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## G = C2×C12⋊D4order 192 = 26·3

### Direct product of C2 and C12⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12⋊D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — S3×C22×C4 — C2×C12⋊D4
 Lower central C3 — C2×C6 — C2×C12⋊D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for C2×C12⋊D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd=b-1, dcd=c-1 >

Subgroups: 1304 in 426 conjugacy classes, 135 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C23×C4, C22×D4, D6⋊C4, C3×C4⋊C4, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, C22×Dic3, C22×C12, C22×C12, S3×C23, S3×C23, C2×C4⋊D4, C12⋊D4, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, C22×D12, C22×D12, C2×C12⋊D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, D12, C22×S3, C4⋊D4, C22×D4, C2×C4○D4, C2×D12, S3×D4, Q83S3, S3×C23, C2×C4⋊D4, C12⋊D4, C22×D12, C2×S3×D4, C2×Q83S3, C2×C12⋊D4

Smallest permutation representation of C2×C12⋊D4
On 96 points
Generators in S96
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 83)(26 84)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 79)(34 80)(35 81)(36 82)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(43 57)(44 58)(45 59)(46 60)(47 49)(48 50)(61 86)(62 87)(63 88)(64 89)(65 90)(66 91)(67 92)(68 93)(69 94)(70 95)(71 96)(72 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 51 86 78)(2 58 87 73)(3 53 88 80)(4 60 89 75)(5 55 90 82)(6 50 91 77)(7 57 92 84)(8 52 93 79)(9 59 94 74)(10 54 95 81)(11 49 96 76)(12 56 85 83)(13 40 70 35)(14 47 71 30)(15 42 72 25)(16 37 61 32)(17 44 62 27)(18 39 63 34)(19 46 64 29)(20 41 65 36)(21 48 66 31)(22 43 67 26)(23 38 68 33)(24 45 69 28)
(1 67)(2 66)(3 65)(4 64)(5 63)(6 62)(7 61)(8 72)(9 71)(10 70)(11 69)(12 68)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 96)(25 79)(26 78)(27 77)(28 76)(29 75)(30 74)(31 73)(32 84)(33 83)(34 82)(35 81)(36 80)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 60)(47 59)(48 58)

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

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

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

48 conjugacy classes

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

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 D12 S3×D4 Q8⋊3S3 kernel C2×C12⋊D4 C12⋊D4 C2×D6⋊C4 C6×C4⋊C4 S3×C22×C4 C22×D12 C2×C4⋊C4 C2×C12 C22×S3 C4⋊C4 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 8 2 1 1 3 1 4 4 4 3 4 8 2 2

Matrix representation of C2×C12⋊D4 in GL6(𝔽13)

 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 12 0 0 0 0 0 0 12
,
 0 12 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 5 12 0 0 0 0 0 8
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 11 0 0 0 0 1 12 0 0 0 0 0 0 12 8 0 0 0 0 3 1
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 12 0 0 0 0 0 3 1

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,12,8],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,11,12,0,0,0,0,0,0,12,3,0,0,0,0,8,1],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,1,0,0,0,0,0,12,0,0,0,0,0,0,12,3,0,0,0,0,0,1] >;

C2×C12⋊D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes D_4
% in TeX

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

G:=SmallGroup(192,1065);
// by ID

G=gap.SmallGroup(192,1065);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,675,297,80,6278]);
// 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^7,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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