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## G = C24.14D6order 192 = 26·3

### 3rd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.14D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.14D6
 Lower central C3 — C2×C6 — C24.14D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.14D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 456 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×10], C22 [×7], C22 [×10], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×20], C23, C23 [×2], C23 [×6], Dic3 [×7], C12 [×3], C2×C6 [×7], C2×C6 [×10], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×6], C2×Dic3 [×9], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×2], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C24.C22, C6.C42 [×2], C2×C4×Dic3, C2×Dic3⋊C4, C2×C6.D4 [×2], C6×C22⋊C4, C24.14D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, S3×C2×C4, C4○D12, S3×D4, D42S3 [×3], C2×C3⋊D4, C24.C22, C23.16D6, C23.8D6, Dic34D4, C23.11D6, C4×C3⋊D4, C23.23D6, D63D4, C24.14D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 2 2 2 2 2 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 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.14D6 C6.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C6.D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 2 1 8 1 2 2 2 1 8 4 4 4 1 3

Matrix representation of C24.14D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 7 1 0 0 0 0 0 0 1 0 0 0 0 0 3 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 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 6 11 0 0 0 0 12 7 0 0 0 0 0 0 11 0 0 0 0 0 6 7
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 4 0 0 0 0 7 6

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,7,0,0,0,0,0,1,0,0,0,0,0,0,1,3,0,0,0,0,0,12],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,6,12,0,0,0,0,11,7,0,0,0,0,0,0,11,6,0,0,0,0,0,7],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,7,0,0,0,0,4,6] >;

C24.14D6 in GAP, Magma, Sage, TeX

C_2^4._{14}D_6
% in TeX

G:=Group("C2^4.14D6");
// GroupNames label

G:=SmallGroup(192,503);
// by ID

G=gap.SmallGroup(192,503);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,387,58,6278]);
// Polycyclic

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

׿
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