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G = D48SD16order 128 = 27

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

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

Aliases: D48SD16, C42.463C23, C4.432+ 1+4, (D4×Q8)⋊7C2, (C8×D4)⋊32C2, C88D437C2, C4⋊C4.262D4, Q85(C4○D4), D42(Q8⋊C4), Q8⋊D434C2, Q8⋊Q841C2, (C4×SD16)⋊39C2, (C2×D4).352D4, D46D4.5C2, C2.44(Q8○D8), C4.45(C2×SD16), D4.D443C2, C4⋊C4.400C23, C4⋊C8.343C22, (C2×C8).348C23, (C2×C4).490C24, (C4×C8).272C22, C22⋊C4.102D4, C4.SD1629C2, C23.471(C2×D4), C4⋊Q8.141C22, C22.6(C2×SD16), C4.Q8.99C22, (C4×D4).331C22, (C2×D4).222C23, C4⋊D4.72C22, (C4×Q8).147C22, (C2×Q8).207C23, C2.126(D45D4), C2.29(C22×SD16), C22⋊Q8.70C22, C23.47D432C2, C22⋊C8.223C22, (C22×C8).354C22, C22.750(C22×D4), D4⋊C4.150C22, (C22×C4).1134C23, Q8⋊C4.112C22, (C2×SD16).155C22, (C22×Q8).337C22, (C2×D4)(Q8⋊C4), C4.215(C2×C4○D4), (C2×C4).167(C2×D4), (C2×Q8⋊C4)⋊41C2, (C2×C4⋊C4).660C22, SmallGroup(128,2030)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D48SD16
C1C2C4C2×C4C22×C4C22×Q8D4×Q8 — D48SD16
C1C2C2×C4 — D48SD16
C1C22C4×D4 — D48SD16
C1C2C2C2×C4 — D48SD16

Generators and relations for D48SD16
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a2b, dcd=c3 >

Subgroups: 408 in 210 conjugacy classes, 96 normal (38 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×7], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×21], D4 [×4], D4 [×6], Q8 [×2], Q8 [×11], C23 [×2], C23, C42, C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], SD16 [×4], C22×C4 [×2], C22×C4 [×5], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8 [×9], C4○D4 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4, Q8⋊C4, Q8⋊C4 [×8], C4⋊C8, C4.Q8, C4.Q8 [×2], C2×C4⋊C4 [×2], C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8 [×3], C22.D4 [×2], C4⋊Q8 [×2], C4⋊Q8, C22×C8 [×2], C2×SD16, C2×SD16 [×2], C22×Q8 [×2], C2×C4○D4, C2×Q8⋊C4 [×2], C8×D4, C4×SD16, Q8⋊D4 [×2], D4.D4, C88D4 [×2], Q8⋊Q8, C23.47D4 [×2], C4.SD16, D46D4, D4×Q8, D48SD16
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, Q8○D8, D48SD16

Smallest permutation representation of D48SD16
On 64 points
Generators in S64
(1 22 27 42)(2 23 28 43)(3 24 29 44)(4 17 30 45)(5 18 31 46)(6 19 32 47)(7 20 25 48)(8 21 26 41)(9 62 34 53)(10 63 35 54)(11 64 36 55)(12 57 37 56)(13 58 38 49)(14 59 39 50)(15 60 40 51)(16 61 33 52)
(1 63)(2 55)(3 57)(4 49)(5 59)(6 51)(7 61)(8 53)(9 41)(10 22)(11 43)(12 24)(13 45)(14 18)(15 47)(16 20)(17 38)(19 40)(21 34)(23 36)(25 52)(26 62)(27 54)(28 64)(29 56)(30 58)(31 50)(32 60)(33 48)(35 42)(37 44)(39 46)
(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 5)(2 8)(4 6)(9 36)(10 39)(11 34)(12 37)(13 40)(14 35)(15 38)(16 33)(17 19)(18 22)(21 23)(26 28)(27 31)(30 32)(41 43)(42 46)(45 47)(49 60)(50 63)(51 58)(52 61)(53 64)(54 59)(55 62)(56 57)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E···4K4L···4P8A8B8C8D8E···8J
order12222222244444···44···488888···8
size11112222822224···48···822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4SD16C4○D42+ 1+4Q8○D8
kernelD48SD16C2×Q8⋊C4C8×D4C4×SD16Q8⋊D4D4.D4C88D4Q8⋊Q8C23.47D4C4.SD16D46D4D4×Q8C22⋊C4C4⋊C4C2×D4D4Q8C4C2
# reps1211212121112118412

Matrix representation of D48SD16 in GL4(𝔽17) generated by

13000
0400
00160
00016
,
0400
13000
0010
0001
,
16000
0100
00710
00120
,
1000
01600
00160
00161
G:=sub<GL(4,GF(17))| [13,0,0,0,0,4,0,0,0,0,16,0,0,0,0,16],[0,13,0,0,4,0,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,1,0,0,0,0,7,12,0,0,10,0],[1,0,0,0,0,16,0,0,0,0,16,16,0,0,0,1] >;

D48SD16 in GAP, Magma, Sage, TeX

D_4\rtimes_8{\rm SD}_{16}
% in TeX

G:=Group("D4:8SD16");
// GroupNames label

G:=SmallGroup(128,2030);
// by ID

G=gap.SmallGroup(128,2030);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,352,346,4037,1027,124]);
// 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=d*b*d=a^2*b,d*c*d=c^3>;
// generators/relations

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