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G = C4⋊D16order 128 = 27

The semidirect product of C4 and D16 acting via D16/C16=C2

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

Aliases: C41D16, C164D4, C8.15D8, C42.336D4, (C4×C16)⋊9C2, C4.7(C2×D8), (C2×D16)⋊6C2, C84D47C2, (C2×C4).83D8, C8.39(C2×D4), (C2×C8).252D4, C2.10(C2×D16), C4.1(C41D4), C2.12(C84D4), (C2×C16).83C22, (C2×C8).544C23, (C4×C8).401C22, (C2×D8).16C22, C22.130(C2×D8), (C2×C4).812(C2×D4), SmallGroup(128,978)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C8 — C4⋊D16
C1C2C4C2×C4C2×C8C4×C8C4×C16 — C4⋊D16
C1C2C4C2×C8 — C4⋊D16
C1C22C42C4×C8 — C4⋊D16
C1C2C2C2C2C4C4C2×C8 — C4⋊D16

Generators and relations for C4⋊D16
 G = < a,b,c | a16=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 392 in 108 conjugacy classes, 40 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×4], C22, C22 [×12], C8 [×4], C2×C4, C2×C4 [×2], D4 [×16], C23 [×4], C16 [×4], C42, C2×C8 [×2], D8 [×12], C2×D4 [×8], C4×C8, C2×C16 [×2], D16 [×8], C41D4 [×2], C2×D8 [×4], C2×D8 [×2], C4×C16, C84D4 [×2], C2×D16 [×4], C4⋊D16
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D8 [×4], C2×D4 [×3], D16 [×4], C41D4, C2×D8 [×2], C84D4, C2×D16 [×2], C4⋊D16

Smallest permutation representation of C4⋊D16
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 31 59 44)(2 32 60 45)(3 17 61 46)(4 18 62 47)(5 19 63 48)(6 20 64 33)(7 21 49 34)(8 22 50 35)(9 23 51 36)(10 24 52 37)(11 25 53 38)(12 26 54 39)(13 27 55 40)(14 28 56 41)(15 29 57 42)(16 30 58 43)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 64)(13 63)(14 62)(15 61)(16 60)(17 29)(18 28)(19 27)(20 26)(21 25)(22 24)(30 32)(33 39)(34 38)(35 37)(40 48)(41 47)(42 46)(43 45)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F8A···8H16A···16P
order122222224···48···816···16
size1111161616162···22···22···2

38 irreducible representations

dim1111222222
type++++++++++
imageC1C2C2C2D4D4D4D8D8D16
kernelC4⋊D16C4×C16C84D4C2×D16C16C42C2×C8C8C2×C4C4
# reps11244114416

Matrix representation of C4⋊D16 in GL4(𝔽17) generated by

61300
4600
001613
0091
,
1000
0100
0014
00816
,
1000
01600
00160
0091
G:=sub<GL(4,GF(17))| [6,4,0,0,13,6,0,0,0,0,16,9,0,0,13,1],[1,0,0,0,0,1,0,0,0,0,1,8,0,0,4,16],[1,0,0,0,0,16,0,0,0,0,16,9,0,0,0,1] >;

C4⋊D16 in GAP, Magma, Sage, TeX

C_4\rtimes D_{16}
% in TeX

G:=Group("C4:D16");
// GroupNames label

G:=SmallGroup(128,978);
// by ID

G=gap.SmallGroup(128,978);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,288,422,100,1123,360,4037,124]);
// Polycyclic

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

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