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## G = C4×D4⋊2S3order 192 = 26·3

### Direct product of C4 and D4⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×D4⋊2S3
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C4×D4⋊2S3
 Lower central C3 — C6 — C4×D4⋊2S3
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for C4×D42S3
G = < a,b,c,d,e | a4=b4=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >

Subgroups: 616 in 310 conjugacy classes, 157 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C22×C12, C6×D4, C4×C4○D4, C4×Dic6, S3×C42, C23.16D6, Dic34D4, Dic6⋊C4, C4⋊C47S3, C2×C4×Dic3, C4×C3⋊D4, D4×Dic3, D4×C12, C2×D42S3, C4×D42S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C23×C4, C2×C4○D4, S3×C2×C4, D42S3, S3×C23, C4×C4○D4, S3×C22×C4, C2×D42S3, S3×C4○D4, C4×D42S3

Smallest permutation representation of C4×D42S3
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 28 41 65)(2 25 42 66)(3 26 43 67)(4 27 44 68)(5 22 77 14)(6 23 78 15)(7 24 79 16)(8 21 80 13)(9 32 45 87)(10 29 46 88)(11 30 47 85)(12 31 48 86)(17 89 49 83)(18 90 50 84)(19 91 51 81)(20 92 52 82)(33 69 75 61)(34 70 76 62)(35 71 73 63)(36 72 74 64)(37 95 53 59)(38 96 54 60)(39 93 55 57)(40 94 56 58)
(1 83)(2 84)(3 81)(4 82)(5 76)(6 73)(7 74)(8 75)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 50)(26 51)(27 52)(28 49)(29 56)(30 53)(31 54)(32 55)(33 80)(34 77)(35 78)(36 79)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 9 23)(2 10 24)(3 11 21)(4 12 22)(5 68 86)(6 65 87)(7 66 88)(8 67 85)(13 43 47)(14 44 48)(15 41 45)(16 42 46)(17 39 73)(18 40 74)(19 37 75)(20 38 76)(25 29 79)(26 30 80)(27 31 77)(28 32 78)(33 51 53)(34 52 54)(35 49 55)(36 50 56)(57 71 83)(58 72 84)(59 69 81)(60 70 82)(61 91 95)(62 92 96)(63 89 93)(64 90 94)
(1 41)(2 42)(3 43)(4 44)(5 31)(6 32)(7 29)(8 30)(9 15)(10 16)(11 13)(12 14)(21 47)(22 48)(23 45)(24 46)(25 66)(26 67)(27 68)(28 65)(33 53)(34 54)(35 55)(36 56)(37 75)(38 76)(39 73)(40 74)(57 71)(58 72)(59 69)(60 70)(61 95)(62 96)(63 93)(64 94)(77 86)(78 87)(79 88)(80 85)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E ··· 4L 4M ··· 4T 4U ··· 4AD 6A 6B 6C 6D 6E 6F 6G 12A 12B 12C 12D 12E ··· 12L order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 ··· 4 4 ··· 4 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 size 1 1 1 1 2 2 2 2 6 6 2 1 1 1 1 2 ··· 2 3 ··· 3 6 ··· 6 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 D6 D6 C4○D4 C4○D4 C4×S3 D4⋊2S3 S3×C4○D4 kernel C4×D4⋊2S3 C4×Dic6 S3×C42 C23.16D6 Dic3⋊4D4 Dic6⋊C4 C4⋊C4⋊7S3 C2×C4×Dic3 C4×C3⋊D4 D4×Dic3 D4×C12 C2×D4⋊2S3 D4⋊2S3 C4×D4 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 Dic3 C12 D4 C4 C2 # reps 1 1 1 2 2 1 1 2 2 1 1 1 16 1 1 2 1 2 1 4 4 8 2 2

Matrix representation of C4×D42S3 in GL4(𝔽13) generated by

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

C4×D42S3 in GAP, Magma, Sage, TeX

C_4\times D_4\rtimes_2S_3
% in TeX

G:=Group("C4xD4:2S3");
// GroupNames label

G:=SmallGroup(192,1095);
// by ID

G=gap.SmallGroup(192,1095);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,794,6278]);
// Polycyclic

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

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