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## G = C2×C8○D12order 192 = 26·3

### Direct product of C2 and C8○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C2×C8○D12
 Chief series C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×C4○D12 — C2×C8○D12
 Lower central C3 — C6 — C2×C8○D12
 Upper central C1 — C2×C8 — C22×C8

Generators and relations for C2×C8○D12
G = < a,b,c,d | a2=b8=d2=1, c6=b4, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b4c5 >

Subgroups: 504 in 266 conjugacy classes, 151 normal (29 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C22×C8, C2×M4(2), C8○D4, C2×C4○D4, S3×C8, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C2×C8○D4, S3×C2×C8, C2×C8⋊S3, C8○D12, C2×C4.Dic3, C22×C24, C2×C4○D12, C2×C8○D12
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C8○D4, C23×C4, S3×C2×C4, S3×C23, C2×C8○D4, C8○D12, S3×C22×C4, C2×C8○D12

Smallest permutation representation of C2×C8○D12
On 96 points
Generators in S96
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(37 83)(38 84)(39 73)(40 74)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 81)(48 82)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)
(1 56 66 40 7 50 72 46)(2 57 67 41 8 51 61 47)(3 58 68 42 9 52 62 48)(4 59 69 43 10 53 63 37)(5 60 70 44 11 54 64 38)(6 49 71 45 12 55 65 39)(13 89 79 28 19 95 73 34)(14 90 80 29 20 96 74 35)(15 91 81 30 21 85 75 36)(16 92 82 31 22 86 76 25)(17 93 83 32 23 87 77 26)(18 94 84 33 24 88 78 27)
(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 34)(2 33)(3 32)(4 31)(5 30)(6 29)(7 28)(8 27)(9 26)(10 25)(11 36)(12 35)(13 56)(14 55)(15 54)(16 53)(17 52)(18 51)(19 50)(20 49)(21 60)(22 59)(23 58)(24 57)(37 82)(38 81)(39 80)(40 79)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 84)(48 83)(61 94)(62 93)(63 92)(64 91)(65 90)(66 89)(67 88)(68 87)(69 86)(70 85)(71 96)(72 95)

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6G 8A ··· 8H 8I 8J 8K 8L 8M ··· 8T 12A ··· 12H 24A ··· 24P order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 8 ··· 8 8 8 8 8 8 ··· 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 6 6 6 6 2 1 1 1 1 2 2 6 6 6 6 2 ··· 2 1 ··· 1 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C4 C4 C4 C4 S3 D6 D6 C4×S3 C4×S3 C8○D4 C8○D12 kernel C2×C8○D12 S3×C2×C8 C2×C8⋊S3 C8○D12 C2×C4.Dic3 C22×C24 C2×C4○D12 C2×Dic6 C2×D12 C4○D12 C2×C3⋊D4 C22×C8 C2×C8 C22×C4 C2×C4 C23 C6 C2 # reps 1 2 2 8 1 1 1 2 2 8 4 1 6 1 6 2 8 16

Matrix representation of C2×C8○D12 in GL5(𝔽73)

 72 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 27 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 51 0 0 0 0 0 51
,
 72 0 0 0 0 0 0 72 0 0 0 1 1 0 0 0 0 0 62 8 0 0 0 3 11
,
 72 0 0 0 0 0 0 72 0 0 0 72 0 0 0 0 0 0 11 65 0 0 0 15 62

G:=sub<GL(5,GF(73))| [72,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[27,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,51,0,0,0,0,0,51],[72,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,62,3,0,0,0,8,11],[72,0,0,0,0,0,0,72,0,0,0,72,0,0,0,0,0,0,11,15,0,0,0,65,62] >;

C2×C8○D12 in GAP, Magma, Sage, TeX

C_2\times C_8\circ D_{12}
% in TeX

G:=Group("C2xC8oD12");
// GroupNames label

G:=SmallGroup(192,1297);
// by ID

G=gap.SmallGroup(192,1297);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,80,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=d^2=1,c^6=b^4,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^4*c^5>;
// generators/relations

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