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## G = (C2×D4)⋊43D6order 192 = 26·3

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

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
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (C2×D4)⋊43D6
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 >

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], C232D6 [×2], D63D4 [×2], D63Q8 [×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

Smallest permutation representation of (C2×D4)⋊43D6
On 48 points
Generators in S48
(1 10)(2 11)(3 12)(4 16)(5 17)(6 18)(7 14)(8 15)(9 13)(19 24)(20 22)(21 23)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
(1 20 4 8)(2 21 5 9)(3 19 6 7)(10 22 16 15)(11 23 17 13)(12 24 18 14)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 47 34 44)(32 48 35 45)(33 43 36 46)
(1 40)(2 38)(3 42)(4 37)(5 41)(6 39)(7 30)(8 28)(9 26)(10 36)(11 34)(12 32)(13 44)(14 48)(15 46)(16 33)(17 31)(18 35)(19 27)(20 25)(21 29)(22 43)(23 47)(24 45)
(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 6)(2 5)(3 4)(7 20)(8 19)(9 21)(10 18)(11 17)(12 16)(13 23)(14 22)(15 24)(25 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 6A 6B 6C 6D ··· 6I 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 ··· 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 4 6 6 6 6 2 1 1 1 1 2 2 4 4 6 6 6 6 12 12 12 12 2 2 2 4 ··· 4 2 2 2 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 S3×C4○D4 kernel (C2×D4)⋊43D6 C23.26D6 C4×C3⋊D4 C23.23D6 C23⋊2D6 D6⋊3D4 D6⋊3Q8 S3×C22×C4 C6×C4○D4 C2×C4○D4 C2×C12 C22×C4 C2×D4 C2×Q8 D6 C2×C4 C2 # reps 1 1 4 2 2 2 2 1 1 1 4 3 3 1 8 8 4

Matrix representation of (C2×D4)⋊43D6 in GL6(𝔽13)

 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 5 0 0 0 0 8 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 10 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 12 0 0 0 0 1 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 12 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 5 12

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] >;

(C2×D4)⋊43D6 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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