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## G = C2×C23.12D6order 192 = 26·3

### Direct product of C2 and C23.12D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C23.12D6
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C2×C4×Dic3 — C2×C23.12D6
 Lower central C3 — C2×C6 — C2×C23.12D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C23.12D6
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, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 744 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C4.4D4, C22×D4, C22×Q8, C4×Dic3, C6.D4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C2×C4.4D4, C2×C4×Dic3, C23.12D6, C2×C6.D4, C22×Dic6, D4×C2×C6, C2×C23.12D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C4.4D4, C22×D4, C2×C4○D4, D42S3, C2×C3⋊D4, S3×C23, C2×C4.4D4, C23.12D6, C2×D42S3, C22×C3⋊D4, C2×C23.12D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 2 2 2 2 2 6 ··· 6 12 12 12 12 2 ··· 2 4 ··· 4 4 4 4 4

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 D4⋊2S3 kernel C2×C23.12D6 C2×C4×Dic3 C23.12D6 C2×C6.D4 C22×Dic6 D4×C2×C6 C22×D4 C2×C12 C22×C4 C2×D4 C24 C2×C6 C2×C4 C22 # reps 1 1 8 4 1 1 1 4 1 4 2 8 8 4

Matrix representation of C2×C23.12D6 in GL5(𝔽13)

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

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

C2×C23.12D6 in GAP, Magma, Sage, TeX

C_2\times C_2^3._{12}D_6
% in TeX

G:=Group("C2xC2^3.12D6");
// GroupNames label

G:=SmallGroup(192,1356);
// by ID

G=gap.SmallGroup(192,1356);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,100,1571,185,6278]);
// 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,e*b*e^-1=b*c=c*b,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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