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## G = C24.73D6order 192 = 26·3

### 2nd non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.73D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.73D6
 Lower central C3 — C2×C6 — C24.73D6
 Upper central C1 — C23 — C23×C4

Generators and relations for C24.73D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 504 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C23×C4, Dic3⋊C4, C6.D4, C6.D4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, C23×C6, C23.8Q8, C6.C42, C2×Dic3⋊C4, C2×C6.D4, C23×C12, C24.73D6
Quotients:

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4×S3 C3⋊D4 C4○D12 kernel C24.73D6 C6.C42 C2×Dic3⋊C4 C2×C6.D4 C23×C12 C6.D4 C23×C4 C2×C12 C22×C6 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C23 C23 C22 # reps 1 2 2 2 1 8 1 4 2 2 2 1 4 8 4 4 4 8

Matrix representation of C24.73D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 12 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 1 0 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 1 0 0 0 0 0 0 1 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 2 11 0 0 0 0 0 6 0 0 0 0 0 0 3 0 0 0 0 0 0 4
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 11 2 0 0 0 0 4 2 0 0 0 0 0 0 0 9 0 0 0 0 3 0

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

C24.73D6 in GAP, Magma, Sage, TeX

`C_2^4._{73}D_6`
`% in TeX`

`G:=Group("C2^4.73D6");`
`// GroupNames label`

`G:=SmallGroup(192,769);`
`// by ID`

`G=gap.SmallGroup(192,769);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,758,58,6278]);`
`// 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,f*a*f^-1=a*c=c*a,a*d=d*a,a*e=e*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=d*e^5>;`
`// generators/relations`

׿
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