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## G = (C2×C4)⋊3D12order 192 = 26·3

### 2nd semidirect product of C2×C4 and D12 acting via D12/C6=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C4)⋊3D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C4)⋊3D12
 Lower central C3 — C22×C6 — (C2×C4)⋊3D12
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C4)⋊3D12
G = < a,b,c,d | a2=b4=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=c-1 >

Subgroups: 808 in 238 conjugacy classes, 63 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C22×D4, D6⋊C4, C3×C4⋊C4, C2×D12, C22×Dic3, C22×C12, S3×C23, C23.10D4, C6.C42, C2×D6⋊C4, C6×C4⋊C4, C22×D12, (C2×C4)⋊3D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C22≀C2, C4⋊D4, C22.D4, C4.4D4, C2×D12, C4○D12, S3×D4, Q83S3, C2×C3⋊D4, C23.10D4, D6.D4, C12⋊D4, C127D4, C232D6, C12.23D4, (C2×C4)⋊3D12

Smallest permutation representation of (C2×C4)⋊3D12
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 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 25)(24 26)(37 75)(38 76)(39 77)(40 78)(41 79)(42 80)(43 81)(44 82)(45 83)(46 84)(47 73)(48 74)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(57 85)(58 86)(59 87)(60 88)
(1 91 79 31)(2 32 80 92)(3 93 81 33)(4 34 82 94)(5 95 83 35)(6 36 84 96)(7 85 73 25)(8 26 74 86)(9 87 75 27)(10 28 76 88)(11 89 77 29)(12 30 78 90)(13 65 59 37)(14 38 60 66)(15 67 49 39)(16 40 50 68)(17 69 51 41)(18 42 52 70)(19 71 53 43)(20 44 54 72)(21 61 55 45)(22 46 56 62)(23 63 57 47)(24 48 58 64)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 85)(14 96)(15 95)(16 94)(17 93)(18 92)(19 91)(20 90)(21 89)(22 88)(23 87)(24 86)(25 59)(26 58)(27 57)(28 56)(29 55)(30 54)(31 53)(32 52)(33 51)(34 50)(35 49)(36 60)(37 47)(38 46)(39 45)(40 44)(41 43)(61 67)(62 66)(63 65)(68 72)(69 71)(73 75)(76 84)(77 83)(78 82)(79 81)

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4F 4G 4H 4I 4J 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 12 12 12 2 4 ··· 4 12 12 12 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 C3⋊D4 C4○D12 S3×D4 Q8⋊3S3 kernel (C2×C4)⋊3D12 C6.C42 C2×D6⋊C4 C6×C4⋊C4 C22×D12 C2×C4⋊C4 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 4 1 1 1 4 4 3 6 4 4 4 2 2

Matrix representation of (C2×C4)⋊3D12 in GL6(𝔽13)

 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 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 2 4 0 0 0 0 9 11 0 0 0 0 0 0 1 11 0 0 0 0 1 12
,
 3 10 0 0 0 0 3 6 0 0 0 0 0 0 3 10 0 0 0 0 3 6 0 0 0 0 0 0 5 3 0 0 0 0 0 8
,
 1 1 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 12 1

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,1,1,0,0,0,0,11,12],[3,3,0,0,0,0,10,6,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,5,0,0,0,0,0,3,8],[1,0,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;

(C2×C4)⋊3D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_3D_{12}
% in TeX

G:=Group("(C2xC4):3D12");
// GroupNames label

G:=SmallGroup(192,550);
// by ID

G=gap.SmallGroup(192,550);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,184,6278]);
// Polycyclic

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

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