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G = C2×Q8⋊3D6  order 192 = 26·3

Direct product of C2 and Q8⋊3D6

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

Aliases: C2×Q8⋊3D6, C24⋊4C23, SD16⋊8D6, D12⋊2C23, C12.6C24, D24⋊20C22, (C2×C8)⋊10D6, C3⋊C8⋊2C23, C8⋊4(C22×S3), (C2×Q8)⋊24D6, C4.43(S3×D4), (C2×D24)⋊26C2, C6⋊3(C8⋊C22), (C6×SD16)⋊5C2, (C2×SD16)⋊4S3, (C4×S3).15D4, D6.50(C2×D4), C12.81(C2×D4), (S3×D4)⋊6C22, C4.6(S3×C23), (C3×Q8)⋊2C23, Q8⋊3(C22×S3), (C2×C24)⋊13C22, C8⋊S3⋊8C22, D4⋊S3⋊10C22, (C2×D4).182D6, (C4×S3).3C23, (C6×Q8)⋊18C22, (C3×D4).4C23, D4.4(C22×S3), (C2×D12)⋊33C22, Dic3.55(C2×D4), Q8⋊2S3⋊8C22, Q8⋊3S3⋊5C22, (C3×SD16)⋊8C22, (C22×S3).98D4, C22.139(S3×D4), C6.107(C22×D4), (C2×C12).523C23, (C2×Dic3).192D4, (C6×D4).164C22, (C2×S3×D4)⋊23C2, C3⋊3(C2×C8⋊C22), C2.80(C2×S3×D4), (C2×C8⋊S3)⋊4C2, (C2×D4⋊S3)⋊27C2, (C2×C3⋊C8)⋊15C22, (C2×C6).396(C2×D4), (C2×Q8⋊3S3)⋊14C2, (C2×Q8⋊2S3)⋊26C2, (S3×C2×C4).156C22, (C2×C4).612(C22×S3), SmallGroup(192,1318)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — C2×Q8⋊3D6
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×S3×D4 — C2×Q8⋊3D6
Lower central
C3 — C6 — C12 — C2×Q8⋊3D6

Subgroups: 952 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×4], C22, C22 [×24], S3 [×6], C6, C6 [×2], C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×10], D4 [×2], D4 [×15], Q8 [×2], Q8, C23 [×12], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×18], C2×C6, C2×C6 [×4], C2×C8, C2×C8, M4(2) [×4], D8 [×8], SD16 [×4], SD16 [×4], C22×C4 [×2], C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×6], C24, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C4×S3 [×4], D12 [×4], D12 [×6], C2×Dic3, C3⋊D4 [×4], C2×C12, C2×C12, C3×D4 [×2], C3×D4, C3×Q8 [×2], C3×Q8, C22×S3, C22×S3 [×10], C22×C6, C2×M4(2), C2×D8 [×2], C2×SD16, C2×SD16, C8⋊C22 [×8], C22×D4, C2×C4○D4, C8⋊S3 [×4], D24 [×4], C2×C3⋊C8, D4⋊S3 [×4], Q8⋊2S3 [×4], C2×C24, C3×SD16 [×4], S3×C2×C4, S3×C2×C4, C2×D12 [×2], C2×D12, S3×D4 [×4], S3×D4 [×2], Q8⋊3S3 [×4], Q8⋊3S3 [×2], C2×C3⋊D4, C6×D4, C6×Q8, S3×C23, C2×C8⋊C22, C2×C8⋊S3, C2×D24, Q8⋊3D6 [×8], C2×D4⋊S3, C2×Q8⋊2S3, C6×SD16, C2×S3×D4, C2×Q8⋊3S3, C2×Q8⋊3D6

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C8⋊C22 [×2], C22×D4, S3×D4 [×2], S3×C23, C2×C8⋊C22, Q8⋊3D6 [×2], C2×S3×D4, C2×Q8⋊3D6

Generators and relations
 G = < a,b,c,d,e | a2=b4=d6=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe=b-1, dcd-1=b-1c, ece=bc, ede=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽73)

7200000
0720000
001000
000100
000010
000001
,
7200000
0720000
0000721
0072727172
00252510
00242510
,
72700000
010000
0039104444
00345044
0068686863
0039683934
,
7200000
2510000
001112
0000721
00720072
00072072
,
100000
48720000
0000172
0072727271
001001
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],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,25,24,0,0,0,72,25,25,0,0,72,71,1,1,0,0,1,72,0,0],[72,0,0,0,0,0,70,1,0,0,0,0,0,0,39,34,68,39,0,0,10,5,68,68,0,0,44,0,68,39,0,0,44,44,63,34],[72,25,0,0,0,0,0,1,0,0,0,0,0,0,1,0,72,0,0,0,1,0,0,72,0,0,1,72,0,0,0,0,2,1,72,72],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,72,71,1,1] >;
 

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F6A6B6C6D6E8A8B8C8D12A12B12C12D24A24B24C24D
order12222222222234444446666688881212121224242424
size111144661212121222244662228844121244884444

36 irreducible representations

dim111111111222222224444
type+++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D4D4D6D6D6D6C8⋊C22S3×D4S3×D4Q8⋊3D6
kernelC2×Q8⋊3D6C2×C8⋊S3C2×D24Q8⋊3D6C2×D4⋊S3C2×Q8⋊2S3C6×SD16C2×S3×D4C2×Q8⋊3S3C2×SD16C4×S3C2×Dic3C22×S3C2×C8SD16C2×D4C2×Q8C6C4C22C2
# reps111811111121114112114

In GAP, Magma, Sage, TeX 

C_2\times Q_8\rtimes_3D_6
 
% in TeX
 
G:=Group("C2xQ8:3D6");
 
// GroupNames label
 
G:=SmallGroup(192,1318);
 
// by ID
 
G=gap.SmallGroup(192,1318);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,185,136,438,235,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=d^6=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e=b^-1,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
 
// generators/relations
 

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