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G = D46Q16order 128 = 27

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

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

Aliases: D46Q16, C42.487C23, C4.732- 1+4, D43(C2.D8), C4⋊C4.270D4, (C4×Q16)⋊14C2, C82Q818C2, (C8×D4).10C2, C4.29(C2×Q16), C4.Q1614C2, (C2×D4).356D4, C2.54(D4○D8), C8.76(C4○D4), (C4×C8).88C22, D43Q8.5C2, C8.18D415C2, C22.6(C2×Q16), C4⋊C8.300C22, C4⋊C4.243C23, (C2×C8).197C23, (C2×C4).530C24, C22⋊C4.114D4, C23.479(C2×D4), C4⋊Q8.163C22, C2.83(D46D4), C2.20(C22×Q16), C2.D8.63C22, (C4×D4).343C22, (C4×Q8).173C22, (C2×Q8).236C23, C22⋊Q8.99C22, C23.48D410C2, C22⋊C8.186C22, (C22×C8).165C22, (C2×Q16).139C22, C22.790(C22×D4), (C22×C4).1162C23, Q8⋊C4.162C22, (C2×C2.D8)⋊30C2, C4.112(C2×C4○D4), (C2×C4).173(C2×D4), (C2×C4⋊C4).682C22, SmallGroup(128,2070)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D46Q16
C1C2C4C2×C4C22×C4C2×C4⋊C4D43Q8 — D46Q16
C1C2C2×C4 — D46Q16
C1C22C4×D4 — D46Q16
C1C2C2C2×C4 — D46Q16

Generators and relations for D46Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c-1 >

Subgroups: 312 in 178 conjugacy classes, 96 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×16], D4 [×4], Q8 [×6], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4, C4⋊C4 [×6], C4⋊C4 [×12], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], Q16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×6], C4⋊C8, C2.D8, C2.D8 [×8], C2×C4⋊C4 [×4], C4×D4, C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×4], C42.C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×Q16, C2×C2.D8 [×2], C8×D4, C4×Q16, C8.18D4 [×2], C4.Q16 [×2], C23.48D4 [×4], C82Q8, D43Q8 [×2], D46Q16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], Q16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×Q16 [×6], C22×D4, C2×C4○D4, 2- 1+4, D46D4, C22×Q16, D4○D8, D46Q16

Smallest permutation representation of D46Q16
On 64 points
Generators in S64
(1 34 18 43)(2 35 19 44)(3 36 20 45)(4 37 21 46)(5 38 22 47)(6 39 23 48)(7 40 24 41)(8 33 17 42)(9 52 60 28)(10 53 61 29)(11 54 62 30)(12 55 63 31)(13 56 64 32)(14 49 57 25)(15 50 58 26)(16 51 59 27)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 56)(10 49)(11 50)(12 51)(13 52)(14 53)(15 54)(16 55)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 36)(25 61)(26 62)(27 63)(28 64)(29 57)(30 58)(31 59)(32 60)
(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 16 5 12)(2 15 6 11)(3 14 7 10)(4 13 8 9)(17 60 21 64)(18 59 22 63)(19 58 23 62)(20 57 24 61)(25 41 29 45)(26 48 30 44)(27 47 31 43)(28 46 32 42)(33 52 37 56)(34 51 38 55)(35 50 39 54)(36 49 40 53)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4K4L···4Q8A8B8C8D8E···8J
order1222222244444···44···488888···8
size1111222222224···48···822224···4

35 irreducible representations

dim1111111112222244
type++++++++++++--+
imageC1C2C2C2C2C2C2C2C2D4D4D4C4○D4Q162- 1+4D4○D8
kernelD46Q16C2×C2.D8C8×D4C4×Q16C8.18D4C4.Q16C23.48D4C82Q8D43Q8C22⋊C4C4⋊C4C2×D4C8D4C4C2
# reps1211224122114812

Matrix representation of D46Q16 in GL4(𝔽17) generated by

1000
0100
0012
001616
,
16000
01600
0012
00016
,
31400
3300
00160
00016
,
01300
13000
0048
001313
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,2,16],[3,3,0,0,14,3,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,13,0,0,0,0,0,4,13,0,0,8,13] >;

D46Q16 in GAP, Magma, Sage, TeX

D_4\rtimes_6Q_{16}
% in TeX

G:=Group("D4:6Q16");
// GroupNames label

G:=SmallGroup(128,2070);
// by ID

G=gap.SmallGroup(128,2070);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,436,346,4037,1027,124]);
// Polycyclic

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

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