Q8.(C2xD4)

Smallest permutation representation of Q8.(C2xD4)
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 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4L4M4N4O4P8A8B8C8D8E8F
order122222222244444···44444888888
size111122444422224···48888444488

32 irreducible representations

dim11111111112224
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4D4Q8oD8
kernelQ8.(C2xD4)(C22xC8):C2C23.24D4C23.38D4C22:Q16D4.7D4C23.38C23C22xQ16C2xC8.C22C2x2- 1+4C2xD4C2xQ8C4oD4C2
# reps11114411113544

Matrix representation of Q8.(C2xD4) in GL6(F17)

1600000
0160000
000100
0016000
000001
0000160
,
1600000
610000
00121200
0012500
005555
00512512
,
100000
010000
000102
00160150
0000016
000010
,
11150000
960000
0014141111
00314611
000033
0000143
,
620000
8110000
0013090
000408
000040
0000013

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.(C2xD4) 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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