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## G = (C2×C12).290D4order 192 = 26·3

### 264th 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).290D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — (C2×C12).290D4
 Lower central C3 — C22×C6 — (C2×C12).290D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C12).290D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab5, dcd=ab6c-1 >

Subgroups: 552 in 186 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C3, C4 [×9], C22 [×3], C22 [×4], C22 [×10], S3 [×2], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×19], C23, C23 [×8], Dic3 [×5], C12 [×4], D6 [×10], C2×C6 [×3], C2×C6 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, C2×Dic3 [×4], C2×Dic3 [×7], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×S3 [×6], C22×C6, C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], Dic3⋊C4 [×4], D6⋊C4 [×6], C3×C4⋊C4 [×2], C22×Dic3, C22×Dic3 [×2], C22×C12, C22×C12 [×2], S3×C23, C23.4Q8, C6.C42, C2×Dic3⋊C4 [×2], C2×D6⋊C4, C2×D6⋊C4 [×2], C6×C4⋊C4, (C2×C12).290D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], Q8 [×2], C23, D6 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×3], C3⋊D4 [×2], C22×S3, C22⋊Q8 [×3], C22.D4 [×3], C41D4, C4○D12 [×2], S3×D4 [×2], S3×Q8, Q83S3, C2×C3⋊D4, C23.4Q8, D6.D4 [×2], D6⋊Q8 [×2], C23.28D6, C123D4, D63Q8, (C2×C12).290D4

Smallest permutation representation of (C2×C12).290D4
On 96 points
Generators in S96
(1 36)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(37 62)(38 63)(39 64)(40 65)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 61)(49 96)(50 85)(51 86)(52 87)(53 88)(54 89)(55 90)(56 91)(57 92)(58 93)(59 94)(60 95)
(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 61 50 73)(2 47 51 24)(3 71 52 83)(4 45 53 22)(5 69 54 81)(6 43 55 20)(7 67 56 79)(8 41 57 18)(9 65 58 77)(10 39 59 16)(11 63 60 75)(12 37 49 14)(13 36 48 85)(15 34 38 95)(17 32 40 93)(19 30 42 91)(21 28 44 89)(23 26 46 87)(25 72 86 84)(27 70 88 82)(29 68 90 80)(31 66 92 78)(33 64 94 76)(35 62 96 74)
(2 29)(3 11)(4 27)(5 9)(6 25)(8 35)(10 33)(12 31)(13 67)(14 47)(15 65)(16 45)(17 63)(18 43)(19 61)(20 41)(21 71)(22 39)(23 69)(24 37)(26 34)(28 32)(38 77)(40 75)(42 73)(44 83)(46 81)(48 79)(49 92)(51 90)(52 60)(53 88)(54 58)(55 86)(57 96)(59 94)(62 84)(64 82)(66 80)(68 78)(70 76)(72 74)(87 95)(89 93)

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

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

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

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 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + - + + - + image C1 C2 C2 C2 C2 S3 D4 D4 Q8 D6 C4○D4 C3⋊D4 C4○D12 S3×D4 S3×Q8 Q8⋊3S3 kernel (C2×C12).290D4 C6.C42 C2×Dic3⋊C4 C2×D6⋊C4 C6×C4⋊C4 C2×C4⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C22 C22 C22 C22 # reps 1 1 2 3 1 1 4 2 2 3 6 4 8 2 1 1

Matrix representation of (C2×C12).290D4 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 6 3 0 0 0 0 10 7 0 0 0 0 0 0 0 5 0 0 0 0 5 0
,
 10 7 0 0 0 0 10 3 0 0 0 0 0 0 5 0 0 0 0 0 6 8 0 0 0 0 0 0 0 1 0 0 0 0 12 0
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 9 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

(C2×C12).290D4 in GAP, Magma, Sage, TeX

(C_2\times C_{12})._{290}D_4
% in TeX

G:=Group("(C2xC12).290D4");
// GroupNames label

G:=SmallGroup(192,552);
// by ID

G=gap.SmallGroup(192,552);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,232,254,387,268,6278]);
// 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*b^6*c^-1>;
// generators/relations

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