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## G = C24.15D6order 192 = 26·3

### 4th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.15D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C24.15D6
 Lower central C3 — C2×C6 — C24.15D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.15D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=db=bd, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, 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, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C42, 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×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×Dic3, Dic3⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C24.C22, C6.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C24.15D6
Quotients:

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

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

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

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

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 C24.15D6 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 C24.15D6 in GL6(𝔽13)

 1 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 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 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 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 3 2 0 0 0 0 0 0 2 0 0 0 0 0 0 6
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 10 4 0 0 0 0 11 3 0 0 0 0 0 0 0 12 0 0 0 0 1 0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,6,3,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,6],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,11,0,0,0,0,4,3,0,0,0,0,0,0,0,1,0,0,0,0,12,0] >;

C24.15D6 in GAP, Magma, Sage, TeX

C_2^4._{15}D_6
% in TeX

G:=Group("C2^4.15D6");
// GroupNames label

G:=SmallGroup(192,504);
// by ID

G=gap.SmallGroup(192,504);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,64,926,219,6278]);
// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*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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