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## G = C24.18D6order 192 = 26·3

### 7th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.18D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — C24.18D6
 Lower central C3 — C22×C6 — C24.18D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.18D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=dc=cd, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde5 >

Subgroups: 472 in 186 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×9], C22 [×3], C22 [×4], C22 [×10], C6 [×3], C6 [×4], C6 [×2], C2×C4 [×2], C2×C4 [×19], C23, C23 [×2], C23 [×6], Dic3 [×6], C12 [×3], C2×C6 [×3], C2×C6 [×4], C2×C6 [×10], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4 [×4], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.4Q8, C6.C42, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C6.D4 [×2], C6×C22⋊C4, C24.18D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8 [×3], C22.D4 [×3], C41D4, C2×Dic6, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.4Q8, Dic3.D4 [×2], C23.9D6 [×2], C12.48D4, C23.23D6, C123D4, C24.18D6

Smallest permutation representation of C24.18D6
On 96 points
Generators in S96
```(2 16)(4 18)(6 20)(8 22)(10 24)(12 14)(25 70)(26 87)(27 72)(28 89)(29 62)(30 91)(31 64)(32 93)(33 66)(34 95)(35 68)(36 85)(38 52)(40 54)(42 56)(44 58)(46 60)(48 50)(61 80)(63 82)(65 84)(67 74)(69 76)(71 78)(73 94)(75 96)(77 86)(79 88)(81 90)(83 92)
(1 57)(2 58)(3 59)(4 60)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 37)(22 38)(23 39)(24 40)(25 86)(26 87)(27 88)(28 89)(29 90)(30 91)(31 92)(32 93)(33 94)(34 95)(35 96)(36 85)(61 80)(62 81)(63 82)(64 83)(65 84)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(1 21)(2 22)(3 23)(4 24)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(11 19)(12 20)(25 83)(26 84)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 57)(38 58)(39 59)(40 60)(41 49)(42 50)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(61 95)(62 96)(63 85)(64 86)(65 87)(66 88)(67 89)(68 90)(69 91)(70 92)(71 93)(72 94)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 13)(12 14)(25 77)(26 78)(27 79)(28 80)(29 81)(30 82)(31 83)(32 84)(33 73)(34 74)(35 75)(36 76)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 85)(70 86)(71 87)(72 88)
(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 72 21 94)(2 71 22 93)(3 70 23 92)(4 69 24 91)(5 68 13 90)(6 67 14 89)(7 66 15 88)(8 65 16 87)(9 64 17 86)(10 63 18 85)(11 62 19 96)(12 61 20 95)(25 53 83 45)(26 52 84 44)(27 51 73 43)(28 50 74 42)(29 49 75 41)(30 60 76 40)(31 59 77 39)(32 58 78 38)(33 57 79 37)(34 56 80 48)(35 55 81 47)(36 54 82 46)```

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 3 4 4 4 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 2 4 4 4 4 12 ··· 12 2 ··· 2 4 4 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 S3×D4 D4⋊2S3 kernel C24.18D6 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 1 2 1 2 1 1 4 2 2 2 1 6 4 4 4 2 2

Matrix representation of C24.18D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 11 0 0 0 0 0 10 6
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 11 1 0 0 0 0 8 2

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,11,10,0,0,0,0,0,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,11,8,0,0,0,0,1,2] >;`

C24.18D6 in GAP, Magma, Sage, TeX

`C_2^4._{18}D_6`
`% in TeX`

`G:=Group("C2^4.18D6");`
`// GroupNames label`

`G:=SmallGroup(192,508);`
`// by ID`

`G=gap.SmallGroup(192,508);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,344,254,387,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=d*c=c*d,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=a*d=d*a,f*a*f^-1=a*b*d,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;`
`// generators/relations`

׿
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