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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, 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, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2.C42, C2×C22⋊C4, C2×C4⋊C4, D6⋊C4, C3×C4⋊C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C23.11D4, C6.C42, C6.C42, C2×D6⋊C4, C2×D6⋊C4, C6×C4⋊C4, (C2×C12).289D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C22.D4, C4.4D4, C422C2, C4○D12, S3×D4, D42S3, Q83S3, C2×C3⋊D4, C23.11D4, D6.D4, C4⋊C4⋊S3, C23.28D6, D63D4, C12.23D4, (C2×C12).289D4

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

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

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

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

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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