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## G = C2×C23.16D6order 192 = 26·3

### Direct product of C2 and C23.16D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C23.16D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C23×Dic3 — C2×C23.16D6
 Lower central C3 — C6 — C2×C23.16D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C2×C23.16D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 616 in 330 conjugacy classes, 167 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, 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×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×Dic3, Dic3⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×Dic3, C22×C12, C23×C6, C2×C42⋊C2, C23.16D6, C2×C4×Dic3, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C23×Dic3, C2×C23.16D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C42⋊C2, C23×C4, C2×C4○D4, S3×C2×C4, D42S3, S3×C23, C2×C42⋊C2, C23.16D6, S3×C22×C4, C2×D42S3, C2×C23.16D6

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 4Q ··· 4AB 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 2 ··· 2 3 ··· 3 6 ··· 6 2 ··· 2 4 4 4 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 D6 C4○D4 C4×S3 D4⋊2S3 kernel C2×C23.16D6 C23.16D6 C2×C4×Dic3 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C23×Dic3 C22×Dic3 C2×C22⋊C4 C22⋊C4 C22×C4 C24 C2×C6 C23 C22 # reps 1 8 2 2 1 1 1 16 1 4 2 1 8 8 4

Matrix representation of C2×C23.16D6 in GL6(𝔽13)

 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 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
,
 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
,
 0 5 0 0 0 0 8 8 0 0 0 0 0 0 0 8 0 0 0 0 5 5 0 0 0 0 0 0 8 10 0 0 0 0 8 5
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 8 8 0 0 0 0 0 5 0 0 0 0 0 0 1 11 0 0 0 0 1 12

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

C2×C23.16D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{16}D_6
% in TeX

G:=Group("C2xC2^3.16D6");
// GroupNames label

G:=SmallGroup(192,1039);
// by ID

G=gap.SmallGroup(192,1039);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,297,80,6278]);
// Polycyclic

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