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G = SD1611D4order 128 = 27

2nd semidirect product of SD16 and D4 acting through Inn(SD16)

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

Aliases: SD1611D4, C42.451C23, C4.1382+ 1+4, C2.68D42, C4⋊C42SD16, (C8×D4)⋊13C2, C43(C4○D8), C88D48C2, C8.85(C2×D4), D4⋊D47C2, D46D46C2, C4⋊D812C2, C85D427C2, C4⋊C4.406D4, Q86D45C2, D4.30(C2×D4), Q8.27(C2×D4), C42Q1612C2, (C4×SD16)⋊36C2, (C2×D4).230D4, D4.7D47C2, C22⋊C4.95D4, C4.98(C22×D4), C4⋊C8.294C22, C4⋊C4.223C23, (C2×C4).482C24, (C2×C8).602C23, (C4×C8).271C22, C23.104(C2×D4), C4⋊Q8.138C22, C2.65(D4○SD16), (C2×D4).216C23, (C4×D4).325C22, (C2×D8).136C22, C41D4.79C22, C4⋊D4.67C22, (C4×Q8).143C22, (C2×Q8).203C23, C4.Q8.165C22, C22⋊Q8.66C22, D4⋊C4.10C22, C22⋊C8.199C22, (C22×C8).193C22, (C2×Q16).131C22, (C2×SD16).94C22, C22.742(C22×D4), (C22×C4).1126C23, Q8⋊C4.178C22, C4⋊C4(C2×SD16), (C2×C4○D8)⋊12C2, C2.55(C2×C4○D8), (C2×C4).920(C2×D4), (C2×C4○D4).193C22, SmallGroup(128,2016)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD1611D4
C1C2C22C2×C4C2×D4C2×C4○D4C2×C4○D8 — SD1611D4
C1C2C2×C4 — SD1611D4
C1C22C4×D4 — SD1611D4
C1C2C2C2×C4 — SD1611D4

Generators and relations for SD1611D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a3, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 512 in 247 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×16], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×19], D4 [×2], D4 [×20], Q8 [×2], Q8 [×5], C23 [×2], C23 [×3], C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], D8 [×4], SD16 [×4], SD16 [×6], Q16 [×4], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×D4 [×8], C2×Q8, C2×Q8 [×2], C4○D4 [×12], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C2×C4⋊C4, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C4⋊D4 [×3], C22⋊Q8 [×2], C22.D4 [×2], C41D4, C41D4, C4⋊Q8, C22×C8 [×2], C2×D8 [×2], C2×SD16, C2×SD16 [×4], C2×Q16 [×2], C4○D8 [×8], C2×C4○D4 [×4], C8×D4, C4×SD16, D4⋊D4 [×2], D4.7D4 [×2], C4⋊D8, C42Q16, C88D4 [×2], C85D4, D46D4, Q86D4, C2×C4○D8 [×2], SD1611D4
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, SD1611D4

Smallest permutation representation of SD1611D4
On 64 points
Generators in S64
(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 39)(2 34)(3 37)(4 40)(5 35)(6 38)(7 33)(8 36)(9 28)(10 31)(11 26)(12 29)(13 32)(14 27)(15 30)(16 25)(17 49)(18 52)(19 55)(20 50)(21 53)(22 56)(23 51)(24 54)(41 63)(42 58)(43 61)(44 64)(45 59)(46 62)(47 57)(48 60)
(1 25 23 62)(2 26 24 63)(3 27 17 64)(4 28 18 57)(5 29 19 58)(6 30 20 59)(7 31 21 60)(8 32 22 61)(9 52 47 40)(10 53 48 33)(11 54 41 34)(12 55 42 35)(13 56 43 36)(14 49 44 37)(15 50 45 38)(16 51 46 39)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 33)(7 34)(8 35)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(17 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A···4H4I4J4K4L4M4N4O8A8B8C8D8E···8J
order12222222224···4444444488888···8
size11114444882···2444888822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D4C4○D82+ 1+4D4○SD16
kernelSD1611D4C8×D4C4×SD16D4⋊D4D4.7D4C4⋊D8C42Q16C88D4C85D4D46D4Q86D4C2×C4○D8C22⋊C4C4⋊C4SD16C2×D4C4C4C2
# reps1112211211122141812

Matrix representation of SD1611D4 in GL4(𝔽17) generated by

16000
01600
00512
0055
,
16000
01600
0033
00314
,
1200
161600
00160
00016
,
1000
161600
00013
0040
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,5,5,0,0,12,5],[16,0,0,0,0,16,0,0,0,0,3,3,0,0,3,14],[1,16,0,0,2,16,0,0,0,0,16,0,0,0,0,16],[1,16,0,0,0,16,0,0,0,0,0,4,0,0,13,0] >;

SD1611D4 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_{11}D_4
% in TeX

G:=Group("SD16:11D4");
// GroupNames label

G:=SmallGroup(128,2016);
// by ID

G=gap.SmallGroup(128,2016);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,352,2019,346,2804,1411,375,172]);
// 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,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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