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## G = (C2×C12)⋊5D4order 192 = 26·3

### 1st semidirect product of C2×C12 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12)⋊5D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C12)⋊5D4
 Lower central C3 — C22×C6 — (C2×C12)⋊5D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C12)⋊5D4
G = < a,b,c,d | a2=b12=c4=d2=1, cbc-1=ab=ba, ac=ca, ad=da, dbd=b-1, dcd=c-1 >

Subgroups: 1168 in 322 conjugacy classes, 69 normal (12 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, 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, C22×D4, D6⋊C4, C2×D12, C22×Dic3, C22×C12, S3×C23, C232D4, C3×C2.C42, C2×D6⋊C4, C22×D12, (C2×C12)⋊5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C22≀C2, C4⋊D4, C41D4, C2×D12, S3×D4, Q83S3, C232D4, C4⋊D12, D6⋊D4, C12⋊D4, (C2×C12)⋊5D4

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

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2M 3 4A ··· 4F 4G 4H 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 12 ··· 12 2 4 ··· 4 12 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + + + + + image C1 C2 C2 C2 S3 D4 D4 D6 C4○D4 D12 S3×D4 Q8⋊3S3 kernel (C2×C12)⋊5D4 C3×C2.C42 C2×D6⋊C4 C22×D12 C2.C42 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 1 3 3 1 6 6 3 2 12 3 1

Matrix representation of (C2×C12)⋊5D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 2 0 0 0 0 0 12 0 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 0 0 0 3 3 0 0 0 0 10 6
,
 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 11 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 12 0

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

(C2×C12)⋊5D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})\rtimes_5D_4
% in TeX

G:=Group("(C2xC12):5D4");
// GroupNames label

G:=SmallGroup(192,230);
// by ID

G=gap.SmallGroup(192,230);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,387,58,6278]);
// Polycyclic

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

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