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G = C3×D4○SD16  order 192 = 26·3

Direct product of C3 and D4○SD16

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C3×D4○SD16, C24.49C23, C12.86C24, 2+ (1+4)⋊9C6, 2- (1+4)⋊5C6, C8○D4⋊8C6, C4○D8⋊5C6, D8⋊5(C2×C6), C8⋊C22⋊5C6, Q16⋊5(C2×C6), C4.46(C6×D4), SD16⋊6(C2×C6), (C2×SD16)⋊6C6, (C3×D4).46D4, D4.12(C3×D4), C8.C22⋊4C6, C4.9(C23×C6), (C3×Q8).46D4, Q8.17(C3×D4), C22.8(C6×D4), (C2×C24)⋊24C22, (C6×SD16)⋊17C2, C12.407(C2×D4), (C3×D8)⋊22C22, M4(2)⋊7(C2×C6), C8.13(C22×C6), (C6×Q8)⋊32C22, D4.6(C22×C6), (C3×Q16)⋊19C22, (C3×D4).39C23, C6.207(C22×D4), (C3×Q8).40C23, Q8.10(C22×C6), (C2×C12).688C23, (C3×SD16)⋊21C22, (C6×D4).226C22, (C3×2- (1+4))⋊7C2, (C3×M4(2))⋊28C22, (C3×2+ (1+4))⋊10C2, (C2×C8)⋊5(C2×C6), C2.31(D4×C2×C6), C4○D4⋊3(C2×C6), (C3×C8○D4)⋊9C2, (C2×Q8)⋊8(C2×C6), (C3×C4○D8)⋊12C2, (C3×C8⋊C22)⋊12C2, (C2×D4).39(C2×C6), (C2×C6).185(C2×D4), (C3×C4○D4)⋊15C22, (C3×C8.C22)⋊11C2, (C2×C4).49(C22×C6), SmallGroup(192,1466)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C4 — C3×D4○SD16
Chief series
C1 — C2 — C4 — C12 — C3×Q8 — C3×SD16 — C6×SD16 — C3×D4○SD16
Lower central
C1 — C2 — C4 — C3×D4○SD16
Upper central
C1 — C6 — C3×C4○D4 — C3×D4○SD16

Subgroups: 410 in 258 conjugacy classes, 158 normal (26 characteristic)
C1, C2, C2 [×7], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×7], C6, C6 [×7], C8, C8 [×3], C2×C4 [×3], C2×C4 [×9], D4, D4 [×6], D4 [×9], Q8 [×2], Q8 [×3], Q8 [×3], C23 [×3], C12, C12 [×3], C12 [×4], C2×C6 [×3], C2×C6 [×7], C2×C8 [×3], M4(2) [×3], D8 [×3], SD16, SD16 [×9], Q16 [×3], C2×D4 [×3], C2×D4 [×3], C2×Q8 [×3], C2×Q8, C4○D4, C4○D4 [×6], C4○D4 [×4], C24, C24 [×3], C2×C12 [×3], C2×C12 [×9], C3×D4, C3×D4 [×6], C3×D4 [×9], C3×Q8 [×2], C3×Q8 [×3], C3×Q8 [×3], C22×C6 [×3], C8○D4, C2×SD16 [×3], C4○D8 [×3], C8⋊C22 [×3], C8.C22 [×3], 2+ (1+4), 2- (1+4), C2×C24 [×3], C3×M4(2) [×3], C3×D8 [×3], C3×SD16, C3×SD16 [×9], C3×Q16 [×3], C6×D4 [×3], C6×D4 [×3], C6×Q8 [×3], C6×Q8, C3×C4○D4, C3×C4○D4 [×6], C3×C4○D4 [×4], D4○SD16, C3×C8○D4, C6×SD16 [×3], C3×C4○D8 [×3], C3×C8⋊C22 [×3], C3×C8.C22 [×3], C3×2+ (1+4), C3×2- (1+4), C3×D4○SD16

Quotients:
C1, C2 [×15], C3, C22 [×35], C6 [×15], D4 [×4], C23 [×15], C2×C6 [×35], C2×D4 [×6], C24, C3×D4 [×4], C22×C6 [×15], C22×D4, C6×D4 [×6], C23×C6, D4○SD16, D4×C2×C6, C3×D4○SD16

Generators and relations
 G = < a,b,c,d,e | a3=b4=c2=e2=1, d4=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d3 >

Smallest permutation representation
►On 48 points
Generators in S48
(1 39 18)(2 40 19)(3 33 20)(4 34 21)(5 35 22)(6 36 23)(7 37 24)(8 38 17)(9 47 26)(10 48 27)(11 41 28)(12 42 29)(13 43 30)(14 44 31)(15 45 32)(16 46 25)

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

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

(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)

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

Matrix representation ►G ⊆ GL4(𝔽73) generated by

64000
06400
00640
00064
,
0010
0001
72000
07200
,
0010
0001
1000
0100
,
66700
6600
00667
0066
,
1000
07200
0010
00072
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[0,0,72,0,0,0,0,72,1,0,0,0,0,1,0,0],[0,0,1,0,0,0,0,1,1,0,0,0,0,1,0,0],[6,6,0,0,67,6,0,0,0,0,6,6,0,0,67,6],[1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
 

66 conjugacy classes

class 1 2A2B2C2D2E2F2G2H3A3B4A4B4C4D4E4F4G4H6A6B6C···6H6I···6P8A8B8C8D8E12A···12H12I···12P24A24B24C24D24E···24J
order1222222223344444444666···66···68888812···1212···122424242424···24
size1122244441122224444112···24···4224442···24···422224···4

66 irreducible representations

dim1111111111111111222244
type++++++++++
imageC1C2C2C2C2C2C2C2C3C6C6C6C6C6C6C6D4D4C3×D4C3×D4D4○SD16C3×D4○SD16
kernelC3×D4○SD16C3×C8○D4C6×SD16C3×C4○D8C3×C8⋊C22C3×C8.C22C3×2+ (1+4)C3×2- (1+4)D4○SD16C8○D4C2×SD16C4○D8C8⋊C22C8.C222+ (1+4)2- (1+4)C3×D4C3×Q8D4Q8C3C1
# reps1133331122666622316224

In GAP, Magma, Sage, TeX 

C_3\times D_4\circ SD_{16}
 
% in TeX
 
G:=Group("C3xD4oSD16");
 
// GroupNames label
 
G:=SmallGroup(192,1466);
 
// by ID
 
G=gap.SmallGroup(192,1466);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701,745,6053,3036,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;
 
// generators/relations
 

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