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G = S3×C8○D4  order 192 = 26·3

Direct product of S3 and C8○D4

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

Aliases: S3×C8○D4, M4(2)⋊27D6, C24.53C23, C12.71C24, (C2×C8)⋊30D6, (S3×D4).2C4, (S3×Q8).2C4, C8○D12⋊16C2, C4○D4.58D6, D4.12(C4×S3), C3⋊C8.36C23, Q8.18(C4×S3), D12.C4⋊14C2, (S3×C8)⋊20C22, (C2×C24)⋊31C22, D12.20(C2×C4), D4⋊2S3.2C4, C8.66(C22×S3), C4.70(S3×C23), C6.34(C23×C4), Q8⋊3S3.2C4, C8⋊S3⋊20C22, (S3×M4(2))⋊12C2, D4.Dic3⋊14C2, (C4×S3).41C23, C12.38(C22×C4), Dic6.21(C2×C4), D6.15(C22×C4), (C2×C12).513C23, C4○D12.51C22, C4.Dic3⋊26C22, (C3×M4(2))⋊32C22, Dic3.15(C22×C4), C3⋊3(C2×C8○D4), (S3×C2×C8)⋊30C2, C4.38(S3×C2×C4), C22.4(S3×C2×C4), (C3×C8○D4)⋊13C2, (C2×C3⋊C8)⋊34C22, (S3×C4○D4).5C2, C3⋊D4.1(C2×C4), C2.35(S3×C22×C4), (C4×S3).18(C2×C4), (C3×D4).16(C2×C4), (C2×C6).4(C22×C4), (C3×Q8).17(C2×C4), (S3×C2×C4).254C22, (C22×S3).47(C2×C4), (C2×C4).606(C22×S3), (C2×Dic3).73(C2×C4), (C3×C4○D4).43C22, SmallGroup(192,1308)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — S3×C8○D4
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — S3×C8○D4
Lower central
C3 — C6 — S3×C8○D4

Subgroups: 512 in 266 conjugacy classes, 149 normal (24 characteristic)
C1, C2, C2 [×8], C3, C4, C4 [×3], C4 [×4], C22 [×3], C22 [×10], S3 [×2], S3 [×3], C6, C6 [×3], C8, C8 [×3], C8 [×4], C2×C4 [×3], C2×C4 [×13], D4 [×3], D4 [×9], Q8, Q8 [×3], C23 [×3], Dic3, Dic3 [×3], C12, C12 [×3], D6, D6 [×3], D6 [×6], C2×C6 [×3], C2×C8 [×3], C2×C8 [×13], M4(2) [×3], M4(2) [×9], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4, C4○D4 [×7], C3⋊C8, C3⋊C8 [×3], C24, C24 [×3], Dic6 [×3], C4×S3, C4×S3 [×9], D12 [×3], C2×Dic3 [×3], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×3], C22×C8 [×3], C2×M4(2) [×3], C8○D4, C8○D4 [×7], C2×C4○D4, S3×C8, S3×C8 [×9], C8⋊S3 [×6], C2×C3⋊C8 [×3], C4.Dic3 [×3], C2×C24 [×3], C3×M4(2) [×3], S3×C2×C4 [×3], C4○D12 [×3], S3×D4 [×3], D4⋊2S3 [×3], S3×Q8, Q8⋊3S3, C3×C4○D4, C2×C8○D4, S3×C2×C8 [×3], C8○D12 [×3], S3×M4(2) [×3], D12.C4 [×3], D4.Dic3, C3×C8○D4, S3×C4○D4, S3×C8○D4

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C8○D4 [×2], C23×C4, S3×C2×C4 [×6], S3×C23, C2×C8○D4, S3×C22×C4, S3×C8○D4

Generators and relations
 G = < a,b,c,d,e | a3=b2=c8=e2=1, d2=c4, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c4d >

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

(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)

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

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

Matrix representation ►G ⊆ GL4(𝔽73) generated by

07200
17200
0010
0001
,
17200
07200
0010
0001
,
27000
02700
00220
00022
,
72000
07200
00072
0010
,
1000
0100
0010
00072
G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,72,72,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,22,0,0,0,0,22],[72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;
 

60 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D8A8B8C8D8E···8J8K8L8M8N8O···8T12A12B12C12D12E24A24B24C24D24E···24J
order122222222234444444444666688888···888888···812121212122424242424···24
size112223366621122233666244411112···233336···62244422224···4

60 irreducible representations

dim11111111111122222224
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D6D6D6C4×S3C4×S3C8○D4S3×C8○D4
kernelS3×C8○D4S3×C2×C8C8○D12S3×M4(2)D12.C4D4.Dic3C3×C8○D4S3×C4○D4S3×D4D4⋊2S3S3×Q8Q8⋊3S3C8○D4C2×C8M4(2)C4○D4D4Q8S3C1
# reps13333111662213316284

In GAP, Magma, Sage, TeX 

S_3\times C_8\circ D_4
 
% in TeX
 
G:=Group("S3xC8oD4");
 
// GroupNames label
 
G:=SmallGroup(192,1308);
 
// by ID
 
G=gap.SmallGroup(192,1308);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,80,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^2=c^8=e^2=1,d^2=c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^4*d>;
 
// generators/relations
 

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