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## G = D4×C3⋊D4order 192 = 26·3

### Direct product of D4 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — D4×C3⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — D4×C3⋊D4
 Lower central C3 — C2×C6 — D4×C3⋊D4
 Upper central C1 — C22 — C22×D4

Generators and relations for D4×C3⋊D4
G = < a,b,c,d,e | a4=b2=c3=d4=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 1240 in 428 conjugacy classes, 123 normal (43 characteristic)
C1, C2 [×3], C2 [×12], C3, C4 [×2], C4 [×7], C22, C22 [×6], C22 [×38], S3 [×4], C6 [×3], C6 [×8], C2×C4 [×2], C2×C4 [×13], D4 [×4], D4 [×30], C23, C23 [×4], C23 [×23], Dic3 [×2], Dic3 [×4], C12 [×2], C12, D6 [×2], D6 [×16], C2×C6, C2×C6 [×6], C2×C6 [×20], C42, C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×D4 [×28], C24 [×2], C24 [×2], C4×S3 [×2], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×2], C2×Dic3 [×4], C3⋊D4 [×4], C3⋊D4 [×18], C2×C12 [×2], C2×C12 [×2], C3×D4 [×4], C3×D4 [×6], C22×S3, C22×S3 [×2], C22×S3 [×10], C22×C6, C22×C6 [×4], C22×C6 [×10], C4×D4 [×2], C22≀C2 [×4], C4⋊D4 [×4], C41D4, C22×D4, C22×D4 [×3], C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C6.D4, C6.D4 [×4], S3×C2×C4, C2×D12, S3×D4 [×4], C22×Dic3 [×2], C2×C3⋊D4, C2×C3⋊D4 [×10], C2×C3⋊D4 [×8], C22×C12, C6×D4 [×2], C6×D4 [×2], C6×D4 [×4], S3×C23 [×2], C23×C6 [×2], D42, C4×C3⋊D4, C127D4, D4×Dic3, C232D6 [×2], D63D4, C23.14D6 [×2], C123D4, C244S3 [×2], C2×S3×D4, C22×C3⋊D4 [×2], D4×C2×C6, D4×C3⋊D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4 [×2], 2+ 1+4, S3×D4 [×2], C2×C3⋊D4 [×6], S3×C23, D42, C2×S3×D4, D46D6, C22×C3⋊D4, D4×C3⋊D4

Smallest permutation representation of D4×C3⋊D4
On 48 points
Generators in S48
(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)
(2 4)(5 7)(9 11)(13 15)(18 20)(21 23)(25 27)(29 31)(34 36)(37 39)(42 44)(46 48)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 34 29)(6 35 30)(7 36 31)(8 33 32)(9 25 44)(10 26 41)(11 27 42)(12 28 43)(21 37 48)(22 38 45)(23 39 46)(24 40 47)
(1 6 41 47)(2 7 42 48)(3 8 43 45)(4 5 44 46)(9 39 20 29)(10 40 17 30)(11 37 18 31)(12 38 19 32)(13 36 27 21)(14 33 28 22)(15 34 25 23)(16 35 26 24)
(1 3)(2 4)(5 48)(6 45)(7 46)(8 47)(9 27)(10 28)(11 25)(12 26)(13 20)(14 17)(15 18)(16 19)(21 29)(22 30)(23 31)(24 32)(33 40)(34 37)(35 38)(36 39)(41 43)(42 44)

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

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

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

45 conjugacy classes

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

45 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C3⋊D4 2+ 1+4 S3×D4 D4⋊6D6 kernel D4×C3⋊D4 C4×C3⋊D4 C12⋊7D4 D4×Dic3 C23⋊2D6 D6⋊3D4 C23.14D6 C12⋊3D4 C24⋊4S3 C2×S3×D4 C22×C3⋊D4 D4×C2×C6 C22×D4 C3⋊D4 C3×D4 C22×C4 C2×D4 C24 D4 C6 C22 C2 # reps 1 1 1 1 2 1 2 1 2 1 2 1 1 4 4 1 4 2 8 1 2 2

Matrix representation of D4×C3⋊D4 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 0 12 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 12
,
 12 12 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 2 4 0 0 2 11 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 12 12 0 0 0 0 12 0 0 0 0 12
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,0,1,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,12],[12,1,0,0,12,0,0,0,0,0,1,0,0,0,0,1],[2,2,0,0,4,11,0,0,0,0,1,0,0,0,0,1],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,0,12] >;

D4×C3⋊D4 in GAP, Magma, Sage, TeX

D_4\times C_3\rtimes D_4
% in TeX

G:=Group("D4xC3:D4");
// GroupNames label

G:=SmallGroup(192,1360);
// by ID

G=gap.SmallGroup(192,1360);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,6278]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^4=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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