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G = C2×S3×SD16  order 192 = 26·3

Direct product of C2, S3 and SD16

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×S3×SD16, C24⋊6C23, C12.5C24, Dic6⋊2C23, D12.3C23, (C2×C8)⋊28D6, C3⋊C8⋊7C23, C8⋊6(C22×S3), (C2×Q8)⋊23D6, C6⋊2(C2×SD16), C4.42(S3×D4), (C4×S3).28D4, D6.64(C2×D4), C12.80(C2×D4), C4.5(S3×C23), Q8⋊2(C22×S3), (S3×Q8)⋊5C22, (C3×Q8)⋊1C23, C3⋊2(C22×SD16), (S3×C8)⋊17C22, (C2×C24)⋊18C22, (C6×SD16)⋊10C2, (C2×D4).181D6, D4.S3⋊9C22, (C6×Q8)⋊17C22, (S3×D4).5C22, (C3×D4).3C23, D4.3(C22×S3), C24⋊C2⋊17C22, (C4×S3).25C23, Dic3.12(C2×D4), Q8⋊2S3⋊7C22, C22.138(S3×D4), C6.106(C22×D4), (C2×C12).522C23, (C2×Dic3).122D4, (C3×SD16)⋊12C22, (C2×Dic6)⋊37C22, (C6×D4).163C22, (C22×S3).111D4, (C2×D12).177C22, (S3×C2×C8)⋊9C2, (C2×S3×Q8)⋊14C2, C2.79(C2×S3×D4), (C2×S3×D4).10C2, (C2×C3⋊C8)⋊36C22, (C2×C24⋊C2)⋊31C2, (C2×D4.S3)⋊27C2, (C2×C6).395(C2×D4), (C2×Q8⋊2S3)⋊25C2, (S3×C2×C4).257C22, (C2×C4).611(C22×S3), SmallGroup(192,1317)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2×S3×SD16
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×S3×SD16
Lower central
C3 — C6 — C12 — C2×S3×SD16

Subgroups: 888 in 298 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×6], C22, C22 [×22], S3 [×4], S3 [×2], C6, C6 [×2], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×11], D4 [×2], D4 [×8], Q8 [×2], Q8 [×8], C23 [×11], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], D6 [×12], C2×C6, C2×C6 [×4], C2×C8, C2×C8 [×5], SD16 [×4], SD16 [×12], C22×C4 [×2], C2×D4, C2×D4 [×8], C2×Q8, C2×Q8 [×8], C24, C3⋊C8 [×2], C24 [×2], Dic6 [×2], Dic6 [×5], C4×S3 [×4], C4×S3 [×4], D12 [×2], D12, C2×Dic3, C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×S3, C22×S3 [×9], C22×C6, C22×C8, C2×SD16, C2×SD16 [×11], C22×D4, C22×Q8, S3×C8 [×4], C24⋊C2 [×4], C2×C3⋊C8, D4.S3 [×4], Q8⋊2S3 [×4], C2×C24, C3×SD16 [×4], C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, S3×D4 [×4], S3×D4 [×2], S3×Q8 [×4], S3×Q8 [×2], C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C22×SD16, S3×C2×C8, C2×C24⋊C2, S3×SD16 [×8], C2×D4.S3, C2×Q8⋊2S3, C6×SD16, C2×S3×D4, C2×S3×Q8, C2×S3×SD16

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], SD16 [×4], C2×D4 [×6], C24, C22×S3 [×7], C2×SD16 [×6], C22×D4, S3×D4 [×2], S3×C23, C22×SD16, S3×SD16 [×2], C2×S3×D4, C2×S3×SD16

Generators and relations
 G = < a,b,c,d,e | a2=b3=c2=d8=e2=1, 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 15)(2 16)(3 9)(4 10)(5 11)(6 12)(7 13)(8 14)(17 27)(18 28)(19 29)(20 30)(21 31)(22 32)(23 25)(24 26)(33 43)(34 44)(35 45)(36 46)(37 47)(38 48)(39 41)(40 42)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
7200000
0720000
0072000
0007200
000001
000010
,
67670000
6670000
0067600
00676700
0000720
0000072
,
7200000
010000
001000
0007200
000010
000001

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[67,6,0,0,0,0,67,67,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D24A24B24C24D
order12222222222234444444466666888888881212121224242424
size1111333344121222244661212222882222666644884444

42 irreducible representations

dim111111111222222222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6SD16S3×D4S3×D4S3×SD16
kernelC2×S3×SD16S3×C2×C8C2×C24⋊C2S3×SD16C2×D4.S3C2×Q8⋊2S3C6×SD16C2×S3×D4C2×S3×Q8C2×SD16C4×S3C2×Dic3C22×S3C2×C8SD16C2×D4C2×Q8D6C4C22C2
# reps111811111121114118114

In GAP, Magma, Sage, TeX 

C_2\times S_3\times SD_{16}
 
% in TeX
 
G:=Group("C2xS3xSD16");
 
// GroupNames label
 
G:=SmallGroup(192,1317);
 
// by ID
 
G=gap.SmallGroup(192,1317);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,185,136,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^8=e^2=1,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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