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## G = C24.55D6order 192 = 26·3

### 2nd non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.55D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C24.55D6
 Lower central C3 — C2×C6 — C24.55D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.55D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 552 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×10], C22 [×3], C22 [×8], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×30], C23, C23 [×6], C23 [×4], Dic3 [×4], Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, C2×Dic3 [×8], C2×Dic3 [×16], C2×C12 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C23×C4, Dic3⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×2], C22×Dic3 [×6], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.7Q8, C6.C42 [×2], C2×Dic3⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.55D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×2], C23, D6 [×3], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.7Q8, C23.16D6, Dic3.D4 [×2], S3×C22⋊C4, C2×Dic3⋊C4, C23.14D6 [×2], C24.55D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 4 4 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 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 Q8 D6 D6 C4○D4 Dic6 C4×S3 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.55D6 C6.C42 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C23×Dic3 C22×Dic3 C2×C22⋊C4 C2×Dic3 C22×C6 C22×C6 C22×C4 C24 C2×C6 C23 C23 C23 C22 C22 # reps 1 2 2 1 1 1 8 1 4 2 2 2 1 4 4 4 4 2 2

Matrix representation of C24.55D6 in GL6(𝔽13)

 12 0 0 0 0 0 5 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
,
 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
,
 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 12 0 0 0 0 0 0 12
,
 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
,
 8 11 0 0 0 0 0 5 0 0 0 0 0 0 7 0 0 0 0 0 0 11 0 0 0 0 0 0 6 0 0 0 0 0 0 11
,
 12 10 0 0 0 0 5 1 0 0 0 0 0 0 0 11 0 0 0 0 7 0 0 0 0 0 0 0 0 2 0 0 0 0 6 0

`G:=sub<GL(6,GF(13))| [12,5,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[8,0,0,0,0,0,11,5,0,0,0,0,0,0,7,0,0,0,0,0,0,11,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[12,5,0,0,0,0,10,1,0,0,0,0,0,0,0,7,0,0,0,0,11,0,0,0,0,0,0,0,0,6,0,0,0,0,2,0] >;`

C24.55D6 in GAP, Magma, Sage, TeX

`C_2^4._{55}D_6`
`% in TeX`

`G:=Group("C2^4.55D6");`
`// GroupNames label`

`G:=SmallGroup(192,501);`
`// by ID`

`G=gap.SmallGroup(192,501);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,387,58,6278]);`
`// 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=f*a*f^-1=a*d=d*a,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^5>;`
`// generators/relations`

׿
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