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## G = C24.32D6order 192 = 26·3

### 21st non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.32D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.32D6
 Lower central C3 — C22×C6 — C24.32D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C24.32D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=c, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 952 in 322 conjugacy classes, 69 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×30], S3 [×2], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×2], C2×C4 [×13], D4 [×24], C23, C23 [×4], C23 [×20], Dic3 [×5], C12 [×2], D6 [×10], C2×C6 [×3], C2×C6 [×4], C2×C6 [×20], C22⋊C4 [×6], C22×C4, C22×C4 [×3], C2×D4 [×18], C24 [×2], C24, C2×Dic3 [×4], C2×Dic3 [×7], C3⋊D4 [×16], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×4], C22×C6 [×12], C2.C42, C2×C22⋊C4 [×3], C22×D4, C22×D4 [×2], D6⋊C4 [×2], C6.D4 [×4], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×12], C22×C12, C6×D4 [×6], S3×C23, C23×C6 [×2], C232D4, C6.C42, C2×D6⋊C4, C2×C6.D4 [×2], C22×C3⋊D4 [×2], D4×C2×C6, C24.32D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×12], C23, D6 [×3], C2×D4 [×6], C4○D4, C3⋊D4 [×6], C22×S3, C22≀C2 [×3], C4⋊D4 [×3], C41D4, S3×D4 [×3], D42S3, C2×C3⋊D4 [×3], C232D4, C232D6 [×2], D63D4, C23.14D6 [×2], C123D4, C244S3, C24.32D6

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

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C ··· 4H 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 12 12 2 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 D4 D4 D6 D6 C4○D4 C3⋊D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.32D6 C6.C42 C2×D6⋊C4 C2×C6.D4 C22×C3⋊D4 D4×C2×C6 C22×D4 C2×Dic3 C2×C12 C22×S3 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 # reps 1 1 1 2 2 1 1 4 2 2 4 1 2 2 4 8 3 1

Matrix representation of C24.32D6 in GL6(𝔽13)

 2 4 0 0 0 0 9 11 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 3 0 0 0 0 0 1
,
 12 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 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 12 0 0 0 0 0 0 12
,
 1 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 12 0 0 0 0 0 0 12
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 5 11 0 0 0 0 12 8
,
 12 12 0 0 0 0 0 1 0 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 0 0 0 8 2 0 0 0 0 0 5

`G:=sub<GL(6,GF(13))| [2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,3,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,5,12,0,0,0,0,11,8],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,8,0,0,0,0,0,2,5] >;`

C24.32D6 in GAP, Magma, Sage, TeX

`C_2^4._{32}D_6`
`% in TeX`

`G:=Group("C2^4.32D6");`
`// GroupNames label`

`G:=SmallGroup(192,782);`
`// by ID`

`G=gap.SmallGroup(192,782);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,387,6278]);`
`// Polycyclic`

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

׿
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