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G = C4⋊C16order 64 = 26

The semidirect product of C4 and C16 acting via C16/C8=C2

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

Aliases: C4⋊C16, C8.7Q8, C8.28D4, C42.7C4, C2.3M5(2), C4.11M4(2), (C4×C8).2C2, (C2×C4).4C8, (C2×C8).7C4, C2.2(C4⋊C8), (C2×C16).2C2, C2.2(C2×C16), C4.19(C4⋊C4), C22.9(C2×C8), (C2×C8).108C22, (C2×C4).82(C2×C4), SmallGroup(64,44)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4⋊C16
C1C2C4C8C2×C8C4×C8 — C4⋊C16
C1C2 — C4⋊C16
C1C2×C8 — C4⋊C16
C1C2C2C2C2C4C4C2×C8 — C4⋊C16

Generators and relations for C4⋊C16
 G = < a,b | a4=b16=1, bab-1=a-1 >

2C4
2C8
2C16
2C16

Smallest permutation representation of C4⋊C16
Regular action on 64 points
Generators in S64
(1 35 53 29)(2 30 54 36)(3 37 55 31)(4 32 56 38)(5 39 57 17)(6 18 58 40)(7 41 59 19)(8 20 60 42)(9 43 61 21)(10 22 62 44)(11 45 63 23)(12 24 64 46)(13 47 49 25)(14 26 50 48)(15 33 51 27)(16 28 52 34)
(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)

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

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

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

C4⋊C16 is a maximal subgroup of
C8.31D8  C8.17Q16  C8.30D8  C4.D16  C8.27D8  C8.16Q16  C4.10D16  C4.6Q32  C4⋊M5(2)  C4⋊C4.7C8  C8.12M4(2)  D4×C16  C169D4  C166D4  Q8×C16  D82D4  Q162D4  D8.4D4  Q16.4D4  D8.5D4  Q16.5D4  D81Q8  Q16⋊Q8  D8⋊Q8  C4.Q32  D8.Q8  Q16.Q8
 C4p⋊C16: C82C16  C8.36D8  C12⋊C16  C203C16  C20⋊C16  C28⋊C16 ...
 C2p.M5(2): D4⋊C16  Q8⋊C16  C42.13C8  C42.6C8  C164Q8  Dic3⋊C16  C40.88D4  C10.M5(2) ...
C4⋊C16 is a maximal quotient of
C22.7M5(2)  C10.M5(2)
 C4p⋊C16: C82C16  C8.36D8  C12⋊C16  C203C16  C20⋊C16  C28⋊C16 ...
 C8p.D4: C4⋊C32  C8.C16  Dic3⋊C16  C40.88D4  Dic7⋊C16 ...

40 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H8A···8H8I8J8K8L16A···16P
order1222444444448···8888816···16
size1111111122221···122222···2

40 irreducible representations

dim11111112222
type++++-
imageC1C2C2C4C4C8C16D4Q8M4(2)M5(2)
kernelC4⋊C16C4×C8C2×C16C42C2×C8C2×C4C4C8C8C4C2
# reps112228161124

Matrix representation of C4⋊C16 in GL3(𝔽17) generated by

100
0113
0916
,
1400
01011
007
G:=sub<GL(3,GF(17))| [1,0,0,0,1,9,0,13,16],[14,0,0,0,10,0,0,11,7] >;

C4⋊C16 in GAP, Magma, Sage, TeX

C_4\rtimes C_{16}
% in TeX

G:=Group("C4:C16");
// GroupNames label

G:=SmallGroup(64,44);
// by ID

G=gap.SmallGroup(64,44);
# by ID

G:=PCGroup([6,-2,2,-2,2,-2,-2,48,73,31,69,88]);
// Polycyclic

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

Export

Subgroup lattice of C4⋊C16 in TeX

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