Copied to
clipboard

G = Q1613D4order 128 = 27

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

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

Aliases: Q1613D4, C42.454C23, C4.1412+ 1+4, C2.71D42, C4⋊C42Q16, (C8×D4)⋊15C2, C44(C4○D8), C8.83(C2×D4), D4⋊D48C2, C84D412C2, C87D422C2, C4⋊C4.407D4, (C4×Q16)⋊10C2, Q86D46C2, Q8.30(C2×D4), C4⋊SD1641C2, (C2×D4).232D4, C2.43(D4○D8), (C4×C8).82C22, C22⋊C4.97D4, C4⋊C8.342C22, C4⋊C4.226C23, (C2×C4).485C24, (C2×C8).570C23, (C2×D8).35C22, C23.105(C2×D4), C4.101(C22×D4), (C2×D4).218C23, (C4×D4).328C22, C4⋊D4.69C22, C41D4.80C22, (C4×Q8).146C22, (C2×Q8).393C23, C2.D8.190C22, C22⋊C8.200C22, (C22×C8).195C22, (C2×Q16).171C22, C22.745(C22×D4), D4⋊C4.117C22, (C22×C4).1129C23, Q8⋊C4.158C22, (C2×SD16).153C22, C4⋊C4(C2×Q16), (C2×C4○D8)⋊14C2, C2.57(C2×C4○D8), (C2×C4).922(C2×D4), (C2×C4○D4).195C22, SmallGroup(128,2019)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — Q1613D4
C1C2C22C2×C4C2×Q8C2×C4○D4C2×C4○D8 — Q1613D4
C1C2C2×C4 — Q1613D4
C1C22C4×D4 — Q1613D4
C1C2C2C2×C4 — Q1613D4

Generators and relations for Q1613D4
 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 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×18], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×28], Q8 [×4], Q8 [×2], C23 [×2], C23 [×4], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×6], SD16 [×8], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×4], C2×D4 [×12], C2×Q8 [×2], C4○D4 [×12], C4×C8, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×2], C4⋊C8, C2.D8, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4 [×4], C4⋊D4 [×4], C41D4 [×2], C41D4 [×2], C22×C8 [×2], C2×D8 [×4], C2×SD16 [×4], C2×Q16, C4○D8 [×8], C2×C4○D4 [×4], C8×D4, C4×Q16, D4⋊D4 [×4], C4⋊SD16 [×2], C87D4 [×2], C84D4, Q86D4 [×2], C2×C4○D8 [×2], Q1613D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C24, C4○D8 [×2], C22×D4 [×2], 2+ 1+4, D42, C2×C4○D8, D4○D8, Q1613D4

Smallest permutation representation of Q1613D4
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 28 5 32)(2 27 6 31)(3 26 7 30)(4 25 8 29)(9 36 13 40)(10 35 14 39)(11 34 15 38)(12 33 16 37)(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 9 23 45)(2 16 24 44)(3 15 17 43)(4 14 18 42)(5 13 19 41)(6 12 20 48)(7 11 21 47)(8 10 22 46)(25 39 56 61)(26 38 49 60)(27 37 50 59)(28 36 51 58)(29 35 52 57)(30 34 53 64)(31 33 54 63)(32 40 55 62)
(1 8)(2 7)(3 6)(4 5)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 48)(16 47)(17 20)(18 19)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 64)(34 63)(35 62)(36 61)(37 60)(38 59)(39 58)(40 57)(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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(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), (1,9,23,45)(2,16,24,44)(3,15,17,43)(4,14,18,42)(5,13,19,41)(6,12,20,48)(7,11,21,47)(8,10,22,46)(25,39,56,61)(26,38,49,60)(27,37,50,59)(28,36,51,58)(29,35,52,57)(30,34,53,64)(31,33,54,63)(32,40,55,62), (1,8)(2,7)(3,6)(4,5)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(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), (1,28,5,32)(2,27,6,31)(3,26,7,30)(4,25,8,29)(9,36,13,40)(10,35,14,39)(11,34,15,38)(12,33,16,37)(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), (1,9,23,45)(2,16,24,44)(3,15,17,43)(4,14,18,42)(5,13,19,41)(6,12,20,48)(7,11,21,47)(8,10,22,46)(25,39,56,61)(26,38,49,60)(27,37,50,59)(28,36,51,58)(29,35,52,57)(30,34,53,64)(31,33,54,63)(32,40,55,62), (1,8)(2,7)(3,6)(4,5)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,48)(16,47)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,64)(34,63)(35,62)(36,61)(37,60)(38,59)(39,58)(40,57)(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)], [(1,28,5,32),(2,27,6,31),(3,26,7,30),(4,25,8,29),(9,36,13,40),(10,35,14,39),(11,34,15,38),(12,33,16,37),(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)], [(1,9,23,45),(2,16,24,44),(3,15,17,43),(4,14,18,42),(5,13,19,41),(6,12,20,48),(7,11,21,47),(8,10,22,46),(25,39,56,61),(26,38,49,60),(27,37,50,59),(28,36,51,58),(29,35,52,57),(30,34,53,64),(31,33,54,63),(32,40,55,62)], [(1,8),(2,7),(3,6),(4,5),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,48),(16,47),(17,20),(18,19),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,64),(34,63),(35,62),(36,61),(37,60),(38,59),(39,58),(40,57),(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+++++++++++++++
imageC1C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4D4○D8
kernelQ1613D4C8×D4C4×Q16D4⋊D4C4⋊SD16C87D4C84D4Q86D4C2×C4○D8C22⋊C4C4⋊C4Q16C2×D4C4C4C2
# reps1114221222141812

Matrix representation of Q1613D4 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] >;

Q1613D4 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

׿
×
𝔽