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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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×S3, C22×S3, C22×C6, C22×C6, C42⋊C2, C4×D4, C22≀C2, C4⋊D4, C22⋊Q8, C22.D4, C23×C4, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, S3×C23, C22.19C24, C23.26D6, C4×C3⋊D4, C23.23D6, C232D6, D63D4, D63Q8, S3×C22×C4, C6×C4○D4, (C2×D4)⋊43D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, C2×C3⋊D4, S3×C23, C22.19C24, S3×C4○D4, 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 7)(5 8)(6 9)(13 20)(14 21)(15 19)(16 24)(17 22)(18 23)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)(31 41)(32 42)(33 37)(34 38)(35 39)(36 40)
(1 19 4 23)(2 20 5 24)(3 21 6 22)(7 18 10 15)(8 16 11 13)(9 17 12 14)(25 40 28 37)(26 41 29 38)(27 42 30 39)(31 43 34 46)(32 44 35 47)(33 45 36 48)
(1 40)(2 38)(3 42)(4 37)(5 41)(6 39)(7 33)(8 31)(9 35)(10 36)(11 34)(12 32)(13 43)(14 47)(15 45)(16 46)(17 44)(18 48)(19 25)(20 29)(21 27)(22 30)(23 28)(24 26)
(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 12)(8 11)(9 10)(13 16)(14 18)(15 17)(19 22)(20 24)(21 23)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 38)(32 37)(33 42)(34 41)(35 40)(36 39)

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

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

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