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## G = C2×C23.9D6order 192 = 26·3

### Direct product of C2 and C23.9D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.9D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×C23.9D6
 Lower central C3 — C2×C6 — C2×C23.9D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.9D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=c, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 872 in 342 conjugacy classes, 119 normal (31 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C2×C22.D4, C23.9D6, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, S3×C22×C4, C22×C3⋊D4, C2×C23.9D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C22×S3, C22.D4, C22×D4, C2×C4○D4, C4○D12, S3×D4, D42S3, S3×C23, C2×C22.D4, C23.9D6, C2×C4○D12, C2×S3×D4, C2×D42S3, C2×C23.9D6

Smallest permutation representation of C2×C23.9D6
On 96 points
Generators in S96
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 13)(10 14)(11 15)(12 16)(25 94)(26 95)(27 96)(28 85)(29 86)(30 87)(31 88)(32 89)(33 90)(34 91)(35 92)(36 93)(37 65)(38 66)(39 67)(40 68)(41 69)(42 70)(43 71)(44 72)(45 61)(46 62)(47 63)(48 64)(49 83)(50 84)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 81)(60 82)
(1 38)(2 77)(3 40)(4 79)(5 42)(6 81)(7 44)(8 83)(9 46)(10 73)(11 48)(12 75)(13 62)(14 51)(15 64)(16 53)(17 66)(18 55)(19 68)(20 57)(21 70)(22 59)(23 72)(24 49)(25 71)(26 60)(27 61)(28 50)(29 63)(30 52)(31 65)(32 54)(33 67)(34 56)(35 69)(36 58)(37 88)(39 90)(41 92)(43 94)(45 96)(47 86)(74 87)(76 89)(78 91)(80 93)(82 95)(84 85)
(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 89)(2 90)(3 91)(4 92)(5 93)(6 94)(7 95)(8 96)(9 85)(10 86)(11 87)(12 88)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(21 36)(22 25)(23 26)(24 27)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 73)(48 74)(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 25 7 31)(2 30 8 36)(3 35 9 29)(4 28 10 34)(5 33 11 27)(6 26 12 32)(13 86 19 92)(14 91 20 85)(15 96 21 90)(16 89 22 95)(17 94 23 88)(18 87 24 93)(37 60 43 54)(38 53 44 59)(39 58 45 52)(40 51 46 57)(41 56 47 50)(42 49 48 55)(61 74 67 80)(62 79 68 73)(63 84 69 78)(64 77 70 83)(65 82 71 76)(66 75 72 81)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 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 6 6 6 6 2 2 2 2 2 4 4 6 6 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 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 kernel C2×C23.9D6 C23.9D6 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 S3×C22×C4 C22×C3⋊D4 C2×C22⋊C4 C22×S3 C22⋊C4 C22×C4 C24 C2×C6 C22 C22 C22 # reps 1 8 1 1 1 1 1 1 1 1 4 4 2 1 8 8 2 2

Matrix representation of C2×C23.9D6 in GL6(𝔽13)

 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
,
 1 11 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 0 1 0 0 0 0 1 0
,
 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 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
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 8 0 0 0 0 0 8 5 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 5 0 0 0 0 0 0 5

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

C2×C23.9D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._9D_6
% in TeX

G:=Group("C2xC2^3.9D6");
// GroupNames label

G:=SmallGroup(192,1047);
// by ID

G=gap.SmallGroup(192,1047);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,100,675,297,6278]);
// Polycyclic

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