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## G = C4○D4⋊3Dic3order 192 = 26·3

### 1st semidirect product of C4○D4 and Dic3 acting via Dic3/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C4○D4⋊3Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C4○D4⋊3Dic3
 Lower central C3 — C6 — C12 — C4○D4⋊3Dic3
 Upper central C1 — C22 — C22×C4 — C2×C4○D4

Generators and relations for C4○D43Dic3
G = < a,b,c,d,e | a4=c2=d6=1, b2=a2, e2=d3, ab=ba, ac=ca, ad=da, eae-1=a-1, cbc=a2b, bd=db, ebe-1=abc, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 360 in 162 conjugacy classes, 71 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C3×Q8, C22×C6, C22×C6, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C2×C3⋊C8, C4.Dic3, C4⋊Dic3, C4⋊Dic3, C22×Dic3, C22×C12, C22×C12, C6×D4, C6×D4, C6×Q8, C3×C4○D4, C3×C4○D4, C23.36D4, D4⋊Dic3, Q82Dic3, C2×C4.Dic3, C2×C4⋊Dic3, C6×C4○D4, C4○D43Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, C6.D4, C22×Dic3, C2×C3⋊D4, C23.36D4, D4⋊D6, Q8.14D6, C2×C6.D4, C4○D43Dic3

Smallest permutation representation of C4○D43Dic3
On 96 points
Generators in S96
(1 51 23 31)(2 52 24 32)(3 53 19 33)(4 54 20 34)(5 49 21 35)(6 50 22 36)(7 39 28 47)(8 40 29 48)(9 41 30 43)(10 42 25 44)(11 37 26 45)(12 38 27 46)(13 65 81 70)(14 66 82 71)(15 61 83 72)(16 62 84 67)(17 63 79 68)(18 64 80 69)(55 92 78 87)(56 93 73 88)(57 94 74 89)(58 95 75 90)(59 96 76 85)(60 91 77 86)
(1 47 23 39)(2 48 24 40)(3 43 19 41)(4 44 20 42)(5 45 21 37)(6 46 22 38)(7 31 28 51)(8 32 29 52)(9 33 30 53)(10 34 25 54)(11 35 26 49)(12 36 27 50)(13 65 81 70)(14 66 82 71)(15 61 83 72)(16 62 84 67)(17 63 79 68)(18 64 80 69)(55 87 78 92)(56 88 73 93)(57 89 74 94)(58 90 75 95)(59 85 76 96)(60 86 77 91)
(1 39)(2 40)(3 41)(4 42)(5 37)(6 38)(7 31)(8 32)(9 33)(10 34)(11 35)(12 36)(13 74)(14 75)(15 76)(16 77)(17 78)(18 73)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(25 54)(26 49)(27 50)(28 51)(29 52)(30 53)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 65 4 62)(2 64 5 61)(3 63 6 66)(7 57 10 60)(8 56 11 59)(9 55 12 58)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 68 22 71)(20 67 23 70)(21 72 24 69)(25 77 28 74)(26 76 29 73)(27 75 30 78)(31 81 34 84)(32 80 35 83)(33 79 36 82)(37 85 40 88)(38 90 41 87)(39 89 42 86)(43 92 46 95)(44 91 47 94)(45 96 48 93)

G:=sub<Sym(96)| (1,51,23,31)(2,52,24,32)(3,53,19,33)(4,54,20,34)(5,49,21,35)(6,50,22,36)(7,39,28,47)(8,40,29,48)(9,41,30,43)(10,42,25,44)(11,37,26,45)(12,38,27,46)(13,65,81,70)(14,66,82,71)(15,61,83,72)(16,62,84,67)(17,63,79,68)(18,64,80,69)(55,92,78,87)(56,93,73,88)(57,94,74,89)(58,95,75,90)(59,96,76,85)(60,91,77,86), (1,47,23,39)(2,48,24,40)(3,43,19,41)(4,44,20,42)(5,45,21,37)(6,46,22,38)(7,31,28,51)(8,32,29,52)(9,33,30,53)(10,34,25,54)(11,35,26,49)(12,36,27,50)(13,65,81,70)(14,66,82,71)(15,61,83,72)(16,62,84,67)(17,63,79,68)(18,64,80,69)(55,87,78,92)(56,88,73,93)(57,89,74,94)(58,90,75,95)(59,85,76,96)(60,86,77,91), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,74)(14,75)(15,76)(16,77)(17,78)(18,73)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,54)(26,49)(27,50)(28,51)(29,52)(30,53)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,65,4,62)(2,64,5,61)(3,63,6,66)(7,57,10,60)(8,56,11,59)(9,55,12,58)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,81,34,84)(32,80,35,83)(33,79,36,82)(37,85,40,88)(38,90,41,87)(39,89,42,86)(43,92,46,95)(44,91,47,94)(45,96,48,93)>;

G:=Group( (1,51,23,31)(2,52,24,32)(3,53,19,33)(4,54,20,34)(5,49,21,35)(6,50,22,36)(7,39,28,47)(8,40,29,48)(9,41,30,43)(10,42,25,44)(11,37,26,45)(12,38,27,46)(13,65,81,70)(14,66,82,71)(15,61,83,72)(16,62,84,67)(17,63,79,68)(18,64,80,69)(55,92,78,87)(56,93,73,88)(57,94,74,89)(58,95,75,90)(59,96,76,85)(60,91,77,86), (1,47,23,39)(2,48,24,40)(3,43,19,41)(4,44,20,42)(5,45,21,37)(6,46,22,38)(7,31,28,51)(8,32,29,52)(9,33,30,53)(10,34,25,54)(11,35,26,49)(12,36,27,50)(13,65,81,70)(14,66,82,71)(15,61,83,72)(16,62,84,67)(17,63,79,68)(18,64,80,69)(55,87,78,92)(56,88,73,93)(57,89,74,94)(58,90,75,95)(59,85,76,96)(60,86,77,91), (1,39)(2,40)(3,41)(4,42)(5,37)(6,38)(7,31)(8,32)(9,33)(10,34)(11,35)(12,36)(13,74)(14,75)(15,76)(16,77)(17,78)(18,73)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(25,54)(26,49)(27,50)(28,51)(29,52)(30,53)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,85)(62,86)(63,87)(64,88)(65,89)(66,90)(67,91)(68,92)(69,93)(70,94)(71,95)(72,96), (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)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,65,4,62)(2,64,5,61)(3,63,6,66)(7,57,10,60)(8,56,11,59)(9,55,12,58)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,68,22,71)(20,67,23,70)(21,72,24,69)(25,77,28,74)(26,76,29,73)(27,75,30,78)(31,81,34,84)(32,80,35,83)(33,79,36,82)(37,85,40,88)(38,90,41,87)(39,89,42,86)(43,92,46,95)(44,91,47,94)(45,96,48,93) );

G=PermutationGroup([[(1,51,23,31),(2,52,24,32),(3,53,19,33),(4,54,20,34),(5,49,21,35),(6,50,22,36),(7,39,28,47),(8,40,29,48),(9,41,30,43),(10,42,25,44),(11,37,26,45),(12,38,27,46),(13,65,81,70),(14,66,82,71),(15,61,83,72),(16,62,84,67),(17,63,79,68),(18,64,80,69),(55,92,78,87),(56,93,73,88),(57,94,74,89),(58,95,75,90),(59,96,76,85),(60,91,77,86)], [(1,47,23,39),(2,48,24,40),(3,43,19,41),(4,44,20,42),(5,45,21,37),(6,46,22,38),(7,31,28,51),(8,32,29,52),(9,33,30,53),(10,34,25,54),(11,35,26,49),(12,36,27,50),(13,65,81,70),(14,66,82,71),(15,61,83,72),(16,62,84,67),(17,63,79,68),(18,64,80,69),(55,87,78,92),(56,88,73,93),(57,89,74,94),(58,90,75,95),(59,85,76,96),(60,86,77,91)], [(1,39),(2,40),(3,41),(4,42),(5,37),(6,38),(7,31),(8,32),(9,33),(10,34),(11,35),(12,36),(13,74),(14,75),(15,76),(16,77),(17,78),(18,73),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(25,54),(26,49),(27,50),(28,51),(29,52),(30,53),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,85),(62,86),(63,87),(64,88),(65,89),(66,90),(67,91),(68,92),(69,93),(70,94),(71,95),(72,96)], [(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),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,65,4,62),(2,64,5,61),(3,63,6,66),(7,57,10,60),(8,56,11,59),(9,55,12,58),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,68,22,71),(20,67,23,70),(21,72,24,69),(25,77,28,74),(26,76,29,73),(27,75,30,78),(31,81,34,84),(32,80,35,83),(33,79,36,82),(37,85,40,88),(38,90,41,87),(39,89,42,86),(43,92,46,95),(44,91,47,94),(45,96,48,93)]])

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D ··· 6I 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12J order 1 2 2 2 2 2 2 2 3 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 1 1 2 2 4 4 2 2 2 2 2 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 4 4 4 4 type + + + + + + + + + + + + - + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 D6 Dic3 C3⋊D4 C3⋊D4 C8⋊C22 C8.C22 D4⋊D6 Q8.14D6 kernel C4○D4⋊3Dic3 D4⋊Dic3 Q8⋊2Dic3 C2×C4.Dic3 C2×C4⋊Dic3 C6×C4○D4 C3×C4○D4 C2×C4○D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×Q8 C4○D4 C2×C4 C23 C6 C6 C2 C2 # reps 1 2 2 1 1 1 8 1 3 1 1 1 1 4 6 2 1 1 2 2

Matrix representation of C4○D43Dic3 in GL6(𝔽73)

 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 66 14 0 0 0 0 59 7
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 66 14 0 0 0 0 59 7 0 0 66 14 0 0 0 0 59 7 0 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 0 7 59 0 0 0 0 14 66 0 0 66 14 0 0 0 0 59 7 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 72 1 0 0 0 0 72 0
,
 10 51 0 0 0 0 61 63 0 0 0 0 0 0 72 60 1 13 0 0 59 1 14 72 0 0 1 13 1 13 0 0 14 72 14 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,0,0,0,0,66,59,0,0,0,0,14,7],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,66,59,0,0,0,0,14,7,0,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,7,14,0,0,0,0,59,66,0,0],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[10,61,0,0,0,0,51,63,0,0,0,0,0,0,72,59,1,14,0,0,60,1,13,72,0,0,1,14,1,14,0,0,13,72,13,72] >;

C4○D43Dic3 in GAP, Magma, Sage, TeX

C_4\circ D_4\rtimes_3{\rm Dic}_3
% in TeX

G:=Group("C4oD4:3Dic3");
// GroupNames label

G:=SmallGroup(192,791);
// by ID

G=gap.SmallGroup(192,791);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,1684,438,102,6278]);
// Polycyclic

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

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