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## G = C22×D4⋊2S3order 192 = 26·3

### Direct product of C22 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C22×D4⋊2S3
 Chief series C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — S3×C22×C4 — C22×D4⋊2S3
 Lower central C3 — C6 — C22×D4⋊2S3
 Upper central C1 — C23 — C22×D4

Generators and relations for C22×D42S3
G = < a,b,c,d,e,f | a2=b2=c4=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, de=ed, fdf=c2d, fef=e-1 >

Subgroups: 1704 in 890 conjugacy classes, 463 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, C24, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, S3×C23, C23×C6, C22×C4○D4, C22×Dic6, S3×C22×C4, C2×D42S3, C23×Dic3, C22×C3⋊D4, D4×C2×C6, C22×D42S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, C25, D42S3, S3×C23, C22×C4○D4, C2×D42S3, S3×C24, C22×D42S3

Smallest permutation representation of C22×D42S3
On 96 points
Generators in S96
(1 76)(2 73)(3 74)(4 75)(5 37)(6 38)(7 39)(8 40)(9 52)(10 49)(11 50)(12 51)(13 24)(14 21)(15 22)(16 23)(17 65)(18 66)(19 67)(20 68)(25 78)(26 79)(27 80)(28 77)(29 96)(30 93)(31 94)(32 95)(33 86)(34 87)(35 88)(36 85)(41 84)(42 81)(43 82)(44 83)(45 54)(46 55)(47 56)(48 53)(57 70)(58 71)(59 72)(60 69)(61 90)(62 91)(63 92)(64 89)
(1 33)(2 34)(3 35)(4 36)(5 16)(6 13)(7 14)(8 15)(9 65)(10 66)(11 67)(12 68)(17 52)(18 49)(19 50)(20 51)(21 39)(22 40)(23 37)(24 38)(25 63)(26 64)(27 61)(28 62)(29 43)(30 44)(31 41)(32 42)(45 70)(46 71)(47 72)(48 69)(53 60)(54 57)(55 58)(56 59)(73 87)(74 88)(75 85)(76 86)(77 91)(78 92)(79 89)(80 90)(81 95)(82 96)(83 93)(84 94)
(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 10)(2 9)(3 12)(4 11)(5 62)(6 61)(7 64)(8 63)(13 27)(14 26)(15 25)(16 28)(17 87)(18 86)(19 85)(20 88)(21 79)(22 78)(23 77)(24 80)(29 45)(30 48)(31 47)(32 46)(33 66)(34 65)(35 68)(36 67)(37 91)(38 90)(39 89)(40 92)(41 72)(42 71)(43 70)(44 69)(49 76)(50 75)(51 74)(52 73)(53 93)(54 96)(55 95)(56 94)(57 82)(58 81)(59 84)(60 83)
(1 14 59)(2 15 60)(3 16 57)(4 13 58)(5 54 35)(6 55 36)(7 56 33)(8 53 34)(9 25 83)(10 26 84)(11 27 81)(12 28 82)(17 92 30)(18 89 31)(19 90 32)(20 91 29)(21 72 76)(22 69 73)(23 70 74)(24 71 75)(37 45 88)(38 46 85)(39 47 86)(40 48 87)(41 49 79)(42 50 80)(43 51 77)(44 52 78)(61 95 67)(62 96 68)(63 93 65)(64 94 66)
(1 74)(2 75)(3 76)(4 73)(5 47)(6 48)(7 45)(8 46)(9 52)(10 49)(11 50)(12 51)(13 69)(14 70)(15 71)(16 72)(17 65)(18 66)(19 67)(20 68)(21 57)(22 58)(23 59)(24 60)(25 44)(26 41)(27 42)(28 43)(29 62)(30 63)(31 64)(32 61)(33 88)(34 85)(35 86)(36 87)(37 56)(38 53)(39 54)(40 55)(77 82)(78 83)(79 84)(80 81)(89 94)(90 95)(91 96)(92 93)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 2P 2Q 2R 2S 3 4A 4B 4C 4D 4E ··· 4L 4M ··· 4T 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 2 ··· 2 6 6 6 6 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 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 D4⋊2S3 kernel C22×D4⋊2S3 C22×Dic6 S3×C22×C4 C2×D4⋊2S3 C23×Dic3 C22×C3⋊D4 D4×C2×C6 C22×D4 C22×C4 C2×D4 C24 C2×C6 C22 # reps 1 1 1 24 2 2 1 1 1 12 2 8 4

Matrix representation of C22×D42S3 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 5 0 0 0 0 0 8
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 8 0 0 0 5 0
,
 1 0 0 0 0 0 12 1 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 12

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

C22×D42S3 in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes_2S_3
% in TeX

G:=Group("C2^2xD4:2S3");
// GroupNames label

G:=SmallGroup(192,1515);
// by ID

G=gap.SmallGroup(192,1515);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,136,1684,235,6278]);
// Polycyclic

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

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