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## G = C24⋊S3order 144 = 24·32

### 5th semidirect product of C24 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C24⋊S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C4×C3⋊S3 — C24⋊S3
 Lower central C32 — C3×C6 — C24⋊S3
 Upper central C1 — C4 — C8

Generators and relations for C24⋊S3
G = < a,b,c | a24=b3=c2=1, ab=ba, cac=a5, cbc=b-1 >

Subgroups: 178 in 60 conjugacy classes, 29 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, S3, C6, C8, C8, C2×C4, C32, Dic3, C12, D6, M4(2), C3⋊S3, C3×C6, C3⋊C8, C24, C4×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, C324C8, C3×C24, C4×C3⋊S3, C24⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), C3⋊S3, C4×S3, C2×C3⋊S3, C8⋊S3, C4×C3⋊S3, C24⋊S3

Smallest permutation representation of C24⋊S3
On 72 points
Generators in S72
```(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)
(1 49 41)(2 50 42)(3 51 43)(4 52 44)(5 53 45)(6 54 46)(7 55 47)(8 56 48)(9 57 25)(10 58 26)(11 59 27)(12 60 28)(13 61 29)(14 62 30)(15 63 31)(16 64 32)(17 65 33)(18 66 34)(19 67 35)(20 68 36)(21 69 37)(22 70 38)(23 71 39)(24 72 40)
(2 6)(3 11)(4 16)(5 21)(8 12)(9 17)(10 22)(14 18)(15 23)(20 24)(25 65)(26 70)(27 51)(28 56)(29 61)(30 66)(31 71)(32 52)(33 57)(34 62)(35 67)(36 72)(37 53)(38 58)(39 63)(40 68)(41 49)(42 54)(43 59)(44 64)(45 69)(46 50)(47 55)(48 60)```

`G:=sub<Sym(72)| (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), (1,49,41)(2,50,42)(3,51,43)(4,52,44)(5,53,45)(6,54,46)(7,55,47)(8,56,48)(9,57,25)(10,58,26)(11,59,27)(12,60,28)(13,61,29)(14,62,30)(15,63,31)(16,64,32)(17,65,33)(18,66,34)(19,67,35)(20,68,36)(21,69,37)(22,70,38)(23,71,39)(24,72,40), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,65)(26,70)(27,51)(28,56)(29,61)(30,66)(31,71)(32,52)(33,57)(34,62)(35,67)(36,72)(37,53)(38,58)(39,63)(40,68)(41,49)(42,54)(43,59)(44,64)(45,69)(46,50)(47,55)(48,60)>;`

`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), (1,49,41)(2,50,42)(3,51,43)(4,52,44)(5,53,45)(6,54,46)(7,55,47)(8,56,48)(9,57,25)(10,58,26)(11,59,27)(12,60,28)(13,61,29)(14,62,30)(15,63,31)(16,64,32)(17,65,33)(18,66,34)(19,67,35)(20,68,36)(21,69,37)(22,70,38)(23,71,39)(24,72,40), (2,6)(3,11)(4,16)(5,21)(8,12)(9,17)(10,22)(14,18)(15,23)(20,24)(25,65)(26,70)(27,51)(28,56)(29,61)(30,66)(31,71)(32,52)(33,57)(34,62)(35,67)(36,72)(37,53)(38,58)(39,63)(40,68)(41,49)(42,54)(43,59)(44,64)(45,69)(46,50)(47,55)(48,60) );`

`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)], [(1,49,41),(2,50,42),(3,51,43),(4,52,44),(5,53,45),(6,54,46),(7,55,47),(8,56,48),(9,57,25),(10,58,26),(11,59,27),(12,60,28),(13,61,29),(14,62,30),(15,63,31),(16,64,32),(17,65,33),(18,66,34),(19,67,35),(20,68,36),(21,69,37),(22,70,38),(23,71,39),(24,72,40)], [(2,6),(3,11),(4,16),(5,21),(8,12),(9,17),(10,22),(14,18),(15,23),(20,24),(25,65),(26,70),(27,51),(28,56),(29,61),(30,66),(31,71),(32,52),(33,57),(34,62),(35,67),(36,72),(37,53),(38,58),(39,63),(40,68),(41,49),(42,54),(43,59),(44,64),(45,69),(46,50),(47,55),(48,60)]])`

C24⋊S3 is a maximal subgroup of
S3×C8⋊S3  C246D6  C24.64D6  D12.4D6  C24.95D6  M4(2)×C3⋊S3  C24.47D6  C248D6  C247D6  C24.32D6  C24.35D6  He35M4(2)  C72⋊S3  C338M4(2)  C339M4(2)  C3315M4(2)
C24⋊S3 is a maximal quotient of
C12.30Dic6  C24⋊Dic3  C12.60D12  C72⋊S3  He36M4(2)  C338M4(2)  C339M4(2)  C3315M4(2)

42 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 8D 12A ··· 12H 24A ··· 24P order 1 2 2 3 3 3 3 4 4 4 6 6 6 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 18 2 2 2 2 1 1 18 2 2 2 2 2 2 18 18 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + image C1 C2 C2 C2 C4 C4 S3 D6 M4(2) C4×S3 C8⋊S3 kernel C24⋊S3 C32⋊4C8 C3×C24 C4×C3⋊S3 C3⋊Dic3 C2×C3⋊S3 C24 C12 C32 C6 C3 # reps 1 1 1 1 2 2 4 4 2 8 16

Matrix representation of C24⋊S3 in GL4(𝔽73) generated by

 0 72 0 0 1 72 0 0 0 0 57 8 0 0 65 65
,
 1 0 0 0 0 1 0 0 0 0 0 72 0 0 1 72
,
 0 1 0 0 1 0 0 0 0 0 72 1 0 0 0 1
`G:=sub<GL(4,GF(73))| [0,1,0,0,72,72,0,0,0,0,57,65,0,0,8,65],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,72],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,1,1] >;`

C24⋊S3 in GAP, Magma, Sage, TeX

`C_{24}\rtimes S_3`
`% in TeX`

`G:=Group("C24:S3");`
`// GroupNames label`

`G:=SmallGroup(144,86);`
`// by ID`

`G=gap.SmallGroup(144,86);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,121,31,50,964,3461]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,c*a*c=a^5,c*b*c=b^-1>;`
`// generators/relations`

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