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G = C169D4order 128 = 27

3rd semidirect product of C16 and D4 acting via D4/C22=C2

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C169D4, C221M5(2), C4⋊C4.9C8, C4⋊C1617C2, C4⋊C8.22C4, (C2×D4).9C8, C2.16(C8×D4), (C4×D4).19C4, C165C410C2, (C8×D4).17C2, C8.139(C2×D4), C4.176(C4×D4), C22⋊C4.5C8, C22⋊C1615C2, (C22×C16)⋊13C2, C4.58(C8○D4), C2.7(D4○C16), C22⋊C8.21C4, C23.23(C2×C8), C2.9(C2×M5(2)), C8.102(C4○D4), (C2×M5(2))⋊19C2, (C2×C16).54C22, C42.171(C2×C4), (C4×C8).323C22, (C2×C8).633C23, C22.52(C22×C8), (C22×C8).502C22, (C2×C4).29(C2×C8), (C2×C8).151(C2×C4), (C22×C4).290(C2×C4), (C2×C4).618(C22×C4), SmallGroup(128,900)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C169D4
C1C2C4C8C2×C8C2×C16C22×C16 — C169D4
C1C22 — C169D4
C1C2×C8 — C169D4
C1C2C2C2C2C4C4C2×C8 — C169D4

Generators and relations for C169D4
 G = < a,b,c | a16=b4=c2=1, bab-1=cac=a9, cbc=b-1 >

Subgroups: 116 in 82 conjugacy classes, 50 normal (46 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×4], D4 [×2], C23 [×2], C16 [×2], C16 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C2×D4, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C16 [×4], C2×C16 [×2], M5(2) [×2], C4×D4, C22×C8 [×2], C165C4, C22⋊C16 [×2], C4⋊C16, C8×D4, C22×C16, C2×M5(2), C169D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, C2×C8 [×6], C22×C4, C2×D4, C4○D4, M5(2) [×2], C4×D4, C22×C8, C8○D4, C8×D4, C2×M5(2), D4○C16, C169D4

Smallest permutation representation of C169D4
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 62 29)(2 48 63 22)(3 41 64 31)(4 34 49 24)(5 43 50 17)(6 36 51 26)(7 45 52 19)(8 38 53 28)(9 47 54 21)(10 40 55 30)(11 33 56 23)(12 42 57 32)(13 35 58 25)(14 44 59 18)(15 37 60 27)(16 46 61 20)
(1 62)(2 55)(3 64)(4 57)(5 50)(6 59)(7 52)(8 61)(9 54)(10 63)(11 56)(12 49)(13 58)(14 51)(15 60)(16 53)(18 26)(20 28)(22 30)(24 32)(34 42)(36 44)(38 46)(40 48)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222224444444448···88888888816···1616···16
size11112241111224441···1222244442···24···4

56 irreducible representations

dim111111111111122222
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C8C8C8D4C4○D4C8○D4M5(2)D4○C16
kernelC169D4C165C4C22⋊C16C4⋊C16C8×D4C22×C16C2×M5(2)C22⋊C8C4⋊C8C4×D4C22⋊C4C4⋊C4C2×D4C16C8C4C22C2
# reps112111142284422488

Matrix representation of C169D4 in GL4(𝔽17) generated by

2000
0200
0092
0038
,
0100
16000
00106
0037
,
16000
0100
00160
0091
G:=sub<GL(4,GF(17))| [2,0,0,0,0,2,0,0,0,0,9,3,0,0,2,8],[0,16,0,0,1,0,0,0,0,0,10,3,0,0,6,7],[16,0,0,0,0,1,0,0,0,0,16,9,0,0,0,1] >;

C169D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_9D_4
% in TeX

G:=Group("C16:9D4");
// GroupNames label

G:=SmallGroup(128,900);
// by ID

G=gap.SmallGroup(128,900);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,100,102,124]);
// Polycyclic

G:=Group<a,b,c|a^16=b^4=c^2=1,b*a*b^-1=c*a*c=a^9,c*b*c=b^-1>;
// generators/relations

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