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## G = C4⋊D4.S3order 192 = 26·3

### 3rd non-split extension by C4⋊D4 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C4⋊D4.S3
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4⋊Dic3 — C2×C4⋊Dic3 — C4⋊D4.S3
 Lower central C3 — C6 — C2×C12 — C4⋊D4.S3
 Upper central C1 — C22 — C22×C4 — C4⋊D4

Generators and relations for C4⋊D4.S3
G = < a,b,c,d,e | a4=b4=c2=d3=1, e2=a2b2, bab-1=cac=eae-1=a-1, ad=da, cbc=b-1, bd=db, ebe-1=a-1b, cd=dc, ece-1=a-1c, ede-1=d-1 >

Subgroups: 304 in 114 conjugacy classes, 43 normal (27 characteristic)
C1, C2 [×3], C2 [×3], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×5], C6 [×3], C6 [×3], C8 [×2], C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, Dic3 [×2], C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×5], C22⋊C4, C4⋊C4, C4⋊C4 [×3], C2×C8 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C3⋊C8 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×4], C22×C6, C22×C6, C22⋊C8, D4⋊C4 [×2], C4.Q8 [×2], C2×C4⋊C4, C4⋊D4, C2×C3⋊C8 [×2], C4⋊Dic3 [×2], C4⋊Dic3, C3×C22⋊C4, C3×C4⋊C4, C22×Dic3, C22×C12, C6×D4, C6×D4, C23.46D4, C12.Q8 [×2], C12.55D4, D4⋊Dic3 [×2], C2×C4⋊Dic3, C3×C4⋊D4, C4⋊D4.S3
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], SD16 [×2], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3, C22.D4, C2×SD16, C8⋊C22, D4.S3 [×2], D42S3 [×2], C2×C3⋊D4, C23.46D4, C2×D4.S3, C23.23D6, D4⋊D6, C4⋊D4.S3

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 D6 C4○D4 SD16 C3⋊D4 C3⋊D4 C8⋊C22 D4⋊2S3 D4.S3 D4⋊D6 kernel C4⋊D4.S3 C12.Q8 C12.55D4 D4⋊Dic3 C2×C4⋊Dic3 C3×C4⋊D4 C4⋊D4 C2×C12 C22×C6 C4⋊C4 C22×C4 C2×D4 C12 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 1 1 1 1 4 4 2 2 1 2 2 2

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

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 72 0
,
 0 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 11 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 11 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 16 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 27 0 0 0 0 27 0 0 0 0 0 0 0 69 14 0 0 0 0 25 4 0 0 0 0 0 0 6 67 0 0 0 0 67 67

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,11,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,11,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,16,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,69,25,0,0,0,0,14,4,0,0,0,0,0,0,6,67,0,0,0,0,67,67] >;`

C4⋊D4.S3 in GAP, Magma, Sage, TeX

`C_4\rtimes D_4.S_3`
`% in TeX`

`G:=Group("C4:D4.S3");`
`// GroupNames label`

`G:=SmallGroup(192,593);`
`// by ID`

`G=gap.SmallGroup(192,593);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,336,254,219,1123,297,136,6278]);`
`// Polycyclic`

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

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