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## G = (C2×C12).33D4order 192 = 26·3

### 7th non-split extension by C2×C12 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — (C2×C12).33D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C12).33D4
 Lower central C3 — C22×C6 — (C2×C12).33D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C12).33D4
G = < a,b,c,d | a2=b12=c4=1, d2=a, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=ac-1 >

Subgroups: 560 in 186 conjugacy classes, 65 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C4⋊Dic3, D6⋊C4, C22×Dic3, C22×C12, S3×C23, C23.4Q8, C3×C2.C42, C2×C4⋊Dic3, C2×D6⋊C4, (C2×C12).33D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, D12, C22×S3, C22⋊Q8, C22.D4, C41D4, C2×D12, D42S3, S3×Q8, C23.4Q8, C4⋊D12, C23.21D6, C4.D12, (C2×C12).33D4

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

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A ··· 4F 4G ··· 4L 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 Q8 D6 C4○D4 D12 D4⋊2S3 S3×Q8 kernel (C2×C12).33D4 C3×C2.C42 C2×C4⋊Dic3 C2×D6⋊C4 C2.C42 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 # reps 1 1 3 3 1 6 2 3 6 12 3 1

Matrix representation of (C2×C12).33D4 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
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 10 10 0 0 0 0 3 7
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,10,3,0,0,0,0,10,7],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

(C2×C12).33D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{33}D_4
% in TeX

G:=Group("(C2xC12).33D4");
// GroupNames label

G:=SmallGroup(192,236);
// by ID

G=gap.SmallGroup(192,236);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,226,6278]);
// Polycyclic

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

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