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G = SD164Q8order 128 = 27

The semidirect product of SD16 and Q8 acting through Inn(SD16)

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

Aliases: SD164Q8, C42.522C23, C4.962- 1+4, C4⋊C43SD16, (C8×Q8)⋊15C2, Q8.8(C2×Q8), C2.39(D4×Q8), C8.38(C2×Q8), C4⋊C4.416D4, Q83Q89C2, C83Q828C2, Q8.Q811C2, D4.10(C2×Q8), D4.Q8.1C2, C4.Q1617C2, C4.77(C4○D8), (C2×Q8).187D4, D43Q8.9C2, C8.5Q811C2, C4.39(C22×Q8), C4⋊C4.270C23, C4⋊C8.328C22, (C2×C4).573C24, (C4×C8).124C22, (C2×C8).374C23, D4⋊Q8.11C2, (C4×SD16).15C2, C4⋊Q8.202C22, C2.D8.72C22, (C4×D4).211C22, (C2×D4).436C23, (C2×Q8).408C23, (C4×Q8).312C22, C4.Q8.116C22, C2.107(D4○SD16), C22.833(C22×D4), C42.C2.72C22, D4⋊C4.173C22, Q8⋊C4.186C22, (C2×SD16).180C22, C2.77(C2×C4○D8), (C2×C4).179(C2×D4), SmallGroup(128,2113)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — SD164Q8
C1C2C4C2×C4C42C4×D4D43Q8 — SD164Q8
C1C2C2×C4 — SD164Q8
C1C22C4×Q8 — SD164Q8
C1C2C2C2×C4 — SD164Q8

Generators and relations for SD164Q8
 G = < a,b,c,d | a8=b2=c4=1, d2=c2, bab=a3, ac=ca, ad=da, bc=cb, dbd-1=a4b, dcd-1=c-1 >

Subgroups: 288 in 169 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×13], C22, C22 [×4], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×4], C2×C4 [×11], D4 [×2], D4, Q8 [×2], Q8 [×6], C23, C42, C42 [×2], C42 [×3], C22⋊C4 [×3], C4⋊C4 [×3], C4⋊C4 [×6], C4⋊C4 [×13], C2×C8 [×2], C2×C8 [×2], SD16 [×4], C22×C4 [×3], C2×D4, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C4×C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C4.Q8, C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4, C4×D4, C4×D4 [×2], C4×Q8 [×2], C4×Q8 [×2], C4×Q8, C22⋊Q8 [×3], C42.C2 [×4], C42.C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C4×SD16, C4×SD16 [×2], C8×Q8, D4⋊Q8, C4.Q16, D4.Q8 [×2], Q8.Q8 [×2], C83Q8, C8.5Q8 [×2], D43Q8, Q83Q8, SD164Q8
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C24, C4○D8 [×2], C22×D4, C22×Q8, 2- 1+4, D4×Q8, C2×C4○D8, D4○SD16, SD164Q8

Smallest permutation representation of SD164Q8
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 18)(2 21)(3 24)(4 19)(5 22)(6 17)(7 20)(8 23)(9 55)(10 50)(11 53)(12 56)(13 51)(14 54)(15 49)(16 52)(25 43)(26 46)(27 41)(28 44)(29 47)(30 42)(31 45)(32 48)(33 57)(34 60)(35 63)(36 58)(37 61)(38 64)(39 59)(40 62)
(1 9 22 51)(2 10 23 52)(3 11 24 53)(4 12 17 54)(5 13 18 55)(6 14 19 56)(7 15 20 49)(8 16 21 50)(25 62 41 38)(26 63 42 39)(27 64 43 40)(28 57 44 33)(29 58 45 34)(30 59 46 35)(31 60 47 36)(32 61 48 37)
(1 28 22 44)(2 29 23 45)(3 30 24 46)(4 31 17 47)(5 32 18 48)(6 25 19 41)(7 26 20 42)(8 27 21 43)(9 33 51 57)(10 34 52 58)(11 35 53 59)(12 36 54 60)(13 37 55 61)(14 38 56 62)(15 39 49 63)(16 40 50 64)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E4A···4H4I···4M4N···4S8A8B8C8D8E···8J
order1222224···44···44···488888···8
size1111442···24···48···822224···4

35 irreducible representations

dim11111111111222244
type++++++++++++-+-
imageC1C2C2C2C2C2C2C2C2C2C2D4Q8D4C4○D82- 1+4D4○SD16
kernelSD164Q8C4×SD16C8×Q8D4⋊Q8C4.Q16D4.Q8Q8.Q8C83Q8C8.5Q8D43Q8Q83Q8C4⋊C4SD16C2×Q8C4C4C2
# reps13111221211341812

Matrix representation of SD164Q8 in GL4(𝔽17) generated by

7700
5000
00160
00016
,
16000
1100
00160
00016
,
1000
0100
0012
001616
,
4800
131300
0040
001313
G:=sub<GL(4,GF(17))| [7,5,0,0,7,0,0,0,0,0,16,0,0,0,0,16],[16,1,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,2,16],[4,13,0,0,8,13,0,0,0,0,4,13,0,0,0,13] >;

SD164Q8 in GAP, Magma, Sage, TeX

{\rm SD}_{16}\rtimes_4Q_8
% in TeX

G:=Group("SD16:4Q8");
// GroupNames label

G:=SmallGroup(128,2113);
// by ID

G=gap.SmallGroup(128,2113);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,568,758,346,80,4037,1027,124]);
// Polycyclic

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

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