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G = D4×SD16order 128 = 27

Direct product of D4 and SD16

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

Aliases: D4×SD16, C42.448C23, C4.1352+ 1+4, D42.3C2, C2.65D42, C814(C2×D4), (D4×Q8)⋊6C2, Q85(C2×D4), (C8×D4)⋊30C2, C42(C2×SD16), C88D433C2, C85D426C2, C4⋊C877C22, (C4×C8)⋊44C22, C4⋊C4.254D4, D4.27(C2×D4), C4⋊Q817C22, C4⋊SD1640C2, Q8⋊D432C2, (C4×SD16)⋊35C2, (C2×D4).348D4, C22⋊C4.93D4, (C4×Q8)⋊22C22, C222(C2×SD16), C4.95(C22×D4), C4.Q869C22, C22⋊SD1633C2, D4.D441C2, C4⋊C4.220C23, C22⋊C869C22, (C2×C8).344C23, (C2×C4).479C24, (C22×C8)⋊38C22, C23.465(C2×D4), C22⋊Q813C22, D4⋊C446C22, C2.63(D4○SD16), Q8⋊C485C22, (C22×SD16)⋊28C2, (C2×SD16)⋊49C22, (C2×D4).418C23, (C4×D4).323C22, C4⋊D4.65C22, C41D4.78C22, (C2×Q8).200C23, (C22×Q8)⋊25C22, C2.27(C22×SD16), C22.739(C22×D4), (C22×C4).1123C23, (C22×D4).403C22, (C2×D4)(C2×SD16), (C2×C4).161(C2×D4), SmallGroup(128,2013)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4×SD16
C1C2C22C2×C4C2×D4C22×D4C22×SD16 — D4×SD16
C1C2C2×C4 — D4×SD16
C1C22C4×D4 — D4×SD16
C1C2C2C2×C4 — D4×SD16

Generators and relations for D4×SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c3 >

Subgroups: 624 in 273 conjugacy classes, 104 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×4], C22 [×22], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×15], D4 [×6], D4 [×16], Q8 [×2], Q8 [×9], C23 [×2], C23 [×13], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], SD16 [×4], SD16 [×14], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×16], C2×Q8, C2×Q8 [×2], C2×Q8 [×8], C24 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8 [×2], C41D4, C4⋊Q8, C4⋊Q8, C22×C8 [×2], C2×SD16, C2×SD16 [×8], C2×SD16 [×8], C22×D4 [×2], C22×D4, C22×Q8 [×2], C8×D4, C4×SD16, Q8⋊D4 [×2], C22⋊SD16 [×2], C4⋊SD16, D4.D4, C88D4 [×2], C85D4, D42, D4×Q8, C22×SD16 [×2], D4×SD16
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], SD16 [×4], C2×D4 [×12], C24, C2×SD16 [×6], C22×D4 [×2], 2+ 1+4, D42, C22×SD16, D4○SD16, D4×SD16

Smallest permutation representation of D4×SD16
On 32 points
Generators in S32
(1 14 20 32)(2 15 21 25)(3 16 22 26)(4 9 23 27)(5 10 24 28)(6 11 17 29)(7 12 18 30)(8 13 19 31)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 25)(7 26)(8 27)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)
(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 20)(2 23)(3 18)(4 21)(5 24)(6 19)(7 22)(8 17)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4SD162+ 1+4D4○SD16
kernelD4×SD16C8×D4C4×SD16Q8⋊D4C22⋊SD16C4⋊SD16D4.D4C88D4C85D4D42D4×Q8C22×SD16C22⋊C4C4⋊C4SD16C2×D4D4C4C2
# reps1112211211122141812

Matrix representation of D4×SD16 in GL4(𝔽17) generated by

1000
0100
00115
00116
,
16000
01600
00115
00016
,
101000
12000
0010
0001
,
1000
161600
00160
00016
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,1,1,0,0,15,16],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,15,16],[10,12,0,0,10,0,0,0,0,0,1,0,0,0,0,1],[1,16,0,0,0,16,0,0,0,0,16,0,0,0,0,16] >;

D4×SD16 in GAP, Magma, Sage, TeX

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

G:=Group("D4xSD16");
// GroupNames label

G:=SmallGroup(128,2013);
// by ID

G=gap.SmallGroup(128,2013);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,2019,346,2804,1411,375,172]);
// 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,b*c=c*b,b*d=d*b,d*c*d=c^3>;
// generators/relations

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