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## G = C2×C12⋊3D4order 192 = 26·3

### Direct product of C2 and C12⋊3D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12⋊3D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C12⋊3D4
 Lower central C3 — C2×C6 — C2×C12⋊3D4
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C123D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 1448 in 498 conjugacy classes, 143 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×8], C22, C22 [×6], C22 [×40], S3 [×4], C6, C6 [×6], C6 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×48], C23, C23 [×4], C23 [×28], Dic3 [×8], C12 [×4], D6 [×20], C2×C6, C2×C6 [×6], C2×C6 [×20], C42 [×4], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×44], C24 [×2], C24 [×2], D12 [×8], C2×Dic3 [×12], C3⋊D4 [×32], C2×C12 [×6], C3×D4 [×8], C22×S3 [×4], C22×S3 [×12], C22×C6, C22×C6 [×4], C22×C6 [×12], C2×C42, C41D4 [×8], C22×D4, C22×D4 [×5], C4×Dic3 [×4], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×16], C2×C3⋊D4 [×16], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23 [×2], C23×C6 [×2], C2×C41D4, C2×C4×Dic3, C123D4 [×8], C22×D12, C22×C3⋊D4 [×4], D4×C2×C6, C2×C123D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×12], C23 [×15], D6 [×7], C2×D4 [×18], C24, C3⋊D4 [×4], C22×S3 [×7], C41D4 [×4], C22×D4 [×3], S3×D4 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C41D4, C123D4 [×4], C2×S3×D4 [×2], C22×C3⋊D4, C2×C123D4

Smallest permutation representation of C2×C123D4
On 96 points
Generators in S96
(1 23)(2 24)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(11 21)(12 22)(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 77)(38 78)(39 79)(40 80)(41 81)(42 82)(43 83)(44 84)(45 73)(46 74)(47 75)(48 76)(61 85)(62 86)(63 87)(64 88)(65 89)(66 90)(67 91)(68 92)(69 93)(70 94)(71 95)(72 96)
(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 68 25 75)(2 61 26 80)(3 66 27 73)(4 71 28 78)(5 64 29 83)(6 69 30 76)(7 62 31 81)(8 67 32 74)(9 72 33 79)(10 65 34 84)(11 70 35 77)(12 63 36 82)(13 90 60 45)(14 95 49 38)(15 88 50 43)(16 93 51 48)(17 86 52 41)(18 91 53 46)(19 96 54 39)(20 89 55 44)(21 94 56 37)(22 87 57 42)(23 92 58 47)(24 85 59 40)
(1 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 36)(9 35)(10 34)(11 33)(12 32)(13 50)(14 49)(15 60)(16 59)(17 58)(18 57)(19 56)(20 55)(21 54)(22 53)(23 52)(24 51)(37 39)(40 48)(41 47)(42 46)(43 45)(61 69)(62 68)(63 67)(64 66)(70 72)(73 83)(74 82)(75 81)(76 80)(77 79)(85 93)(86 92)(87 91)(88 90)(94 96)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 12 12 12 12 2 2 2 2 2 6 ··· 6 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 D4 D6 D6 D6 C3⋊D4 S3×D4 kernel C2×C12⋊3D4 C2×C4×Dic3 C12⋊3D4 C22×D12 C22×C3⋊D4 D4×C2×C6 C22×D4 C2×Dic3 C2×C12 C22×C4 C2×D4 C24 C2×C4 C22 # reps 1 1 8 1 4 1 1 8 4 1 4 2 8 4

Matrix representation of C2×C123D4 in GL5(𝔽13)

 12 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 1 1 0 0 0 12 0 0 0 0 0 0 12 2 0 0 0 12 1
,
 12 0 0 0 0 0 2 4 0 0 0 2 11 0 0 0 0 0 12 2 0 0 0 12 1
,
 12 0 0 0 0 0 12 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,12,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,2,1],[12,0,0,0,0,0,2,2,0,0,0,4,11,0,0,0,0,0,12,12,0,0,0,2,1],[12,0,0,0,0,0,12,1,0,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,12] >;

C2×C123D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes_3D_4
% in TeX

G:=Group("C2xC12:3D4");
// GroupNames label

G:=SmallGroup(192,1362);
// by ID

G=gap.SmallGroup(192,1362);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,297,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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