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## G = Dic3×C22⋊C4order 192 = 26·3

### Direct product of Dic3 and C22⋊C4

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic3×C22⋊C4
G = < a,b,c,d,e | a6=c2=d2=e4=1, b2=a3, bab-1=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 552 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C23×C4, C4×Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C4×C22⋊C4, C6.C42, C2×C4×Dic3, C2×C6.D4, C6×C22⋊C4, C23×Dic3, Dic3×C22⋊C4
Quotients:

Smallest permutation representation of Dic3×C22⋊C4
On 96 points
Generators in S96
(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 58 4 55)(2 57 5 60)(3 56 6 59)(7 64 10 61)(8 63 11 66)(9 62 12 65)(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 83 34 80)(32 82 35 79)(33 81 36 84)(37 86 40 89)(38 85 41 88)(39 90 42 87)(43 92 46 95)(44 91 47 94)(45 96 48 93)
(1 11)(2 12)(3 7)(4 8)(5 9)(6 10)(13 79)(14 80)(15 81)(16 82)(17 83)(18 84)(19 27)(20 28)(21 29)(22 30)(23 25)(24 26)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 63)(56 64)(57 65)(58 66)(59 61)(60 62)(67 74)(68 75)(69 76)(70 77)(71 78)(72 73)(85 95)(86 96)(87 91)(88 92)(89 93)(90 94)
(1 23)(2 24)(3 19)(4 20)(5 21)(6 22)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 87)(14 88)(15 89)(16 90)(17 85)(18 86)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(37 52)(38 53)(39 54)(40 49)(41 50)(42 51)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(61 78)(62 73)(63 74)(64 75)(65 76)(66 77)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(1 42 8 32)(2 37 9 33)(3 38 10 34)(4 39 11 35)(5 40 12 36)(6 41 7 31)(13 74 94 70)(14 75 95 71)(15 76 96 72)(16 77 91 67)(17 78 92 68)(18 73 93 69)(19 53 30 43)(20 54 25 44)(21 49 26 45)(22 50 27 46)(23 51 28 47)(24 52 29 48)(55 90 66 79)(56 85 61 80)(57 86 62 81)(58 87 63 82)(59 88 64 83)(60 89 65 84)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 S3 D4 Dic3 D6 D6 C4○D4 C4×S3 S3×D4 D4⋊2S3 kernel Dic3×C22⋊C4 C6.C42 C2×C4×Dic3 C2×C6.D4 C6×C22⋊C4 C23×Dic3 C6.D4 C3×C22⋊C4 C22×Dic3 C2×C22⋊C4 C2×Dic3 C22⋊C4 C22×C4 C24 C2×C6 C23 C22 C22 # reps 1 2 2 1 1 1 8 8 8 1 4 4 2 1 4 8 2 2

Matrix representation of Dic3×C22⋊C4 in GL6(𝔽13)

 1 12 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 6 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 8 5 0 0 0 0 0 0 7 6 0 0 0 0 9 6 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1
,
 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 12 0 0 0 0 0 0 12
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,10,6,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,8,0,0,0,0,0,5,0,0,0,0,0,0,7,9,0,0,0,0,6,6,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

Dic3×C22⋊C4 in GAP, Magma, Sage, TeX

{\rm Dic}_3\times C_2^2\rtimes C_4
% in TeX

G:=Group("Dic3xC2^2:C4");
// GroupNames label

G:=SmallGroup(192,500);
// by ID

G=gap.SmallGroup(192,500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,387,100,6278]);
// Polycyclic

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

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