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G = D45Q16order 128 = 27

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

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

Aliases: D45Q16, C42.464C23, C4.442+ 1+4, (C8×D4).9C2, (D4×Q8).5C2, C4⋊C4.263D4, (C4×Q16)⋊11C2, C4.28(C2×Q16), D43(Q8⋊C4), C4.Q1612C2, C42Q1614C2, (C2×D4).353D4, (C4×C8).83C22, D43Q8.4C2, C22⋊Q169C2, C8.18D412C2, C22.5(C2×Q16), C4⋊C4.401C23, C4⋊C8.296C22, (C2×C8).183C23, (C2×C4).491C24, Q8.16(C4○D4), C22⋊C4.103D4, C4.SD1614C2, C23.472(C2×D4), C4⋊Q8.142C22, C2.19(C22×Q16), C2.D8.53C22, C2.68(D4○SD16), (C4×D4).332C22, C23.48D48C2, (C4×Q8).148C22, (C2×Q8).208C23, C2.127(D45D4), C22⋊Q8.71C22, C22⋊C8.184C22, (C22×C8).160C22, Q8⋊C4.11C22, (C2×Q16).132C22, C22.751(C22×D4), (C22×C4).1135C23, (C22×Q8).338C22, C4.216(C2×C4○D4), (C2×C4).168(C2×D4), (C2×Q8⋊C4)⋊25C2, (C2×C4⋊C4).661C22, SmallGroup(128,2031)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D45Q16
C1C2C4C2×C4C22×C4C22×Q8D4×Q8 — D45Q16
C1C2C2×C4 — D45Q16
C1C22C4×D4 — D45Q16
C1C2C2C2×C4 — D45Q16

Generators and relations for D45Q16
 G = < a,b,c,d | a4=b2=c8=1, d2=c4, bab=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd-1=c-1 >

Subgroups: 352 in 194 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×4], C22 [×4], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×4], Q8 [×2], Q8 [×12], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C2×Q8 [×9], C4×C8, C22⋊C8 [×2], Q8⋊C4 [×2], Q8⋊C4 [×8], C4⋊C8, C2.D8, C2.D8 [×2], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×4], C42.C2, C4⋊Q8 [×2], C4⋊Q8, C22×C8 [×2], C2×Q16, C2×Q16 [×2], C22×Q8 [×2], C2×Q8⋊C4 [×2], C8×D4, C4×Q16, C22⋊Q16 [×2], C42Q16, C8.18D4 [×2], C4.Q16, C23.48D4 [×2], C4.SD16, D4×Q8, D43Q8, D45Q16
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, D45D4, C22×Q16, D4○SD16, D45Q16

Smallest permutation representation of D45Q16
On 64 points
Generators in S64
(1 15 59 40)(2 16 60 33)(3 9 61 34)(4 10 62 35)(5 11 63 36)(6 12 64 37)(7 13 57 38)(8 14 58 39)(17 28 45 53)(18 29 46 54)(19 30 47 55)(20 31 48 56)(21 32 41 49)(22 25 42 50)(23 26 43 51)(24 27 44 52)
(1 36)(2 12)(3 38)(4 14)(5 40)(6 16)(7 34)(8 10)(9 57)(11 59)(13 61)(15 63)(17 32)(18 50)(19 26)(20 52)(21 28)(22 54)(23 30)(24 56)(25 46)(27 48)(29 42)(31 44)(33 64)(35 58)(37 60)(39 62)(41 53)(43 55)(45 49)(47 51)
(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 41 13 45)(10 48 14 44)(11 47 15 43)(12 46 16 42)(17 34 21 38)(18 33 22 37)(19 40 23 36)(20 39 24 35)(49 57 53 61)(50 64 54 60)(51 63 55 59)(52 62 56 58)

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim1111111111112222244
type+++++++++++++++-+
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4Q16C4○D42+ 1+4D4○SD16
kernelD45Q16C2×Q8⋊C4C8×D4C4×Q16C22⋊Q16C42Q16C8.18D4C4.Q16C23.48D4C4.SD16D4×Q8D43Q8C22⋊C4C4⋊C4C2×D4D4Q8C4C2
# reps1211212121112118412

Matrix representation of D45Q16 in GL4(𝔽17) generated by

1000
0100
0012
001616
,
16000
01600
0012
00016
,
14300
141400
0048
001313
,
161000
10100
0010
0001
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],[14,14,0,0,3,14,0,0,0,0,4,13,0,0,8,13],[16,10,0,0,10,1,0,0,0,0,1,0,0,0,0,1] >;

D45Q16 in GAP, Magma, Sage, TeX

D_4\rtimes_5Q_{16}
% in TeX

G:=Group("D4:5Q16");
// GroupNames label

G:=SmallGroup(128,2031);
// by ID

G=gap.SmallGroup(128,2031);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,352,346,4037,1027,124]);
// 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,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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