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G = C16:D6  order 192 = 26·3

1st semidirect product of C16 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C16:1D6, D48:2C2, C8.3D12, C24.2D4, C48:1C22, C4.14D24, C12.14D8, D24:8C22, M5(2):1S3, C22.5D24, C24.59C23, Dic12:7C22, (C2xC6).6D8, C4oD24:9C2, C48:C2:1C2, C6.13(C2xD8), (C2xC8).73D6, (C2xD24):11C2, C3:1(C16:C22), (C2xC4).41D12, C2.15(C2xD24), C4.40(C2xD12), (C2xC12).128D4, C12.283(C2xD4), (C3xM5(2)):1C2, C8.49(C22xS3), (C2xC24).59C22, SmallGroup(192,467)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — C16:D6
Chief series
C1 — C3 — C6 — C12 — C24 — D24 — C2xD24 — C16:D6
Lower central
C3 — C6 — C12 — C24 — C16:D6
Upper central
C1 — C2 — C2xC4 — C2xC8 — M5(2)

Generators and relations for C16:D6
 G = < a,b,c | a16=b6=c2=1, bab-1=a9, cac=a7, cbc=b-1 >

Subgroups: 424 in 90 conjugacy classes, 35 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2xC4, C2xC4, D4, Q8, C23, Dic3, C12, D6, C2xC6, C16, C2xC8, D8, SD16, Q16, C2xD4, C4oD4, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C22xS3, M5(2), D16, SD32, C2xD8, C4oD8, C48, C24:C2, D24, D24, D24, Dic12, C2xC24, C2xD12, C4oD12, C16:C22, D48, C48:C2, C3xM5(2), C2xD24, C4oD24, C16:D6
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2xD4, D12, C22xS3, C2xD8, D24, C2xD12, C16:C22, C2xD24, C16:D6

Smallest permutation representation of C16:D6
►On 48 points
Generators in S48
(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)

(1 46 27)(2 39 28 10 47 20)(3 48 29)(4 41 30 12 33 22)(5 34 31)(6 43 32 14 35 24)(7 36 17)(8 45 18 16 37 26)(9 38 19)(11 40 21)(13 42 23)(15 44 25)

(1 27)(2 18)(3 25)(4 32)(5 23)(6 30)(7 21)(8 28)(9 19)(10 26)(11 17)(12 24)(13 31)(14 22)(15 29)(16 20)(33 35)(34 42)(36 40)(37 47)(39 45)(41 43)(44 48)
 
G:=sub<Sym(48)| (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), (1,46,27)(2,39,28,10,47,20)(3,48,29)(4,41,30,12,33,22)(5,34,31)(6,43,32,14,35,24)(7,36,17)(8,45,18,16,37,26)(9,38,19)(11,40,21)(13,42,23)(15,44,25), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48)>;
 
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), (1,46,27)(2,39,28,10,47,20)(3,48,29)(4,41,30,12,33,22)(5,34,31)(6,43,32,14,35,24)(7,36,17)(8,45,18,16,37,26)(9,38,19)(11,40,21)(13,42,23)(15,44,25), (1,27)(2,18)(3,25)(4,32)(5,23)(6,30)(7,21)(8,28)(9,19)(10,26)(11,17)(12,24)(13,31)(14,22)(15,29)(16,20)(33,35)(34,42)(36,40)(37,47)(39,45)(41,43)(44,48) );
 
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)], [(1,46,27),(2,39,28,10,47,20),(3,48,29),(4,41,30,12,33,22),(5,34,31),(6,43,32,14,35,24),(7,36,17),(8,45,18,16,37,26),(9,38,19),(11,40,21),(13,42,23),(15,44,25)], [(1,27),(2,18),(3,25),(4,32),(5,23),(6,30),(7,21),(8,28),(9,19),(10,26),(11,17),(12,24),(13,31),(14,22),(15,29),(16,20),(33,35),(34,42),(36,40),(37,47),(39,45),(41,43),(44,48)]])
 

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C6A6B8A8B8C12A12B12C16A16B16C16D24A24B24C24D24E24F48A···48H
order1222223444668881212121616161624242424242448···48
size112242424222242422422444442222444···4

36 irreducible representations

dim1111112222222222244
type+++++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D8D8D12D12D24D24C16:C22C16:D6
kernelC16:D6D48C48:C2C3xM5(2)C2xD24C4oD24M5(2)C24C2xC12C16C2xC8C12C2xC6C8C2xC4C4C22C3C1
# reps1221111112122224424

Matrix representation of C16:D6 ►in GL6(F97)

2790000
18810000
000010
000001
00909000
0079000
,
010000
9610000
001000
000100
0000960
0000096
,
9610000
010000
001000
0009600
0000790
00009090

G:=sub<GL(6,GF(97))| [2,18,0,0,0,0,79,81,0,0,0,0,0,0,0,0,90,7,0,0,0,0,90,90,0,0,1,0,0,0,0,0,0,1,0,0],[0,96,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,96],[96,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,96,0,0,0,0,0,0,7,90,0,0,0,0,90,90] >;
 

C16:D6 in GAP, Magma, Sage, TeX 

C_{16}\rtimes D_6
 
% in TeX
 
G:=Group("C16:D6");
 
// GroupNames label
 
G:=SmallGroup(192,467);
 
// by ID
 
G=gap.SmallGroup(192,467);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,254,387,142,675,192,1684,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c|a^16=b^6=c^2=1,b*a*b^-1=a^9,c*a*c=a^7,c*b*c=b^-1>;
 
// generators/relations
 

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