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G = SD323C4order 128 = 27

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

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

Aliases: SD323C4, C42.132D4, C165(C2×C4), Q163(C2×C4), D8.3(C2×C4), C165C41C2, C4.28(C4×D4), C2.16(C4×D8), (C4×Q16)⋊37C2, C163C411C2, (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.Q3216C2, 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

C1C8 — SD323C4
C1C2C4C2×C4C2×C8C4×C8C4×D8 — SD323C4
C1C2C4C8 — SD323C4
C1C22C42C4×C8 — SD323C4
C1C2C2C2C2C4C4C2×C8 — SD323C4

Generators and relations for SD323C4
 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 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C8 [×2], C8, C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×3], C23, C16 [×2], C16, C42, C42, C22⋊C4, C4⋊C4 [×3], C2×C8 [×2], D8 [×2], D8, Q16 [×2], Q16, C22×C4, C2×D4, C2×Q8, C4×C8, D4⋊C4, Q8⋊C4, C2.D8 [×2], C2×C16 [×2], SD32 [×4], C4×D4, C4×Q8, C2×D8, C2×Q16, C165C4, C2.D16, C2.Q32, C163C4, C4×D8, C4×Q16, C2×SD32, SD323C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D8 [×2], C22×C4, C2×D4, C4○D4, C4×D4, C2×D8, C4○D8, C4×D8, C16⋊C22, Q32⋊C2, SD323C4

Smallest permutation representation of SD323C4
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 48)(2 39)(3 46)(4 37)(5 44)(6 35)(7 42)(8 33)(9 40)(10 47)(11 38)(12 45)(13 36)(14 43)(15 34)(16 41)(17 58)(18 49)(19 56)(20 63)(21 54)(22 61)(23 52)(24 59)(25 50)(26 57)(27 64)(28 55)(29 62)(30 53)(31 60)(32 51)
(1 26 48 57)(2 19 33 50)(3 28 34 59)(4 21 35 52)(5 30 36 61)(6 23 37 54)(7 32 38 63)(8 25 39 56)(9 18 40 49)(10 27 41 58)(11 20 42 51)(12 29 43 60)(13 22 44 53)(14 31 45 62)(15 24 46 55)(16 17 47 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), (1,48)(2,39)(3,46)(4,37)(5,44)(6,35)(7,42)(8,33)(9,40)(10,47)(11,38)(12,45)(13,36)(14,43)(15,34)(16,41)(17,58)(18,49)(19,56)(20,63)(21,54)(22,61)(23,52)(24,59)(25,50)(26,57)(27,64)(28,55)(29,62)(30,53)(31,60)(32,51), (1,26,48,57)(2,19,33,50)(3,28,34,59)(4,21,35,52)(5,30,36,61)(6,23,37,54)(7,32,38,63)(8,25,39,56)(9,18,40,49)(10,27,41,58)(11,20,42,51)(12,29,43,60)(13,22,44,53)(14,31,45,62)(15,24,46,55)(16,17,47,64)>;

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

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

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
kernelSD323C4C165C4C2.D16C2.Q32C163C4C4×D8C4×Q16C2×SD32SD32C42C2×C8C8C2×C4C4C2C2
# reps1111111181124422

Matrix representation of SD323C4 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] >;

SD323C4 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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