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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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C42, C41D4, C22×D4, C22×D4, C4×Dic3, C2×D12, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, C2×C41D4, C2×C4×Dic3, C123D4, C22×D12, C22×C3⋊D4, D4×C2×C6, C2×C123D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C41D4, C22×D4, S3×D4, C2×C3⋊D4, S3×C23, C2×C41D4, C123D4, C2×S3×D4, C22×C3⋊D4, C2×C123D4

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

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

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

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

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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