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## G = C2×D4⋊Dic3order 192 = 26·3

### Direct product of C2 and D4⋊Dic3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C2×D4⋊Dic3
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C2×D4⋊Dic3
 Lower central C3 — C6 — C12 — C2×D4⋊Dic3
 Upper central C1 — C23 — C22×C4 — C22×D4

Generators and relations for C2×D4⋊Dic3
G = < a,b,c,d,e | a2=b4=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 488 in 202 conjugacy classes, 87 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×2], C22, C22 [×6], C22 [×16], C6 [×3], C6 [×4], C6 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], D4 [×4], D4 [×6], C23, C23 [×10], Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×6], C2×C6 [×16], C4⋊C4 [×3], C2×C8 [×4], C22×C4, C22×C4, C2×D4 [×6], C2×D4 [×3], C24, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×4], C3×D4 [×4], C3×D4 [×6], C22×C6, C22×C6 [×10], D4⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×D4, C2×C3⋊C8 [×2], C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C22×Dic3, C22×C12, C6×D4 [×6], C6×D4 [×3], C23×C6, C2×D4⋊C4, D4⋊Dic3 [×4], C22×C3⋊C8, C2×C4⋊Dic3, D4×C2×C6, C2×D4⋊Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, D4⋊C4 [×4], C2×C22⋊C4, C2×D8, C2×SD16, D4⋊S3 [×2], D4.S3 [×2], C6.D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×D4⋊C4, D4⋊Dic3 [×4], C2×D4⋊S3, C2×D4.S3, C2×C6.D4, C2×D4⋊Dic3

Smallest permutation representation of C2×D4⋊Dic3
On 96 points
Generators in S96
(1 41)(2 42)(3 37)(4 38)(5 39)(6 40)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)(25 53)(26 54)(27 49)(28 50)(29 51)(30 52)(55 82)(56 83)(57 84)(58 79)(59 80)(60 81)(61 88)(62 89)(63 90)(64 85)(65 86)(66 87)(67 94)(68 95)(69 96)(70 91)(71 92)(72 93)
(1 29 19 12)(2 30 20 7)(3 25 21 8)(4 26 22 9)(5 27 23 10)(6 28 24 11)(13 87 79 95)(14 88 80 96)(15 89 81 91)(16 90 82 92)(17 85 83 93)(18 86 84 94)(31 39 49 44)(32 40 50 45)(33 41 51 46)(34 42 52 47)(35 37 53 48)(36 38 54 43)(55 71 76 63)(56 72 77 64)(57 67 78 65)(58 68 73 66)(59 69 74 61)(60 70 75 62)
(1 12)(2 7)(3 8)(4 9)(5 10)(6 11)(19 29)(20 30)(21 25)(22 26)(23 27)(24 28)(31 39)(32 40)(33 41)(34 42)(35 37)(36 38)(43 54)(44 49)(45 50)(46 51)(47 52)(48 53)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)
(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 90 4 87)(2 89 5 86)(3 88 6 85)(7 81 10 84)(8 80 11 83)(9 79 12 82)(13 29 16 26)(14 28 17 25)(15 27 18 30)(19 92 22 95)(20 91 23 94)(21 96 24 93)(31 57 34 60)(32 56 35 59)(33 55 36 58)(37 61 40 64)(38 66 41 63)(39 65 42 62)(43 68 46 71)(44 67 47 70)(45 72 48 69)(49 78 52 75)(50 77 53 74)(51 76 54 73)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D6 D8 SD16 C3⋊D4 C3⋊D4 D4⋊S3 D4.S3 kernel C2×D4⋊Dic3 D4⋊Dic3 C22×C3⋊C8 C2×C4⋊Dic3 D4×C2×C6 C6×D4 C22×D4 C2×C12 C22×C6 C22×C4 C2×D4 C2×D4 C2×C6 C2×C6 C2×C4 C23 C22 C22 # reps 1 4 1 1 1 8 1 3 1 1 4 2 4 4 6 2 2 2

Matrix representation of C2×D4⋊Dic3 in GL6(𝔽73)

 72 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 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 8 0 0 0 0 0 0 64 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 27 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 67 0 0 0 0 67 67

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,8,0,0,0,0,0,0,64,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,67,0,0,0,0,67,67] >;

C2×D4⋊Dic3 in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes {\rm Dic}_3
% in TeX

G:=Group("C2xD4:Dic3");
// GroupNames label

G:=SmallGroup(192,773);
// by ID

G=gap.SmallGroup(192,773);
# by ID

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

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

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