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G = D47SD16order 128 = 27

1st semidirect product of D4 and SD16 acting through Inn(D4)

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

Aliases: D47SD16, C42.460C23, C4.402+ 1+4, D42.4C2, (C8×D4)⋊31C2, C88D434C2, C4⋊C878C22, C4⋊C4.259D4, (C4×C8)⋊45C22, D43Q82C2, C4⋊Q821C22, D43(D4⋊C4), C4⋊SD1642C2, D42Q839C2, (C4×SD16)⋊37C2, (C2×D4).351D4, C2.44(D4○D8), C4.4D829C2, C22⋊C4.99D4, (C4×Q8)⋊24C22, C4.44(C2×SD16), C4.Q836C22, D4.16(C4○D4), C22⋊SD1634C2, C4⋊C4.399C23, C22⋊C870C22, (C2×C8).346C23, (C2×C4).487C24, (C22×C8)⋊39C22, C23.470(C2×D4), C22⋊Q815C22, D4⋊C448C22, C22.5(C2×SD16), Q8⋊C462C22, (C2×SD16)⋊50C22, (C4×D4).329C22, (C2×D4).220C23, C4⋊D4.70C22, C41D4.83C22, (C2×Q8).204C23, C2.123(D45D4), C2.28(C22×SD16), C23.46D431C2, C22.747(C22×D4), (C22×C4).1131C23, (C22×D4).406C22, (C2×C4⋊C4)⋊56C22, C4.212(C2×C4○D4), (C2×C4).164(C2×D4), (C2×D4⋊C4)⋊40C2, SmallGroup(128,2027)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D47SD16
C1C2C4C2×C4C22×C4C22×D4D42 — D47SD16
C1C2C2×C4 — D47SD16
C1C22C4×D4 — D47SD16
C1C2C2C2×C4 — D47SD16

Generators and relations for D47SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 552 in 231 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×4], C22 [×22], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×13], D4 [×6], D4 [×16], Q8 [×3], C23 [×2], C23 [×13], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×6], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], SD16 [×4], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×16], C2×Q8, C2×Q8, C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×8], Q8⋊C4, C4⋊C8, C4.Q8, C4.Q8 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8 [×2], C42.C2, C41D4, C4⋊Q8, C22×C8 [×2], C2×SD16, C2×SD16 [×2], C22×D4 [×2], C22×D4, C2×D4⋊C4 [×2], C8×D4, C4×SD16, C22⋊SD16 [×2], C4⋊SD16, C88D4 [×2], D42Q8, C23.46D4 [×2], C4.4D8, D42, D43Q8, D47SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C4○D4 [×2], C24, C2×SD16 [×6], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C22×SD16, D4○D8, D47SD16

Smallest permutation representation of D47SD16
On 32 points
Generators in S32
(1 9 28 19)(2 10 29 20)(3 11 30 21)(4 12 31 22)(5 13 32 23)(6 14 25 24)(7 15 26 17)(8 16 27 18)
(1 23)(2 14)(3 17)(4 16)(5 19)(6 10)(7 21)(8 12)(9 32)(11 26)(13 28)(15 30)(18 31)(20 25)(22 27)(24 29)
(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)
(1 32)(2 27)(3 30)(4 25)(5 28)(6 31)(7 26)(8 29)(9 23)(10 18)(11 21)(12 24)(13 19)(14 22)(15 17)(16 20)

G:=sub<Sym(32)| (1,9,28,19)(2,10,29,20)(3,11,30,21)(4,12,31,22)(5,13,32,23)(6,14,25,24)(7,15,26,17)(8,16,27,18), (1,23)(2,14)(3,17)(4,16)(5,19)(6,10)(7,21)(8,12)(9,32)(11,26)(13,28)(15,30)(18,31)(20,25)(22,27)(24,29), (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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20)>;

G:=Group( (1,9,28,19)(2,10,29,20)(3,11,30,21)(4,12,31,22)(5,13,32,23)(6,14,25,24)(7,15,26,17)(8,16,27,18), (1,23)(2,14)(3,17)(4,16)(5,19)(6,10)(7,21)(8,12)(9,32)(11,26)(13,28)(15,30)(18,31)(20,25)(22,27)(24,29), (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), (1,32)(2,27)(3,30)(4,25)(5,28)(6,31)(7,26)(8,29)(9,23)(10,18)(11,21)(12,24)(13,19)(14,22)(15,17)(16,20) );

G=PermutationGroup([(1,9,28,19),(2,10,29,20),(3,11,30,21),(4,12,31,22),(5,13,32,23),(6,14,25,24),(7,15,26,17),(8,16,27,18)], [(1,23),(2,14),(3,17),(4,16),(5,19),(6,10),(7,21),(8,12),(9,32),(11,26),(13,28),(15,30),(18,31),(20,25),(22,27),(24,29)], [(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)], [(1,32),(2,27),(3,30),(4,25),(5,28),(6,31),(7,26),(8,29),(9,23),(10,18),(11,21),(12,24),(13,19),(14,22),(15,17),(16,20)])

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E···4I4J4K4L4M8A8B8C8D8E···8J
order12222222222244444···4444488888···8
size11112222448822224···4888822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4SD16C4○D42+ 1+4D4○D8
kernelD47SD16C2×D4⋊C4C8×D4C4×SD16C22⋊SD16C4⋊SD16C88D4D42Q8C23.46D4C4.4D8D42D43Q8C22⋊C4C4⋊C4C2×D4D4D4C4C2
# reps1211212121112118412

Matrix representation of D47SD16 in GL4(𝔽17) generated by

1200
161600
00160
00016
,
1200
01600
0010
0001
,
13900
4400
00512
0055
,
16000
01600
00160
0001
G:=sub<GL(4,GF(17))| [1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,2,16,0,0,0,0,1,0,0,0,0,1],[13,4,0,0,9,4,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

D47SD16 in GAP, Magma, Sage, TeX

D_4\rtimes_7{\rm SD}_{16}
% in TeX

G:=Group("D4:7SD16");
// GroupNames label

G:=SmallGroup(128,2027);
// by ID

G=gap.SmallGroup(128,2027);
# by ID

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

G:=Group<a,b,c,d|a^4=b^2=c^8=d^2=1,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=c^3>;
// generators/relations

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