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## G = C42⋊3Dic3order 192 = 26·3

### 1st semidirect product of C42 and Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C42⋊3Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×C12 — C23.26D6 — C42⋊3Dic3
 Lower central C3 — C12 — C42⋊3Dic3
 Upper central C1 — C4 — C42⋊C2

Generators and relations for C423Dic3
G = < a,b,c,d | a4=b4=c6=1, d2=c3, ab=ba, cac-1=ab2, dad-1=ab-1, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 216 in 94 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C22×C6, C42⋊C2, C42⋊C2, C2×M4(2), C2×C3⋊C8, C4.Dic3, C4×Dic3, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C4.9C42, C2×C4.Dic3, C23.26D6, C3×C42⋊C2, C423Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4.9C42, C6.C42, C423Dic3

Smallest permutation representation of C423Dic3
On 48 points
Generators in S48
(1 36 22 42)(2 34 23 40)(3 32 24 38)(4 26 18 47)(5 30 16 45)(6 28 17 43)(7 29 14 44)(8 27 15 48)(9 25 13 46)(10 33 21 39)(11 31 19 37)(12 35 20 41)
(1 6 10 9)(2 4 11 7)(3 5 12 8)(13 22 17 21)(14 23 18 19)(15 24 16 20)(25 36 28 33)(26 31 29 34)(27 32 30 35)(37 44 40 47)(38 45 41 48)(39 46 42 43)
(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 17)(2 16)(3 18)(4 20)(5 19)(6 21)(7 24)(8 23)(9 22)(10 13)(11 15)(12 14)(25 43 28 46)(26 48 29 45)(27 47 30 44)(31 38 34 41)(32 37 35 40)(33 42 36 39)

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12N order 1 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 ··· 12 size 1 1 2 2 2 2 1 1 2 2 2 4 4 4 4 12 12 12 12 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 2 2 4 4 type + + + + + + - + - + - + image C1 C2 C2 C2 C4 C4 C4 S3 D4 Q8 D4 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 C3⋊D4 C4.9C42 C42⋊3Dic3 kernel C42⋊3Dic3 C2×C4.Dic3 C23.26D6 C3×C42⋊C2 C2×C3⋊C8 C4×Dic3 C4×C12 C42⋊C2 C2×C12 C2×C12 C22×C6 C42 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C3 C1 # reps 1 1 1 1 4 4 4 1 2 1 1 2 1 2 4 2 2 2 2 4

Matrix representation of C423Dic3 in GL4(𝔽73) generated by

 0 0 1 0 0 0 0 1 30 60 0 0 13 43 0 0
,
 46 0 0 0 0 46 0 0 0 0 46 0 0 0 0 46
,
 0 72 0 0 1 1 0 0 0 0 0 1 0 0 72 72
,
 13 43 0 0 30 60 0 0 0 0 59 66 0 0 7 14
G:=sub<GL(4,GF(73))| [0,0,30,13,0,0,60,43,1,0,0,0,0,1,0,0],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,1,0,0,72,1,0,0,0,0,0,72,0,0,1,72],[13,30,0,0,43,60,0,0,0,0,59,7,0,0,66,14] >;

C423Dic3 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C4^2:3Dic3");
// GroupNames label

G:=SmallGroup(192,90);
// by ID

G=gap.SmallGroup(192,90);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1123,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=c^3,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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