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## G = C24.23D6order 192 = 26·3

### 12nd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.23D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.23D6
 Lower central C3 — C2×C6 — C24.23D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.23D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 808 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×6], C3, C4 [×8], C22 [×7], C22 [×26], S3 [×4], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×24], D4 [×8], C23, C23 [×2], C23 [×16], Dic3 [×5], C12 [×3], D6 [×4], D6 [×12], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, C4×S3 [×8], C2×Dic3 [×4], C2×Dic3 [×7], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C22×S3 [×6], C22×S3 [×4], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4 [×2], S3×C2×C4 [×6], C22×Dic3 [×3], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C23.23D4, C6.C42 [×2], C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C22×C4, C22×C3⋊D4, C24.23D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, S3×C2×C4, C4○D12, S3×D4 [×3], D42S3, C2×C3⋊D4, C23.23D4, S3×C22⋊C4, Dic34D4, C23.9D6, Dic3⋊D4, C4×C3⋊D4, C232D6, D63D4, C24.23D6

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

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 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 C2 C4 S3 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.23D6 C6.C42 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 S3×C22×C4 C22×C3⋊D4 C2×C3⋊D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 1 1 1 8 1 2 2 4 2 1 4 4 4 4 3 1

Matrix representation of C24.23D6 in GL6(𝔽13)

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

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

C24.23D6 in GAP, Magma, Sage, TeX

`C_2^4._{23}D_6`
`% in TeX`

`G:=Group("C2^4.23D6");`
`// GroupNames label`

`G:=SmallGroup(192,515);`
`// by ID`

`G=gap.SmallGroup(192,515);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,387,58,6278]);`
`// Polycyclic`

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

׿
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