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G = C16.A4order 192 = 26·3

The central extension by C16 of A4

non-abelian, soluble

Aliases: C16.A4, Q8.C24, SL2(𝔽3).2C8, D4○C16⋊C3, C8.7(C2×A4), C2.3(C8×A4), C4.5(C4×A4), C8○D4.2C6, C4.A4.3C4, C8.A4.3C2, C4○D4.2C12, SmallGroup(192,204)

Series: Derived Chief Lower central Upper central

C1C2Q8 — C16.A4
C1C2Q8C4○D4C8○D4C8.A4 — C16.A4
Q8 — C16.A4
C1C16

Generators and relations for C16.A4
 G = < a,b,c,d | a16=d3=1, b2=c2=a8, ab=ba, ac=ca, ad=da, cbc-1=a8b, dbd-1=a8bc, dcd-1=b >

6C2
4C3
3C22
3C4
4C6
3D4
3C8
3C2×C4
4C12
3M4(2)
3C16
3C2×C8
4C24
3M5(2)
3C2×C16
4C48

Smallest permutation representation of C16.A4
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 61 9 53)(2 62 10 54)(3 63 11 55)(4 64 12 56)(5 49 13 57)(6 50 14 58)(7 51 15 59)(8 52 16 60)(17 43 25 35)(18 44 26 36)(19 45 27 37)(20 46 28 38)(21 47 29 39)(22 48 30 40)(23 33 31 41)(24 34 32 42)
(1 27 9 19)(2 28 10 20)(3 29 11 21)(4 30 12 22)(5 31 13 23)(6 32 14 24)(7 17 15 25)(8 18 16 26)(33 57 41 49)(34 58 42 50)(35 59 43 51)(36 60 44 52)(37 61 45 53)(38 62 46 54)(39 63 47 55)(40 64 48 56)
(17 43 51)(18 44 52)(19 45 53)(20 46 54)(21 47 55)(22 48 56)(23 33 57)(24 34 58)(25 35 59)(26 36 60)(27 37 61)(28 38 62)(29 39 63)(30 40 64)(31 41 49)(32 42 50)

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

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

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

56 conjugacy classes

class 1 2A2B3A3B4A4B4C6A6B8A8B8C8D8E8F12A12B12C12D16A···16H16I16J16K16L24A···24H48A···48P
order12233444668888881212121216···161616161624···2448···48
size116441164411116644441···166664···44···4

56 irreducible representations

dim1111111123333
type++++
imageC1C2C3C4C6C8C12C24C16.A4A4C2×A4C4×A4C8×A4
kernelC16.A4C8.A4D4○C16C4.A4C8○D4SL2(𝔽3)C4○D4Q8C1C16C8C4C2
# reps11222448241124

Matrix representation of C16.A4 in GL2(𝔽17) generated by

70
07
,
410
013
,
130
104
,
09
1516
G:=sub<GL(2,GF(17))| [7,0,0,7],[4,0,10,13],[13,10,0,4],[0,15,9,16] >;

C16.A4 in GAP, Magma, Sage, TeX

C_{16}.A_4
% in TeX

G:=Group("C16.A4");
// GroupNames label

G:=SmallGroup(192,204);
// by ID

G=gap.SmallGroup(192,204);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-2,42,58,248,851,172,1524,285,124]);
// Polycyclic

G:=Group<a,b,c,d|a^16=d^3=1,b^2=c^2=a^8,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a^8*b,d*b*d^-1=a^8*b*c,d*c*d^-1=b>;
// generators/relations

Export

Subgroup lattice of C16.A4 in TeX

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