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## G = C22×D6⋊C4order 192 = 26·3

### Direct product of C22 and D6⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×D6⋊C4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C24 — C22×D6⋊C4
 Lower central C3 — C6 — C22×D6⋊C4
 Upper central C1 — C24 — C23×C4

Generators and relations for C22×D6⋊C4
G = < a,b,c,d,e | a2=b2=c6=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c3d >

Subgroups: 1784 in 674 conjugacy classes, 247 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C23×C4, C23×C4, C25, D6⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, S3×C23, C23×C6, C22×C22⋊C4, C2×D6⋊C4, C23×Dic3, C23×C12, S3×C24, C22×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C24, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, S3×C23, C22×C22⋊C4, C2×D6⋊C4, S3×C22×C4, C22×D12, C22×C3⋊D4, C22×D6⋊C4

Smallest permutation representation of C22×D6⋊C4
On 96 points
Generators in S96
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 52)(8 53)(9 54)(10 49)(11 50)(12 51)(13 70)(14 71)(15 72)(16 67)(17 68)(18 69)(19 64)(20 65)(21 66)(22 61)(23 62)(24 63)(25 82)(26 83)(27 84)(28 79)(29 80)(30 81)(31 76)(32 77)(33 78)(34 73)(35 74)(36 75)(37 94)(38 95)(39 96)(40 91)(41 92)(42 93)(43 88)(44 89)(45 90)(46 85)(47 86)(48 87)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 42)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 32)(8 31)(9 36)(10 35)(11 34)(12 33)(13 41)(14 40)(15 39)(16 38)(17 37)(18 42)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(49 74)(50 73)(51 78)(52 77)(53 76)(54 75)(55 80)(56 79)(57 84)(58 83)(59 82)(60 81)(61 89)(62 88)(63 87)(64 86)(65 85)(66 90)(67 95)(68 94)(69 93)(70 92)(71 91)(72 96)
(1 95 11 89)(2 96 12 90)(3 91 7 85)(4 92 8 86)(5 93 9 87)(6 94 10 88)(13 73 19 79)(14 74 20 80)(15 75 21 81)(16 76 22 82)(17 77 23 83)(18 78 24 84)(25 67 31 61)(26 68 32 62)(27 69 33 63)(28 70 34 64)(29 71 35 65)(30 72 36 66)(37 49 43 55)(38 50 44 56)(39 51 45 57)(40 52 46 58)(41 53 47 59)(42 54 48 60)

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

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

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

72 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2W 3 4A ··· 4H 4I ··· 4P 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 ··· 6 2 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 kernel C22×D6⋊C4 C2×D6⋊C4 C23×Dic3 C23×C12 S3×C24 S3×C23 C23×C4 C22×C6 C22×C4 C24 C23 C23 C23 # reps 1 12 1 1 1 16 1 8 6 1 8 8 8

Matrix representation of C22×D6⋊C4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 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
,
 1 0 0 0 0 0 0 12 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
,
 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 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 8 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8

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

C22×D6⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times D_6\rtimes C_4
% in TeX

G:=Group("C2^2xD6:C4");
// GroupNames label

G:=SmallGroup(192,1346);
// by ID

G=gap.SmallGroup(192,1346);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,1123,80,6278]);
// Polycyclic

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

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