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G = Q16:13D4order 128 = 27

2nd semidirect product of Q16 and D4 acting through Inn(Q16)

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

Aliases: Q16:13D4, C42.454C23, C4.1412+ 1+4, C2.71D42, C4:C4o2Q16, (C8xD4):15C2, C4:4(C4oD8), C8.83(C2xD4), D4:D4:8C2, C8:4D4:12C2, C8:7D4:22C2, C4:C4.407D4, (C4xQ16):10C2, Q8:6D4:6C2, Q8.30(C2xD4), C4:SD16:41C2, (C2xD4).232D4, C2.43(D4oD8), (C4xC8).82C22, C22:C4.97D4, C4:C8.342C22, C4:C4.226C23, (C2xC4).485C24, (C2xC8).570C23, (C2xD8).35C22, C23.105(C2xD4), C4.101(C22xD4), (C2xD4).218C23, (C4xD4).328C22, C4:D4.69C22, C4:1D4.80C22, (C4xQ8).146C22, (C2xQ8).393C23, C2.D8.190C22, C22:C8.200C22, (C22xC8).195C22, (C2xQ16).171C22, C22.745(C22xD4), D4:C4.117C22, (C22xC4).1129C23, Q8:C4.158C22, (C2xSD16).153C22, C4:C4o(C2xQ16), (C2xC4oD8):14C2, C2.57(C2xC4oD8), (C2xC4).922(C2xD4), (C2xC4oD4).195C22, SmallGroup(128,2019)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — Q16:13D4
C1C2C22C2xC4C2xQ8C2xC4oD4C2xC4oD8 — Q16:13D4
C1C2C2xC4 — Q16:13D4
C1C22C4xD4 — Q16:13D4
C1C2C2C2xC4 — Q16:13D4

Generators and relations for Q16:13D4
 G = < a,b,c,d | a8=c4=d2=1, b2=a4, bab-1=cac-1=dad=a-1, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 552 in 252 conjugacy classes, 96 normal (24 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, Q8, Q8, C23, C23, C42, C42, C22:C4, C22:C4, C4:C4, C4:C4, C4:C4, C2xC8, C2xC8, C2xC8, D8, SD16, Q16, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C2xQ8, C4oD4, C4xC8, C22:C8, D4:C4, Q8:C4, C4:C8, C2.D8, C4xD4, C4xD4, C4xQ8, C4:D4, C4:D4, C4:1D4, C4:1D4, C22xC8, C2xD8, C2xSD16, C2xQ16, C4oD8, C2xC4oD4, C8xD4, C4xQ16, D4:D4, C4:SD16, C8:7D4, C8:4D4, Q8:6D4, C2xC4oD8, Q16:13D4
Quotients: C1, C2, C22, D4, C23, C2xD4, C24, C4oD8, C22xD4, 2+ 1+4, D42, C2xC4oD8, D4oD8, Q16:13D4

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I···4M4N4O8A8B8C8D8E···8J
order12222222224···44···44488888···8
size11114488882···24···48822224···4

35 irreducible representations

dim1111111112222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4oD82+ 1+4D4oD8
kernelQ16:13D4C8xD4C4xQ16D4:D4C4:SD16C8:7D4C8:4D4Q8:6D4C2xC4oD8C22:C4C4:C4Q16C2xD4C4C4C2
# reps1114221222141812

Matrix representation of Q16:13D4 in GL4(F17) generated by

3300
14300
0010
0001
,
121200
12500
00160
00016
,
5500
51200
0042
00013
,
14300
3300
0010
001316
G:=sub<GL(4,GF(17))| [3,14,0,0,3,3,0,0,0,0,1,0,0,0,0,1],[12,12,0,0,12,5,0,0,0,0,16,0,0,0,0,16],[5,5,0,0,5,12,0,0,0,0,4,0,0,0,2,13],[14,3,0,0,3,3,0,0,0,0,1,13,0,0,0,16] >;

Q16:13D4 in GAP, Magma, Sage, TeX

Q_{16}\rtimes_{13}D_4
% in TeX

G:=Group("Q16:13D4");
// GroupNames label

G:=SmallGroup(128,2019);
// by ID

G=gap.SmallGroup(128,2019);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,352,2019,346,248,2804,1411,375,172]);
// 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,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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