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G = SD1610D4order 128 = 27

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

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

Aliases: SD1610D4, C42.449C23, C4.1362+ 1+4, C2.66D42, (C8×D4)⋊11C2, C88D47C2, C8.84(C2×D4), C22⋊D88C2, D4⋊D46C2, D45D47C2, C4⋊C869C22, C4⋊C4.255D4, (C4×C8)⋊19C22, Q85D46C2, (C4×SD16)⋊9C2, D4.28(C2×D4), C22⋊C42SD16, Q8.26(C2×D4), D4.7D45C2, D4.2D46C2, (C2×D4).228D4, C8.12D48C2, C223(C4○D8), C22⋊Q167C2, Q8.D45C2, (C4×Q8)⋊23C22, C4.96(C22×D4), C4.Q870C22, C4⋊C4.221C23, C22⋊C862C22, (C2×C4).480C24, (C2×C8).179C23, (C22×C8)⋊19C22, C22⋊C4.193D4, (C2×Q16)⋊48C22, (C2×D8).34C22, C23.466(C2×D4), C22⋊Q814C22, D4⋊C447C22, C2.64(D4○SD16), Q8⋊C472C22, (C22×SD16)⋊29C2, (C2×SD16)⋊81C22, (C4×D4).324C22, (C2×D4).215C23, C4⋊D4.66C22, C4.4D416C22, (C2×Q8).201C23, C22.740(C22×D4), (C22×C4).1124C23, (C22×D4).404C22, (C22×Q8).334C22, (C2×C4○D8)⋊10C2, C2.53(C2×C4○D8), C22⋊C4(C2×SD16), (C2×C4).918(C2×D4), (C2×C4○D4)⋊18C22, SmallGroup(128,2014)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD1610D4
C1C2C22C2×C4C2×D4C22×D4C22×SD16 — SD1610D4
C1C2C2×C4 — SD1610D4
C1C22C4×D4 — SD1610D4
C1C2C2C2×C4 — SD1610D4

Generators and relations for SD1610D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 544 in 252 conjugacy classes, 96 normal (84 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×19], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×16], D4 [×2], D4 [×16], Q8 [×2], Q8 [×7], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×C8 [×4], D8 [×3], SD16 [×4], SD16 [×8], Q16 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8 [×3], C2×Q8 [×4], C4○D4 [×6], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2×C22⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C22≀C2, C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C22.D4, C4.4D4 [×2], C4.4D4, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×5], C2×SD16 [×4], C2×Q16 [×2], C4○D8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×2], C8×D4, C4×SD16, C22⋊D8, D4⋊D4, C22⋊Q16, D4.7D4, D4.2D4, Q8.D4, C88D4 [×2], C8.12D4, D45D4, Q85D4, C22×SD16, C2×C4○D8, SD1610D4
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○SD16, SD1610D4

Smallest permutation representation of SD1610D4
On 32 points
Generators in S32
(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 18)(2 21)(3 24)(4 19)(5 22)(6 17)(7 20)(8 23)(9 25)(10 28)(11 31)(12 26)(13 29)(14 32)(15 27)(16 30)
(1 16 18 26)(2 9 19 27)(3 10 20 28)(4 11 21 29)(5 12 22 30)(6 13 23 31)(7 14 24 32)(8 15 17 25)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 31)(7 32)(8 25)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A···4F4G4H4I4J4K4L4M4N8A8B8C8D8E···8J
order122222222224···44444444488888···8
size111122444882···24444888822224···4

35 irreducible representations

dim1111111111111112222244
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4D4○SD16
kernelSD1610D4C8×D4C4×SD16C22⋊D8D4⋊D4C22⋊Q16D4.7D4D4.2D4Q8.D4C88D4C8.12D4D45D4Q85D4C22×SD16C2×C4○D8C22⋊C4C4⋊C4SD16C2×D4C22C4C2
# reps1111111112111112141812

Matrix representation of SD1610D4 in GL4(𝔽17) generated by

1000
0100
0077
0050
,
16000
01600
0010
001616
,
16100
15100
0048
001313
,
16100
0100
0048
001313
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,7,5,0,0,7,0],[16,0,0,0,0,16,0,0,0,0,1,16,0,0,0,16],[16,15,0,0,1,1,0,0,0,0,4,13,0,0,8,13],[16,0,0,0,1,1,0,0,0,0,4,13,0,0,8,13] >;

SD1610D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_{10}D_4
% in TeX

G:=Group("SD16:10D4");
// GroupNames label

G:=SmallGroup(128,2014);
// by ID

G=gap.SmallGroup(128,2014);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,2019,346,2804,1411,375,172]);
// Polycyclic

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

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