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## G = C2×C12⋊7D4order 192 = 26·3

### Direct product of C2 and C12⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12⋊7D4
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C22×D12 — C2×C12⋊7D4
 Lower central C3 — C2×C6 — C2×C12⋊7D4
 Upper central C1 — C23 — C23×C4

Generators and relations for C2×C127D4
G = < a,b,c,d | a2=b12=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1176 in 426 conjugacy classes, 143 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×8], C3, C4 [×4], C4 [×6], C22, C22 [×10], C22 [×32], S3 [×4], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×18], D4 [×24], C23, C23 [×6], C23 [×20], Dic3 [×4], C12 [×4], C12 [×2], D6 [×20], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×D4 [×24], C24, C24 [×2], D12 [×8], C2×Dic3 [×4], C2×Dic3 [×4], C3⋊D4 [×16], C2×C12 [×8], C2×C12 [×10], C22×S3 [×4], C22×S3 [×12], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4, C4⋊D4 [×8], C23×C4, C22×D4 [×3], C4⋊Dic3 [×4], D6⋊C4 [×8], C2×D12 [×4], C2×D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×8], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], S3×C23 [×2], C23×C6, C2×C4⋊D4, C2×C4⋊Dic3, C2×D6⋊C4 [×2], C127D4 [×8], C22×D12, C22×C3⋊D4 [×2], C23×C12, C2×C127D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×8], C23 [×15], D6 [×7], C2×D4 [×12], C4○D4 [×2], C24, D12 [×4], C3⋊D4 [×4], C22×S3 [×7], C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×D12 [×6], C4○D12 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C4⋊D4, C127D4 [×4], C22×D12, C2×C4○D12, C22×C3⋊D4, C2×C127D4

Smallest permutation representation of C2×C127D4
On 96 points
Generators in S96
(1 85)(2 86)(3 87)(4 88)(5 89)(6 90)(7 91)(8 92)(9 93)(10 94)(11 95)(12 96)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 46)(26 47)(27 48)(28 37)(29 38)(30 39)(31 40)(32 41)(33 42)(34 43)(35 44)(36 45)(61 74)(62 75)(63 76)(64 77)(65 78)(66 79)(67 80)(68 81)(69 82)(70 83)(71 84)(72 73)
(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 40 17 72)(2 39 18 71)(3 38 19 70)(4 37 20 69)(5 48 21 68)(6 47 22 67)(7 46 23 66)(8 45 24 65)(9 44 13 64)(10 43 14 63)(11 42 15 62)(12 41 16 61)(25 59 79 91)(26 58 80 90)(27 57 81 89)(28 56 82 88)(29 55 83 87)(30 54 84 86)(31 53 73 85)(32 52 74 96)(33 51 75 95)(34 50 76 94)(35 49 77 93)(36 60 78 92)
(1 59)(2 58)(3 57)(4 56)(5 55)(6 54)(7 53)(8 52)(9 51)(10 50)(11 49)(12 60)(13 95)(14 94)(15 93)(16 92)(17 91)(18 90)(19 89)(20 88)(21 87)(22 86)(23 85)(24 96)(25 40)(26 39)(27 38)(28 37)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(61 78)(62 77)(63 76)(64 75)(65 74)(66 73)(67 84)(68 83)(69 82)(70 81)(71 80)(72 79)

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 3 4A ··· 4H 4I 4J 4K 4L 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 12 12 12 12 2 2 ··· 2 12 12 12 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 D12 C4○D12 kernel C2×C12⋊7D4 C2×C4⋊Dic3 C2×D6⋊C4 C12⋊7D4 C22×D12 C22×C3⋊D4 C23×C12 C23×C4 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 # reps 1 1 2 8 1 2 1 1 4 4 6 1 4 8 8 8

Matrix representation of C2×C127D4 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 8 0 0 0 8 0 0 0 0 0 0 6 10 0 0 0 3 3
,
 1 0 0 0 0 0 0 12 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 12 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 12 0

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,8,0,0,0,8,0,0,0,0,0,0,6,3,0,0,0,10,3],[1,0,0,0,0,0,0,1,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,12,0] >;

C2×C127D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}\rtimes_7D_4
% in TeX

G:=Group("C2xC12:7D4");
// GroupNames label

G:=SmallGroup(192,1349);
// by ID

G=gap.SmallGroup(192,1349);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,184,675,6278]);
// 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=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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