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## G = (C2×C4)⋊9D12order 192 = 26·3

### 1st semidirect product of C2×C4 and D12 acting via D12/D6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — (C2×C4)⋊9D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×D12 — (C2×C4)⋊9D12
 Lower central C3 — C2×C6 — (C2×C4)⋊9D12
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C4)⋊9D12
G = < a,b,c,d | a2=b4=c12=d2=1, cbc-1=dbd=ab=ba, ac=ca, ad=da, dcd=c-1 >

Subgroups: 896 in 286 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×6], C3, C4 [×8], C22 [×7], C22 [×26], S3 [×6], C6 [×7], C2×C4 [×4], C2×C4 [×22], D4 [×8], C23, C23 [×18], Dic3 [×3], C12 [×5], D6 [×4], D6 [×22], C2×C6 [×7], C22⋊C4 [×6], C22×C4 [×3], C22×C4 [×8], C2×D4 [×8], C24 [×2], C4×S3 [×8], D12 [×8], C2×Dic3 [×2], C2×Dic3 [×5], C2×C12 [×4], C2×C12 [×7], C22×S3 [×8], C22×S3 [×10], C22×C6, C2.C42, C2.C42, C2×C22⋊C4 [×3], C23×C4, C22×D4, D6⋊C4 [×6], S3×C2×C4 [×6], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C22×C12 [×3], S3×C23 [×2], C23.23D4, C6.C42, C3×C2.C42, C2×D6⋊C4 [×3], S3×C22×C4, C22×D12, (C2×C4)⋊9D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], D12 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, S3×C2×C4, C2×D12, C4○D12, S3×D4 [×3], Q83S3, C23.23D4, C4×D12, S3×C22⋊C4, D6⋊D4, Dic3⋊D4, Dic35D4, D6.D4, C12⋊D4, (C2×C4)⋊9D12

Smallest permutation representation of (C2×C4)⋊9D12
On 96 points
Generators in S96
(1 47)(2 48)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 26)(14 27)(15 28)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 25)(49 84)(50 73)(51 74)(52 75)(53 76)(54 77)(55 78)(56 79)(57 80)(58 81)(59 82)(60 83)(61 87)(62 88)(63 89)(64 90)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 85)(72 86)
(1 23 60 65)(2 25 49 92)(3 13 50 67)(4 27 51 94)(5 15 52 69)(6 29 53 96)(7 17 54 71)(8 31 55 86)(9 19 56 61)(10 33 57 88)(11 21 58 63)(12 35 59 90)(14 74 68 38)(16 76 70 40)(18 78 72 42)(20 80 62 44)(22 82 64 46)(24 84 66 48)(26 73 93 37)(28 75 95 39)(30 77 85 41)(32 79 87 43)(34 81 89 45)(36 83 91 47)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 36)(14 35)(15 34)(16 33)(17 32)(18 31)(19 30)(20 29)(21 28)(22 27)(23 26)(24 25)(37 47)(38 46)(39 45)(40 44)(41 43)(50 60)(51 59)(52 58)(53 57)(54 56)(61 85)(62 96)(63 95)(64 94)(65 93)(66 92)(67 91)(68 90)(69 89)(70 88)(71 87)(72 86)(73 83)(74 82)(75 81)(76 80)(77 79)

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 6A ··· 6G 12A ··· 12L order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 6 6 6 6 12 12 2 2 2 2 2 4 4 4 4 6 6 6 6 12 12 2 ··· 2 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D4 D6 C4○D4 C4×S3 D12 C4○D12 S3×D4 Q8⋊3S3 kernel (C2×C4)⋊9D12 C6.C42 C3×C2.C42 C2×D6⋊C4 S3×C22×C4 C22×D12 C2×D12 C2.C42 C2×Dic3 C2×C12 C22×S3 C22×C4 C2×C6 C2×C4 C2×C4 C22 C22 C22 # reps 1 1 1 3 1 1 8 1 2 2 4 3 4 4 4 4 3 1

Matrix representation of (C2×C4)⋊9D12 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
,
 5 2 0 0 0 0 1 8 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 12 0 0 0 0 0 5 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 10 3 0 0 0 0 10 7
,
 1 0 0 0 0 0 8 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 1 0 0 0 0 0 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],[5,1,0,0,0,0,2,8,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[12,5,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,10,10,0,0,0,0,3,7],[1,8,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,1,12] >;

(C2×C4)⋊9D12 in GAP, Magma, Sage, TeX

(C_2\times C_4)\rtimes_9D_{12}
% in TeX

G:=Group("(C2xC4):9D12");
// GroupNames label

G:=SmallGroup(192,224);
// by ID

G=gap.SmallGroup(192,224);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,387,58,6278]);
// Polycyclic

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

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