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## G = D4⋊(C4×S3)  order 192 = 26·3

### 2nd semidirect product of D4 and C4×S3 acting via C4×S3/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — D4⋊(C4×S3)
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — S3×C2×C4 — C2×D4⋊2S3 — D4⋊(C4×S3)
 Lower central C3 — C6 — C12 — D4⋊(C4×S3)
 Upper central C1 — C22 — C2×C4 — D4⋊C4

Generators and relations for D4⋊(C4×S3)
G = < a,b,c,d,e | a4=b2=c4=d3=e2=1, bab=cac-1=a-1, ad=da, ae=ea, cbc-1=a-1b, bd=db, ebe=a2b, cd=dc, ce=ec, ede=d-1 >

Subgroups: 456 in 162 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×C6, D4⋊C4, D4⋊C4, Q8⋊C4, C2×C4⋊C4, C2×M4(2), C2×C4○D4, C8⋊S3, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, S3×C2×C4, D42S3, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, C23.36D4, C6.SD16, C2.Dic12, D4⋊Dic3, C3×D4⋊C4, S3×C4⋊C4, C2×C8⋊S3, C2×D42S3, D4⋊(C4×S3)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, C23.36D4, S3×C22⋊C4, D8⋊S3, D4.D6, D4⋊(C4×S3)

Smallest permutation representation of D4⋊(C4×S3)
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 57)(2 60)(3 59)(4 58)(5 55)(6 54)(7 53)(8 56)(9 51)(10 50)(11 49)(12 52)(13 67)(14 66)(15 65)(16 68)(17 63)(18 62)(19 61)(20 64)(21 70)(22 69)(23 72)(24 71)(25 79)(26 78)(27 77)(28 80)(29 75)(30 74)(31 73)(32 76)(33 82)(34 81)(35 84)(36 83)(37 91)(38 90)(39 89)(40 92)(41 87)(42 86)(43 85)(44 88)(45 94)(46 93)(47 96)(48 95)
(1 46 22 34)(2 45 23 33)(3 48 24 36)(4 47 21 35)(5 76 85 67)(6 75 86 66)(7 74 87 65)(8 73 88 68)(9 78 89 63)(10 77 90 62)(11 80 91 61)(12 79 92 64)(13 56 32 44)(14 55 29 43)(15 54 30 42)(16 53 31 41)(17 52 26 40)(18 51 27 39)(19 50 28 38)(20 49 25 37)(57 94 69 82)(58 93 70 81)(59 96 71 84)(60 95 72 83)
(1 17 16)(2 18 13)(3 19 14)(4 20 15)(5 95 10)(6 96 11)(7 93 12)(8 94 9)(21 25 30)(22 26 31)(23 27 32)(24 28 29)(33 39 44)(34 40 41)(35 37 42)(36 38 43)(45 51 56)(46 52 53)(47 49 54)(48 50 55)(57 63 68)(58 64 65)(59 61 66)(60 62 67)(69 78 73)(70 79 74)(71 80 75)(72 77 76)(81 92 87)(82 89 88)(83 90 85)(84 91 86)
(5 12)(6 9)(7 10)(8 11)(13 18)(14 19)(15 20)(16 17)(25 30)(26 31)(27 32)(28 29)(37 42)(38 43)(39 44)(40 41)(49 54)(50 55)(51 56)(52 53)(57 59)(58 60)(61 68)(62 65)(63 66)(64 67)(69 71)(70 72)(73 80)(74 77)(75 78)(76 79)(81 83)(82 84)(85 92)(86 89)(87 90)(88 91)(93 95)(94 96)

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 24A 24B 24C 24D order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 8 8 8 8 12 12 12 12 24 24 24 24 size 1 1 1 1 4 4 6 6 2 2 2 4 4 6 6 12 12 12 12 2 2 2 8 8 4 4 12 12 4 4 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 D6 D6 C4×S3 C8⋊C22 C8.C22 S3×D4 S3×D4 D8⋊S3 D4.D6 kernel D4⋊(C4×S3) C6.SD16 C2.Dic12 D4⋊Dic3 C3×D4⋊C4 S3×C4⋊C4 C2×C8⋊S3 C2×D4⋊2S3 D4⋊2S3 D4⋊C4 C4×S3 C2×Dic3 C22×S3 C4⋊C4 C2×C8 C2×D4 D4 C6 C6 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 2 1 1 1 1 1 4 1 1 1 1 2 2

Matrix representation of D4⋊(C4×S3) in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 72 0 0 0 0 1 0 72 0 0 2 0 72 0 0 0 0 2 0 72
,
 70 1 0 0 0 0 65 3 0 0 0 0 0 0 40 66 67 12 0 0 7 33 61 6 0 0 68 10 33 7 0 0 63 5 66 40
,
 27 71 0 0 0 0 0 46 0 0 0 0 0 0 52 0 24 0 0 0 0 52 0 24 0 0 6 0 21 0 0 0 0 6 0 21
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0
,
 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72 0 0 0 0 0 0 1 72 0 0 0 0 0 72

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,2,0,0,0,0,1,0,2,0,0,72,0,72,0,0,0,0,72,0,72],[70,65,0,0,0,0,1,3,0,0,0,0,0,0,40,7,68,63,0,0,66,33,10,5,0,0,67,61,33,66,0,0,12,6,7,40],[27,0,0,0,0,0,71,46,0,0,0,0,0,0,52,0,6,0,0,0,0,52,0,6,0,0,24,0,21,0,0,0,0,24,0,21],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,72,72] >;

D4⋊(C4×S3) in GAP, Magma, Sage, TeX

D_4\rtimes (C_4\times S_3)
% in TeX

G:=Group("D4:(C4xS3)");
// GroupNames label

G:=SmallGroup(192,330);
// by ID

G=gap.SmallGroup(192,330);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,219,58,851,438,102,6278]);
// Polycyclic

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

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