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## G = C2×D4⋊D6order 192 = 26·3

### Direct product of C2 and D4⋊D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D4⋊D6
 Chief series C1 — C3 — C6 — C12 — D12 — C2×D12 — C22×D12 — C2×D4⋊D6
 Lower central C3 — C6 — C12 — C2×D4⋊D6
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for C2×D4⋊D6
G = < a,b,c,d,e | a2=b4=c2=d6=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b2c, ece=b-1c, ede=d-1 >

Subgroups: 872 in 298 conjugacy classes, 111 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×8], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×2], C22 [×22], S3 [×4], C6, C6 [×2], C6 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×5], D4 [×2], D4 [×15], Q8 [×2], Q8, C23, C23 [×11], C12 [×2], C12 [×2], C12 [×2], D6 [×16], C2×C6, C2×C6 [×2], C2×C6 [×6], C2×C8 [×2], M4(2) [×4], D8 [×8], SD16 [×8], C22×C4, C22×C4, C2×D4, C2×D4 [×10], C2×Q8, C4○D4 [×4], C4○D4 [×2], C24, C3⋊C8 [×4], D12 [×4], D12 [×6], C2×C12 [×2], C2×C12 [×4], C2×C12 [×5], C3×D4 [×2], C3×D4 [×5], C3×Q8 [×2], C3×Q8, C22×S3 [×10], C22×C6, C22×C6, C2×M4(2), C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C22×D4, C2×C4○D4, C2×C3⋊C8 [×2], C4.Dic3 [×4], D4⋊S3 [×8], Q82S3 [×8], C2×D12 [×6], C2×D12 [×3], C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4 [×4], C3×C4○D4 [×2], S3×C23, C2×C8⋊C22, C2×C4.Dic3, C2×D4⋊S3 [×2], C2×Q82S3 [×2], D4⋊D6 [×8], C22×D12, C6×C4○D4, C2×D4⋊D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C8⋊C22 [×2], C22×D4, C2×C3⋊D4 [×6], S3×C23, C2×C8⋊C22, D4⋊D6 [×2], C22×C3⋊D4, C2×D4⋊D6

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

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D ··· 6I 8A 8B 8C 8D 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 6 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 4 4 12 12 12 12 2 2 2 2 2 4 4 2 2 2 4 ··· 4 12 12 12 12 2 2 2 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 D6 C3⋊D4 C3⋊D4 C8⋊C22 D4⋊D6 kernel C2×D4⋊D6 C2×C4.Dic3 C2×D4⋊S3 C2×Q8⋊2S3 D4⋊D6 C22×D12 C6×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C6 C2 # reps 1 1 2 2 8 1 1 1 3 1 1 1 1 4 6 2 2 4

Matrix representation of C2×D4⋊D6 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 66 14 0 0 0 0 59 7 0 0 0 0 0 0 7 59 0 0 0 0 14 66
,
 43 60 0 0 0 0 13 30 0 0 0 0 0 0 0 0 66 14 0 0 0 0 59 7 0 0 7 59 0 0 0 0 14 66 0 0
,
 1 72 0 0 0 0 1 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 72 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 66 14 0 0 0 0 7 7

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,0,0,0,0,7,14,0,0,0,0,59,66],[43,13,0,0,0,0,60,30,0,0,0,0,0,0,0,0,7,14,0,0,0,0,59,66,0,0,66,59,0,0,0,0,14,7,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,66,7,0,0,0,0,14,7] >;

C2×D4⋊D6 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes D_6
% in TeX

G:=Group("C2xD4:D6");
// GroupNames label

G:=SmallGroup(192,1379);
// by ID

G=gap.SmallGroup(192,1379);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,1684,235,102,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^6=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

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