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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 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×26], Q8 [×8], C23, C23 [×6], C23 [×4], Dic3 [×8], C12 [×4], C12 [×2], C2×C6, C2×C6 [×10], C2×C6 [×12], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C2×Q8 [×8], C24, Dic6 [×8], C2×Dic3 [×8], C2×Dic3 [×8], C2×C12 [×8], C2×C12 [×10], C22×C6, C22×C6 [×6], C22×C6 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic3⋊C4 [×8], C4⋊Dic3 [×4], C6.D4 [×8], C2×Dic6 [×4], C2×Dic6 [×4], C22×Dic3 [×4], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], C23×C6, C2×C22⋊Q8, C2×Dic3⋊C4 [×2], C12.48D4 [×8], C2×C4⋊Dic3, C2×C6.D4 [×2], C22×Dic6, C23×C12, C2×C12.48D4
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], Q8 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, Dic6 [×4], C3⋊D4 [×4], C22×S3 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C2×Dic6 [×6], C4○D12 [×2], C2×C3⋊D4 [×6], S3×C23, C2×C22⋊Q8, C12.48D4 [×4], 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 14)(2 15)(3 16)(4 17)(5 18)(6 19)(7 20)(8 21)(9 22)(10 23)(11 24)(12 13)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 49)(36 50)(37 89)(38 90)(39 91)(40 92)(41 93)(42 94)(43 95)(44 96)(45 85)(46 86)(47 87)(48 88)(61 73)(62 74)(63 75)(64 76)(65 77)(66 78)(67 79)(68 80)(69 81)(70 82)(71 83)(72 84)
(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 68 89 29)(2 67 90 28)(3 66 91 27)(4 65 92 26)(5 64 93 25)(6 63 94 36)(7 62 95 35)(8 61 96 34)(9 72 85 33)(10 71 86 32)(11 70 87 31)(12 69 88 30)(13 81 48 56)(14 80 37 55)(15 79 38 54)(16 78 39 53)(17 77 40 52)(18 76 41 51)(19 75 42 50)(20 74 43 49)(21 73 44 60)(22 84 45 59)(23 83 46 58)(24 82 47 57)
(1 35 7 29)(2 34 8 28)(3 33 9 27)(4 32 10 26)(5 31 11 25)(6 30 12 36)(13 50 19 56)(14 49 20 55)(15 60 21 54)(16 59 22 53)(17 58 23 52)(18 57 24 51)(37 74 43 80)(38 73 44 79)(39 84 45 78)(40 83 46 77)(41 82 47 76)(42 81 48 75)(61 96 67 90)(62 95 68 89)(63 94 69 88)(64 93 70 87)(65 92 71 86)(66 91 72 85)

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

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

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

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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