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G = C166D4order 128 = 27

3rd semidirect product of C16 and D4 acting via D4/C4=C2

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

Aliases: C166D4, C41M5(2), C4⋊C1618C2, (C4×C16)⋊16C2, C4⋊C4.10C8, C4⋊C8.23C4, C2.17(C8×D4), (C8×D4).18C2, (C4×D4).20C4, (C2×D4).10C8, C4.177(C4×D4), C8.140(C2×D4), C22⋊C4.6C8, C22⋊C1616C2, C2.8(D4○C16), C4.59(C8○D4), C22⋊C8.22C4, C23.11(C2×C8), C8.103(C4○D4), (C2×M5(2))⋊20C2, C42.273(C2×C4), (C4×C8).376C22, (C2×C8).634C23, (C2×C16).55C22, C2.10(C2×M5(2)), C22.53(C22×C8), (C22×C8).420C22, (C2×C4).30(C2×C8), (C2×C8).163(C2×C4), (C2×C4).619(C22×C4), (C22×C4).291(C2×C4), SmallGroup(128,901)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C166D4
C1C2C4C8C2×C8C2×C16C2×M5(2) — C166D4
C1C22 — C166D4
C1C2×C8 — C166D4
C1C2C2C2C2C4C4C2×C8 — C166D4

Generators and relations for C166D4
 G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a9, cbc=b-1 >

Subgroups: 116 in 81 conjugacy classes, 50 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C16 [×2], C16 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], C22×C4 [×2], C2×D4, C4×C8, C22⋊C8 [×2], C4⋊C8, C2×C16 [×2], C2×C16 [×2], M5(2) [×4], C4×D4, C22×C8 [×2], C4×C16, C22⋊C16 [×2], C4⋊C16, C8×D4, C2×M5(2) [×2], C166D4
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, C166D4

Smallest permutation representation of C166D4
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 64 21 44)(2 49 22 45)(3 50 23 46)(4 51 24 47)(5 52 25 48)(6 53 26 33)(7 54 27 34)(8 55 28 35)(9 56 29 36)(10 57 30 37)(11 58 31 38)(12 59 32 39)(13 60 17 40)(14 61 18 41)(15 62 19 42)(16 63 20 43)
(1 44)(2 37)(3 46)(4 39)(5 48)(6 41)(7 34)(8 43)(9 36)(10 45)(11 38)(12 47)(13 40)(14 33)(15 42)(16 35)(17 60)(18 53)(19 62)(20 55)(21 64)(22 57)(23 50)(24 59)(25 52)(26 61)(27 54)(28 63)(29 56)(30 49)(31 58)(32 51)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I8J8K8L8M8N8O8P16A···16P16Q···16X
order12222244444444448···88888888816···1616···16
size11114411112222441···1222244442···24···4

56 irreducible representations

dim11111111111122222
type+++++++
imageC1C2C2C2C2C2C4C4C4C8C8C8D4C4○D4M5(2)C8○D4D4○C16
kernelC166D4C4×C16C22⋊C16C4⋊C16C8×D4C2×M5(2)C22⋊C8C4⋊C8C4×D4C22⋊C4C4⋊C4C2×D4C16C8C4C4C2
# reps11211242284422848

Matrix representation of C166D4 in GL4(𝔽17) generated by

0100
8000
00160
00016
,
01100
3000
001015
0087
,
01100
14000
001015
0077
G:=sub<GL(4,GF(17))| [0,8,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,3,0,0,11,0,0,0,0,0,10,8,0,0,15,7],[0,14,0,0,11,0,0,0,0,0,10,7,0,0,15,7] >;

C166D4 in GAP, Magma, Sage, TeX

C_{16}\rtimes_6D_4
% in TeX

G:=Group("C16:6D4");
// GroupNames label

G:=SmallGroup(128,901);
// by ID

G=gap.SmallGroup(128,901);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,1430,723,100,102,124]);
// Polycyclic

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

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