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G = C2×S3×D8  order 192 = 26·3

Direct product of C2, S3 and D8

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

Aliases: C2×S3×D8, C24⋊3C23, D12⋊1C23, C12.1C24, D24⋊15C22, C6⋊2(C2×D8), (C6×D8)⋊7C2, (C2×C8)⋊26D6, C3⋊C8⋊6C23, C3⋊2(C22×D8), (C2×D4)⋊27D6, C8⋊5(C22×S3), C4.39(S3×D4), (C2×D24)⋊19C2, D4⋊S3⋊8C22, D6.63(C2×D4), (C4×S3).26D4, C12.76(C2×D4), (S3×D4)⋊4C22, (C3×D4)⋊1C23, (C3×D8)⋊9C22, D4⋊1(C22×S3), C4.1(S3×C23), (S3×C8)⋊13C22, (C2×C24)⋊11C22, (C6×D4)⋊18C22, (C2×D12)⋊32C22, (C4×S3).23C23, Dic3.11(C2×D4), C6.102(C22×D4), C22.135(S3×D4), (C2×C12).518C23, (C2×Dic3).121D4, (C22×S3).110D4, (S3×C2×C8)⋊4C2, (C2×S3×D4)⋊21C2, C2.75(C2×S3×D4), (C2×D4⋊S3)⋊25C2, (C2×C3⋊C8)⋊35C22, (C2×C6).391(C2×D4), (S3×C2×C4).255C22, (C2×C4).608(C22×S3), SmallGroup(192,1313)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2×S3×D8
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×S3×D8
Lower central
C3 — C6 — C12 — C2×S3×D8

Subgroups: 1176 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×12], C3, C4 [×2], C4 [×2], C22, C22 [×38], S3 [×4], S3 [×4], C6, C6 [×2], C6 [×4], C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], D4 [×4], D4 [×16], C23 [×21], Dic3 [×2], C12 [×2], D6 [×6], D6 [×24], C2×C6, C2×C6 [×8], C2×C8, C2×C8 [×5], D8 [×4], D8 [×12], C22×C4, C2×D4 [×2], C2×D4 [×16], C24 [×2], C3⋊C8 [×2], C24 [×2], C4×S3 [×4], D12 [×4], D12 [×2], C2×Dic3, C3⋊D4 [×8], C2×C12, C3×D4 [×4], C3×D4 [×2], C22×S3, C22×S3 [×18], C22×C6 [×2], C22×C8, C2×D8, C2×D8 [×11], C22×D4 [×2], S3×C8 [×4], D24 [×4], C2×C3⋊C8, D4⋊S3 [×8], C2×C24, C3×D8 [×4], S3×C2×C4, C2×D12 [×2], S3×D4 [×8], S3×D4 [×4], C2×C3⋊D4 [×2], C6×D4 [×2], S3×C23 [×2], C22×D8, S3×C2×C8, C2×D24, S3×D8 [×8], C2×D4⋊S3 [×2], C6×D8, C2×S3×D4 [×2], C2×S3×D8

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], D8 [×4], C2×D4 [×6], C24, C22×S3 [×7], C2×D8 [×6], C22×D4, S3×D4 [×2], S3×C23, C22×D8, S3×D8 [×2], C2×S3×D4, C2×S3×D8

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=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
100000
010000
001000
000100
0000072
0000172
,
7200000
0720000
0072000
0007200
000001
000010
,
16570000
16160000
00575700
00165700
000010
000001
,
100000
0720000
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],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,57,16,0,0,0,0,57,57,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D6A6B6C6D6E6F6G8A8B8C8D8E8F8G8H12A12B24A24B24C24D
order122222222222222234444666666688888888121224242424
size1111333344441212121222266222888822226666444444

42 irreducible representations

dim111111122222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2S3D4D4D4D6D6D6D8S3×D4S3×D4S3×D8
kernelC2×S3×D8S3×C2×C8C2×D24S3×D8C2×D4⋊S3C6×D8C2×S3×D4C2×D8C4×S3C2×Dic3C22×S3C2×C8D8C2×D4D6C4C22C2
# reps111821212111428114

In GAP, Magma, Sage, TeX 

C_2\times S_3\times D_8
 
% in TeX
 
G:=Group("C2xS3xD8");
 
// GroupNames label
 
G:=SmallGroup(192,1313);
 
// by ID
 
G=gap.SmallGroup(192,1313);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,185,438,235,102,6278]);
 
// 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^-1>;
 
// generators/relations
 

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