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## G = (C2×C12).56D4order 192 = 26·3

### 30th 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).56D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C12).56D4
 Lower central C3 — C22×C6 — (C2×C12).56D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).56D4
G = < a,b,c,d | a2=b4=c12=1, d2=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b2c-1 >

Subgroups: 552 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×9], C22 [×7], C22 [×10], S3 [×2], C6 [×7], C2×C4 [×4], C2×C4 [×17], C23, C23 [×8], Dic3 [×4], C12 [×5], D6 [×10], C2×C6 [×7], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×4], C2×C12 [×7], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×6], C3×C4⋊C4 [×2], C22×Dic3 [×3], C22×C12 [×3], S3×C23, C23.Q8, C6.C42, C2×Dic3⋊C4, C2×C4⋊Dic3, C2×D6⋊C4 [×3], C6×C4⋊C4, (C2×C12).56D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, C2×D12, C4○D12, S3×D4, D42S3, S3×Q8, Q83S3, C2×C3⋊D4, C23.Q8, C12⋊D4, D6⋊Q8, C4.D12, C4⋊C4⋊S3, C127D4, C23.14D6, D63Q8, (C2×C12).56D4

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

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

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

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

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 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + - - + image C1 C2 C2 C2 C2 C2 S3 D4 D4 Q8 D6 C4○D4 D12 C3⋊D4 C4○D12 S3×D4 D4⋊2S3 S3×Q8 Q8⋊3S3 kernel (C2×C12).56D4 C6.C42 C2×Dic3⋊C4 C2×C4⋊Dic3 C2×D6⋊C4 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 C22 C22 # reps 1 1 1 1 3 1 1 2 4 2 3 6 4 4 4 1 1 1 1

Matrix representation of (C2×C12).56D4 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 0 1 0 0 0 0 12 0
,
 10 3 0 0 0 0 10 7 0 0 0 0 0 0 3 10 0 0 0 0 3 6 0 0 0 0 0 0 5 0 0 0 0 0 0 8
,
 10 3 0 0 0 0 6 3 0 0 0 0 0 0 3 10 0 0 0 0 7 10 0 0 0 0 0 0 5 0 0 0 0 0 0 8

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[10,10,0,0,0,0,3,7,0,0,0,0,0,0,3,3,0,0,0,0,10,6,0,0,0,0,0,0,5,0,0,0,0,0,0,8],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,3,7,0,0,0,0,10,10,0,0,0,0,0,0,5,0,0,0,0,0,0,8] >;

(C2×C12).56D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{56}D_4
% in TeX

G:=Group("(C2xC12).56D4");
// GroupNames label

G:=SmallGroup(192,553);
// by ID

G=gap.SmallGroup(192,553);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,184,6278]);
// Polycyclic

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

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