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## G = (C2×C12).289D4order 192 = 26·3

### 263rd 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).289D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C12).289D4
 Lower central C3 — C22×C6 — (C2×C12).289D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).289D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab5, dcd=ac-1 >

Subgroups: 520 in 170 conjugacy classes, 57 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×10], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×17], C23, C23 [×8], Dic3 [×3], C12 [×4], D6 [×10], C2×C6 [×3], C2×C6 [×4], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, C2×Dic3 [×9], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42 [×3], C2×C22⋊C4 [×3], C2×C4⋊C4, D6⋊C4 [×6], C3×C4⋊C4 [×2], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C23.11D4, C6.C42, C6.C42 [×2], C2×D6⋊C4, C2×D6⋊C4 [×2], C6×C4⋊C4, (C2×C12).289D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4 [×5], C3⋊D4 [×2], C22×S3, C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C4○D12 [×2], S3×D4, D42S3, Q83S3 [×2], C2×C3⋊D4, C23.11D4, D6.D4 [×2], C4⋊C4⋊S3 [×2], C23.28D6, D63D4, C12.23D4, (C2×C12).289D4

Smallest permutation representation of (C2×C12).289D4
On 96 points
Generators in S96
(1 67)(2 68)(3 69)(4 70)(5 71)(6 72)(7 61)(8 62)(9 63)(10 64)(11 65)(12 66)(13 59)(14 60)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 40)(26 41)(27 42)(28 43)(29 44)(30 45)(31 46)(32 47)(33 48)(34 37)(35 38)(36 39)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 85)(82 86)(83 87)(84 88)
(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 60 43 88)(2 13 44 83)(3 58 45 86)(4 23 46 81)(5 56 47 96)(6 21 48 79)(7 54 37 94)(8 19 38 77)(9 52 39 92)(10 17 40 75)(11 50 41 90)(12 15 42 73)(14 28 84 67)(16 26 74 65)(18 36 76 63)(20 34 78 61)(22 32 80 71)(24 30 82 69)(25 91 64 51)(27 89 66 49)(29 87 68 59)(31 85 70 57)(33 95 72 55)(35 93 62 53)
(2 72)(3 11)(4 70)(5 9)(6 68)(8 66)(10 64)(12 62)(13 79)(14 88)(15 77)(16 86)(17 75)(18 96)(19 73)(20 94)(21 83)(22 92)(23 81)(24 90)(25 40)(26 30)(27 38)(29 48)(31 46)(32 36)(33 44)(35 42)(39 47)(41 45)(49 93)(50 82)(51 91)(52 80)(53 89)(54 78)(55 87)(56 76)(57 85)(58 74)(59 95)(60 84)(63 71)(65 69)

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

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

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

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 2 4 4 4 type + + + + + + + + + - + image C1 C2 C2 C2 S3 D4 D4 D6 C4○D4 C3⋊D4 C4○D12 S3×D4 D4⋊2S3 Q8⋊3S3 kernel (C2×C12).289D4 C6.C42 C2×D6⋊C4 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 3 3 1 1 2 2 3 10 4 8 1 1 2

Matrix representation of (C2×C12).289D4 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 1 0 0 0 0 0 0 1
,
 4 2 0 0 0 0 11 2 0 0 0 0 0 0 1 11 0 0 0 0 1 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12
,
 5 0 0 0 0 0 8 8 0 0 0 0 0 0 8 10 0 0 0 0 8 5 0 0 0 0 0 0 2 4 0 0 0 0 2 11
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 12 12

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

(C2×C12).289D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{289}D_4
% in TeX

G:=Group("(C2xC12).289D4");
// GroupNames label

G:=SmallGroup(192,551);
// by ID

G=gap.SmallGroup(192,551);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,387,100,6278]);
// 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=a*b^-1,d*b*d=a*b^5,d*c*d=a*c^-1>;
// generators/relations

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