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G = (C2×D4)⋊43D6  order 192 = 26·3

11st semidirect product of C2×D4 and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4)⋊43D6, (C2×C12)⋊15D4, (C2×Q8)⋊35D6, D6⋊6(C4○D4), (C22×C4)⋊32D6, C23⋊2D6⋊34C2, D6⋊3D4⋊46C2, D6⋊3Q8⋊47C2, C12.265(C2×D4), (C6×D4)⋊47C22, (C6×Q8)⋊39C22, (C2×C6).314C24, C4⋊Dic3⋊80C22, C6.164(C22×D4), (C2×C12).651C23, Dic3⋊C4⋊76C22, D6⋊C4.159C22, (C22×C12)⋊42C22, C3⋊8(C22.19C24), (C4×Dic3)⋊60C22, C23.23D6⋊34C2, C6.D4⋊65C22, C23.26D6⋊38C2, C23.147(C22×S3), (C22×C6).240C23, C22.323(S3×C23), (S3×C23).115C22, (C22×S3).243C23, (C2×Dic3).293C23, (C22×Dic3).236C22, (C6×C4○D4)⋊6C2, (S3×C22×C4)⋊7C2, (C2×C4○D4)⋊10S3, (C4×C3⋊D4)⋊60C2, (C2×C6).80(C2×D4), (C2×C4)⋊14(C3⋊D4), C6.215(C2×C4○D4), C2.103(S3×C4○D4), C4.100(C2×C3⋊D4), (S3×C2×C4).263C22, C22.23(C2×C3⋊D4), C2.37(C22×C3⋊D4), (C2×C4).639(C22×S3), (C2×C3⋊D4).140C22, SmallGroup(192,1387)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — (C2×D4)⋊43D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — (C2×D4)⋊43D6
Lower central
C3 — C2×C6 — (C2×D4)⋊43D6

Subgroups: 808 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], S3 [×4], C6, C6 [×2], C6 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], Dic3 [×6], C12 [×4], C12 [×2], D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×8], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, C4×S3 [×8], C2×Dic3 [×6], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×6], C2×C12 [×4], C3×D4 [×6], C3×Q8 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×2], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×Dic3 [×2], Dic3⋊C4 [×4], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×6], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C2×C3⋊D4 [×4], C22×C12, C22×C12 [×2], C6×D4, C6×D4 [×2], C6×Q8, C3×C4○D4 [×4], S3×C23, C22.19C24, C23.26D6, C4×C3⋊D4 [×4], C23.23D6 [×2], C23⋊2D6 [×2], D6⋊3D4 [×2], D6⋊3Q8 [×2], S3×C22×C4, C6×C4○D4, (C2×D4)⋊43D6

Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, C2×C4○D4 [×2], C2×C3⋊D4 [×6], S3×C23, C22.19C24, S3×C4○D4 [×2], C22×C3⋊D4, (C2×D4)⋊43D6

Generators and relations
 G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ece=ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b2c, ede=d-1 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

100000
010000
001000
000100
0000120
0000012
,
500000
080000
001000
000100
000010
000001
,
050000
800000
0012000
0001200
0000110
0000012
,
100000
0120000
00121200
001000
000010
000001
,
1200000
0120000
00121200
000100
000010
0000512

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,10,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,5,0,0,0,0,0,12] >;
 

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K 3 4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P6A6B6C6D···6I12A12B12C12D12E···12J
order122222222222344444444444444446666···61212121212···12
size1111224466662111122446666121212122224···422224···4

48 irreducible representations

dim11111111122222224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D6D6D6C4○D4C3⋊D4S3×C4○D4
kernel(C2×D4)⋊43D6C23.26D6C4×C3⋊D4C23.23D6C23⋊2D6D6⋊3D4D6⋊3Q8S3×C22×C4C6×C4○D4C2×C4○D4C2×C12C22×C4C2×D4C2×Q8D6C2×C4C2
# reps11422221114331884

In GAP, Magma, Sage, TeX 

(C_2\times D_4)\rtimes_{43}D_6
 
% in TeX
 
G:=Group("(C2xD4):43D6");
 
// GroupNames label
 
G:=SmallGroup(192,1387);
 
// by ID
 
G=gap.SmallGroup(192,1387);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,570,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,e*c*e=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,d*c*d^-1=b^2*c,e*d*e=d^-1>;
 
// generators/relations
 

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