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## G = C2×C12.48D4order 192 = 26·3

### Direct product of C2 and C12.48D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C2×C12.48D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C22×Dic3 — C22×Dic6 — C2×C12.48D4
 Lower central C3 — C2×C6 — C2×C12.48D4
 Upper central C1 — C23 — C23×C4

Generators and relations for C2×C12.48D4
G = < a,b,c,d | a2=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 664 in 322 conjugacy classes, 143 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C23, C23, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×Q8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22⋊Q8, C23×C4, C22×Q8, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C2×Dic6, C22×Dic3, C22×C12, C22×C12, C22×C12, C23×C6, C2×C22⋊Q8, C2×Dic3⋊C4, C12.48D4, C2×C4⋊Dic3, C2×C6.D4, C22×Dic6, C23×C12, C2×C12.48D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2×D4, C2×Q8, C4○D4, C24, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C22×D4, C22×Q8, C2×C4○D4, C2×Dic6, C4○D12, C2×C3⋊D4, S3×C23, C2×C22⋊Q8, C12.48D4, C22×Dic6, C2×C4○D12, C22×C3⋊D4, C2×C12.48D4

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

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A ··· 4H 4I ··· 4P 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 2 ··· 2 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 Q8 D6 D6 C4○D4 C3⋊D4 Dic6 C4○D12 kernel C2×C12.48D4 C2×Dic3⋊C4 C12.48D4 C2×C4⋊Dic3 C2×C6.D4 C22×Dic6 C23×C12 C23×C4 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 # reps 1 2 8 1 2 1 1 1 4 4 6 1 4 8 8 8

Matrix representation of C2×C12.48D4 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 0 0 0 0 0 0 3 0 0 0 0 0 0 10 0 0 0 0 0 0 4 0 0 0 0 0 0 6 0 0 0 0 0 0 11
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,3,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,6,0,0,0,0,0,0,11],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0] >;

C2×C12.48D4 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{48}D_4
% in TeX

G:=Group("C2xC12.48D4");
// GroupNames label

G:=SmallGroup(192,1343);
// by ID

G=gap.SmallGroup(192,1343);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,184,675,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

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