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G = Q16:12D4order 128 = 27

1st semidirect product of Q16 and D4 acting through Inn(Q16)

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

Aliases: Q16:12D4, C42.452C23, C4.1392+ 1+4, C2.69D42, (C8xD4):14C2, C8.81(C2xD4), C22:C4o2Q16, C8:7D4:10C2, C4:C4.256D4, (C4xQ16):24C2, Q8:5D4:7C2, Q8.28(C2xD4), Q8:D4:33C2, D4.7D4:8C2, (C2xD4).231D4, C8.12D4:9C2, C22:4(C4oD8), Q8.D4:6C2, C2.42(Q8oD8), C4.99(C22xD4), C8.18D4:22C2, C4:C8.317C22, C4:C4.224C23, (C2xC8).345C23, (C4xC8).119C22, (C2xC4).483C24, C22:C4.194D4, (C22xQ16):12C2, C23.467(C2xD4), (C2xD4).217C23, (C4xD4).326C22, (C2xD8).137C22, C4:D4.68C22, (C4xQ8).144C22, (C2xQ8).391C23, C2.D8.188C22, C22:Q8.67C22, D4:C4.11C22, C22:C8.222C22, (C22xC8).194C22, (C2xQ16).170C22, (C2xSD16).95C22, C4.4D4.56C22, C22.743(C22xD4), (C22xC4).1127C23, Q8:C4.179C22, (C22xQ8).335C22, C22:C4o(C2xQ16), (C2xC4oD8):13C2, C2.56(C2xC4oD8), (C2xC4).921(C2xD4), (C2xC4oD4).194C22, SmallGroup(128,2017)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — Q16:12D4
C1C2C22C2xC4C2xQ8C22xQ8C22xQ16 — Q16:12D4
C1C2C2xC4 — Q16:12D4
C1C22C4xD4 — Q16:12D4
C1C2C2C2xC4 — Q16:12D4

Generators and relations for Q16:12D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 472 in 241 conjugacy classes, 96 normal (44 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, D8, SD16, Q16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C2xQ8, C4oD4, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C2.D8, C4xD4, C4xD4, C4xQ8, C4:D4, C4:D4, C22:Q8, C22:Q8, C4.4D4, C4.4D4, C22xC8, C2xD8, C2xSD16, C2xQ16, C2xQ16, C2xQ16, C4oD8, C22xQ8, C2xC4oD4, C8xD4, C4xQ16, Q8:D4, D4.7D4, Q8.D4, C8:7D4, C8.18D4, C8.12D4, Q8:5D4, C22xQ16, C2xC4oD8, Q16:12D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, D42, C2xC4oD8, Q8oD8, Q16:12D4

Smallest permutation representation of Q16:12D4
On 64 points
Generators in S64
(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 32 13 28)(10 31 14 27)(11 30 15 26)(12 29 16 25)(17 49 21 53)(18 56 22 52)(19 55 23 51)(20 54 24 50)(41 62 45 58)(42 61 46 57)(43 60 47 64)(44 59 48 63)
(1 11 19 48)(2 10 20 47)(3 9 21 46)(4 16 22 45)(5 15 23 44)(6 14 24 43)(7 13 17 42)(8 12 18 41)(25 56 58 36)(26 55 59 35)(27 54 60 34)(28 53 61 33)(29 52 62 40)(30 51 63 39)(31 50 64 38)(32 49 57 37)
(2 8)(3 7)(4 6)(9 42)(10 41)(11 48)(12 47)(13 46)(14 45)(15 44)(16 43)(17 21)(18 20)(22 24)(25 64)(26 63)(27 62)(28 61)(29 60)(30 59)(31 58)(32 57)(34 40)(35 39)(36 38)(50 56)(51 55)(52 54)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G···4L4M4N4O4P8A8B8C8D8E···8J
order1222222224···44···4444488888···8
size1111224882···24···4888822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4oD82+ 1+4Q8oD8
kernelQ16:12D4C8xD4C4xQ16Q8:D4D4.7D4Q8.D4C8:7D4C8.18D4C8.12D4Q8:5D4C22xQ16C2xC4oD8C22:C4C4:C4Q16C2xD4C22C4C2
# reps1112221112112141812

Matrix representation of Q16:12D4 in GL4(F17) generated by

16000
01600
00011
00311
,
16000
01600
00138
0004
,
01600
1000
00160
00161
,
1000
01600
0010
00116
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,0,3,0,0,11,11],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,8,4],[0,1,0,0,16,0,0,0,0,0,16,16,0,0,0,1],[1,0,0,0,0,16,0,0,0,0,1,1,0,0,0,16] >;

Q16:12D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_{12}D_4
% in TeX

G:=Group("Q16:12D4");
// GroupNames label

G:=SmallGroup(128,2017);
// by ID

G=gap.SmallGroup(128,2017);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,248,2804,1411,375,172]);
// Polycyclic

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

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