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## G = C23.54D12order 192 = 26·3

### 20th non-split extension by C23 of D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C23.54D12
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C2×C4○D12 — C23.54D12
 Lower central C3 — C6 — C12 — C23.54D12
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C23.54D12
G = < a,b,c,d,e | a2=b2=c2=1, d12=c, e2=cb=bc, ab=ba, dad-1=ac=ca, ae=ea, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd11 >

Subgroups: 472 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, Dic3 [×4], C12 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×2], C2×C6 [×2], C4⋊C4 [×3], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], C24 [×2], Dic6 [×2], Dic6, C4×S3 [×4], D12 [×2], D12, C2×Dic3 [×5], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C4⋊Dic3 [×2], C4⋊Dic3, C2×C24 [×2], C3×M4(2) [×2], C2×Dic6, S3×C2×C4, C2×D12, C4○D12 [×4], C4○D12 [×2], C22×Dic3, C2×C3⋊D4, C22×C12, C23.36D4, C2.Dic12 [×2], C2.D24 [×2], C2×C4⋊Dic3, C6×M4(2), C2×C4○D12, C23.54D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C8⋊C22, C8.C22, D6⋊C4 [×4], S3×C2×C4, C2×D12, C2×C3⋊D4, C23.36D4, C8⋊D6, C8.D6, C2×D6⋊C4, C23.54D12

Smallest permutation representation of C23.54D12
On 96 points
Generators in S96
```(1 89)(2 78)(3 91)(4 80)(5 93)(6 82)(7 95)(8 84)(9 73)(10 86)(11 75)(12 88)(13 77)(14 90)(15 79)(16 92)(17 81)(18 94)(19 83)(20 96)(21 85)(22 74)(23 87)(24 76)(25 52)(26 65)(27 54)(28 67)(29 56)(30 69)(31 58)(32 71)(33 60)(34 49)(35 62)(36 51)(37 64)(38 53)(39 66)(40 55)(41 68)(42 57)(43 70)(44 59)(45 72)(46 61)(47 50)(48 63)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 71)(11 72)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 61)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 85)(32 86)(33 87)(34 88)(35 89)(36 90)(37 91)(38 92)(39 93)(40 94)(41 95)(42 96)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)
(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 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(49 61)(50 62)(51 63)(52 64)(53 65)(54 66)(55 67)(56 68)(57 69)(58 70)(59 71)(60 72)(73 85)(74 86)(75 87)(76 88)(77 89)(78 90)(79 91)(80 92)(81 93)(82 94)(83 95)(84 96)
(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 49 50 24)(2 23 51 72)(3 71 52 22)(4 21 53 70)(5 69 54 20)(6 19 55 68)(7 67 56 18)(8 17 57 66)(9 65 58 16)(10 15 59 64)(11 63 60 14)(12 13 61 62)(25 74 91 32)(26 31 92 73)(27 96 93 30)(28 29 94 95)(33 90 75 48)(34 47 76 89)(35 88 77 46)(36 45 78 87)(37 86 79 44)(38 43 80 85)(39 84 81 42)(40 41 82 83)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E ··· 4J 6A 6B 6C 6D 6E 8A 8B 8C 8D 12A 12B 12C 12D 12E 12F 24A ··· 24H order 1 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 6 6 6 6 8 8 8 8 12 12 12 12 12 12 24 ··· 24 size 1 1 1 1 2 2 12 12 2 2 2 2 2 12 ··· 12 2 2 2 4 4 4 4 4 4 2 2 2 2 4 4 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4×S3 D12 C3⋊D4 D12 C8⋊C22 C8.C22 C8⋊D6 C8.D6 kernel C23.54D12 C2.Dic12 C2.D24 C2×C4⋊Dic3 C6×M4(2) C2×C4○D12 C4○D12 C2×M4(2) C2×C12 C22×C6 C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C23 C6 C6 C2 C2 # reps 1 2 2 1 1 1 8 1 3 1 2 1 4 2 4 2 1 1 2 2

Matrix representation of C23.54D12 in GL6(𝔽73)

 72 0 0 0 0 0 0 72 0 0 0 0 0 0 39 41 57 13 0 0 32 7 60 70 0 0 54 11 34 32 0 0 62 65 41 66
,
 72 0 0 0 0 0 0 72 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 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 43 30 0 0 0 0 43 13 0 0 0 0 0 0 0 72 0 2 0 0 1 72 71 2 0 0 7 69 0 1 0 0 4 3 72 1
,
 43 30 0 0 0 0 60 30 0 0 0 0 0 0 1 72 71 2 0 0 0 72 0 2 0 0 1 72 72 1 0 0 0 72 0 1

`G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,39,32,54,62,0,0,41,7,11,65,0,0,57,60,34,41,0,0,13,70,32,66],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[43,43,0,0,0,0,30,13,0,0,0,0,0,0,0,1,7,4,0,0,72,72,69,3,0,0,0,71,0,72,0,0,2,2,1,1],[43,60,0,0,0,0,30,30,0,0,0,0,0,0,1,0,1,0,0,0,72,72,72,72,0,0,71,0,72,0,0,0,2,2,1,1] >;`

C23.54D12 in GAP, Magma, Sage, TeX

`C_2^3._{54}D_{12}`
`% in TeX`

`G:=Group("C2^3.54D12");`
`// GroupNames label`

`G:=SmallGroup(192,692);`
`// by ID`

`G=gap.SmallGroup(192,692);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,387,142,1123,136,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^12=c,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=a*c=c*a,a*e=e*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*d^11>;`
`// generators/relations`

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