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## G = C24.17D6order 192 = 26·3

### 6th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.17D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×Dic3⋊C4 — C24.17D6
 Lower central C3 — C22×C6 — C24.17D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.17D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 472 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×9], C22 [×7], C22 [×10], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×19], C23, C23 [×2], C23 [×6], Dic3 [×6], C12 [×3], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], Dic3⋊C4 [×4], C4⋊Dic3 [×2], C6.D4 [×4], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.Q8, C6.C42, C2×Dic3⋊C4 [×2], C2×C4⋊Dic3, C2×C6.D4 [×2], C6×C22⋊C4, C24.17D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], Dic6 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×Dic6, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.Q8, Dic3.D4 [×2], C23.8D6, Dic3⋊D4, C12.48D4, D63D4, C23.14D6, C24.17D6

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 S3 D4 D4 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 S3×D4 D4⋊2S3 kernel C24.17D6 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 1 2 1 2 1 1 4 2 2 2 1 6 4 4 4 2 2

Matrix representation of C24.17D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 9 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
,
 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 0 0 0 0 0 1 4
,
 5 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 3 8 0 0 0 0 12 10

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,9,0,0,0,0,0,12],[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],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,1,0,0,0,0,0,4],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,3,12,0,0,0,0,8,10] >;`

C24.17D6 in GAP, Magma, Sage, TeX

`C_2^4._{17}D_6`
`% in TeX`

`G:=Group("C2^4.17D6");`
`// GroupNames label`

`G:=SmallGroup(192,507);`
`// by ID`

`G=gap.SmallGroup(192,507);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,344,254,387,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d*c=c*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*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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