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## G = C24.21D6order 192 = 26·3

### 10th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.21D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — C24.21D6
 Lower central C3 — C22×C6 — C24.21D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.21D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=d, 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=ce5 >

Subgroups: 440 in 170 conjugacy classes, 59 normal (27 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×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.11D4, C6.C42, C6.C42, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C24.21D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C22.D4, C4.4D4, C422C2, C2×D12, C4○D12, D42S3, C2×C3⋊D4, C23.11D4, C23.8D6, C23.21D6, C127D4, C23.23D6, C23.12D6, C24.21D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 D12 C4○D12 D4⋊2S3 kernel C24.21D6 C6.C42 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C2×C22⋊C4 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 # reps 1 3 1 2 1 1 2 2 2 1 10 4 4 4 4

Matrix representation of C24.21D6 in GL6(𝔽13)

 12 4 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 10 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
,
 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 1 0 0 0 0 0 0 1
,
 5 6 0 0 0 0 0 8 0 0 0 0 0 0 12 5 0 0 0 0 10 1 0 0 0 0 0 0 7 0 0 0 0 0 9 2
,
 4 12 0 0 0 0 2 9 0 0 0 0 0 0 5 1 0 0 0 0 0 8 0 0 0 0 0 0 2 9 0 0 0 0 4 11

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,4,1,0,0,0,0,0,0,12,10,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,6,8,0,0,0,0,0,0,12,10,0,0,0,0,5,1,0,0,0,0,0,0,7,9,0,0,0,0,0,2],[4,2,0,0,0,0,12,9,0,0,0,0,0,0,5,0,0,0,0,0,1,8,0,0,0,0,0,0,2,4,0,0,0,0,9,11] >;`

C24.21D6 in GAP, Magma, Sage, TeX

`C_2^4._{21}D_6`
`% in TeX`

`G:=Group("C2^4.21D6");`
`// GroupNames label`

`G:=SmallGroup(192,512);`
`// by ID`

`G=gap.SmallGroup(192,512);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,387,6278]);`
`// Polycyclic`

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

׿
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