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G = SD32⋊3C4  order 128 = 27

1st semidirect product of SD32 and C4 acting via C4/C2=C2

p-group, metabelian, nilpotent (class 4), monomial

Aliases: SD32⋊3C4, C42.132D4, C16⋊5(C2×C4), Q16⋊3(C2×C4), D8.3(C2×C4), C16⋊5C4⋊1C2, C4.28(C4×D4), C2.16(C4×D8), (C4×Q16)⋊37C2, C16⋊3C4⋊11C2, (C4×D8).19C2, (C2×C4).109D8, (C2×C8).210D4, C2.D16.6C2, C4.14(C4○D8), C8.41(C4○D4), C8.38(C22×C4), (C2×SD32).1C2, C22.64(C2×D8), C2.Q32⋊16C2, C2.4(C16⋊C22), (C2×C8).505C23, (C2×C16).19C22, (C4×C8).218C22, C2.4(Q32⋊C2), (C2×D8).105C22, C2.D8.153C22, (C2×Q16).104C22, (C2×C4).771(C2×D4), SmallGroup(128,907)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C8 — SD32⋊3C4
Chief series
C1 — C2 — C4 — C2×C4 — C2×C8 — C4×C8 — C4×D8 — SD32⋊3C4
Lower central
C1 — C2 — C4 — C8 — SD32⋊3C4
Upper central
C1 — C22 — C42 — C4×C8 — SD32⋊3C4
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C4 — C2×C8 — SD32⋊3C4

Generators and relations for SD32⋊3C4
 G = < a,b,c | a16=b2=c4=1, bab=a7, cac-1=a9, bc=cb >

Subgroups: 196 in 80 conjugacy classes, 40 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C8, C2×C4, C2×C4, D4, Q8, C23, C16, C16, C42, C42, C22⋊C4, C4⋊C4, C2×C8, D8, D8, Q16, Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8, C2×C16, SD32, C4×D4, C4×Q8, C2×D8, C2×Q16, C16⋊5C4, C2.D16, C2.Q32, C16⋊3C4, C4×D8, C4×Q16, C2×SD32, SD32⋊3C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D8, C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C16⋊C22, Q32⋊C2, SD32⋊3C4

Smallest permutation representation of SD32⋊3C4
►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 46)(3 37)(4 44)(5 35)(6 42)(7 33)(8 40)(9 47)(10 38)(11 45)(12 36)(13 43)(14 34)(15 41)(16 48)(17 63)(18 54)(19 61)(20 52)(21 59)(22 50)(23 57)(24 64)(25 55)(26 62)(27 53)(28 60)(29 51)(30 58)(31 49)(32 56)

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G···4L8A8B8C8D8E8F16A···16H
order1222224···44···488888816···16
size1111882···28···82222444···4

32 irreducible representations

dim1111111112222244
type++++++++++++-
imageC1C2C2C2C2C2C2C2C4D4D4C4○D4D8C4○D8C16⋊C22Q32⋊C2
kernelSD32⋊3C4C16⋊5C4C2.D16C2.Q32C16⋊3C4C4×D8C4×Q16C2×SD32SD32C42C2×C8C8C2×C4C4C2C2
# reps1111111181124422

Matrix representation of SD32⋊3C4 ►in GL6(𝔽17)

680000
6110000
00144121
0013141612
00516313
001543
,
1600000
1010000
001000
0001600
000010
0000016
,
400000
040000
000010
000001
001000
000100

G:=sub<GL(6,GF(17))| [6,6,0,0,0,0,8,11,0,0,0,0,0,0,14,13,5,1,0,0,4,14,16,5,0,0,12,16,3,4,0,0,1,12,13,3],[16,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

SD32⋊3C4 in GAP, Magma, Sage, TeX 

{\rm SD}_{32}\rtimes_3C_4
 
% in TeX
 
G:=Group("SD32:3C4");
 
// GroupNames label
 
G:=SmallGroup(128,907);
 
// by ID
 
G=gap.SmallGroup(128,907);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,1123,570,360,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^2=c^4=1,b*a*b=a^7,c*a*c^-1=a^9,b*c=c*b>;
 
// generators/relations
 

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