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## G = C4×C4○D12order 192 = 26·3

### Direct product of C4 and C4○D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C4×C4○D12
 Chief series C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C2×C4○D12 — C4×C4○D12
 Lower central C3 — C6 — C4×C4○D12
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4○D12
G = < a,b,c,d | a4=b4=d2=1, c6=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c5 >

Subgroups: 632 in 310 conjugacy classes, 159 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C42, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C22×S3, C22×C6, C2×C42, C2×C42, C42⋊C2, C4×D4, C4×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, C4×C12, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C4×C4○D4, C4×Dic6, S3×C42, C422S3, C4×D12, C23.26D6, C4×C3⋊D4, C2×C4×C12, C2×C4○D12, C4×C4○D12
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C23×C4, C2×C4○D4, S3×C2×C4, C4○D12, S3×C23, C4×C4○D4, S3×C22×C4, C2×C4○D12, C4×C4○D12

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

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

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

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

72 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A ··· 4L 4M ··· 4R 4S ··· 4AD 6A ··· 6G 12A ··· 12X order 1 2 2 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 6 6 6 6 2 1 ··· 1 2 ··· 2 6 ··· 6 2 ··· 2 2 ··· 2

72 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C4 S3 D6 D6 C4○D4 C4×S3 C4○D12 kernel C4×C4○D12 C4×Dic6 S3×C42 C42⋊2S3 C4×D12 C23.26D6 C4×C3⋊D4 C2×C4×C12 C2×C4○D12 C4○D12 C2×C42 C42 C22×C4 C12 C2×C4 C4 # reps 1 2 2 2 2 1 4 1 1 16 1 4 3 8 8 16

Matrix representation of C4×C4○D12 in GL3(𝔽13) generated by

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

C4×C4○D12 in GAP, Magma, Sage, TeX

C_4\times C_4\circ D_{12}
% in TeX

G:=Group("C4xC4oD12");
// GroupNames label

G:=SmallGroup(192,1033);
// by ID

G=gap.SmallGroup(192,1033);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,80,6278]);
// Polycyclic

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

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