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## G = C2×C6×C4○D4order 192 = 26·3

### Direct product of C2×C6 and C4○D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C2×C6×C4○D4
 Chief series C1 — C2 — C6 — C2×C6 — C3×D4 — C3×C4○D4 — C6×C4○D4 — C2×C6×C4○D4
 Lower central C1 — C2 — C2×C6×C4○D4
 Upper central C1 — C22×C12 — C2×C6×C4○D4

Generators and relations for C2×C6×C4○D4
G = < a,b,c,d,e | a2=b6=c4=e2=1, d2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 1010 in 890 conjugacy classes, 770 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C6, C2×C4, D4, Q8, C23, C23, C23, C12, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C22×C12, C22×C12, C6×D4, C6×Q8, C3×C4○D4, C23×C6, C22×C4○D4, C23×C12, D4×C2×C6, Q8×C2×C6, C6×C4○D4, C2×C6×C4○D4
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C2×C4○D4, C25, C3×C4○D4, C23×C6, C22×C4○D4, C6×C4○D4, C24×C6, C2×C6×C4○D4

Smallest permutation representation of C2×C6×C4○D4
On 96 points
Generators in S96
(1 30)(2 25)(3 26)(4 27)(5 28)(6 29)(7 80)(8 81)(9 82)(10 83)(11 84)(12 79)(13 21)(14 22)(15 23)(16 24)(17 19)(18 20)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)
(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 51 15 44)(2 52 16 45)(3 53 17 46)(4 54 18 47)(5 49 13 48)(6 50 14 43)(7 66 95 59)(8 61 96 60)(9 62 91 55)(10 63 92 56)(11 64 93 57)(12 65 94 58)(19 38 26 31)(20 39 27 32)(21 40 28 33)(22 41 29 34)(23 42 30 35)(24 37 25 36)(67 86 74 79)(68 87 75 80)(69 88 76 81)(70 89 77 82)(71 90 78 83)(72 85 73 84)
(1 75 15 68)(2 76 16 69)(3 77 17 70)(4 78 18 71)(5 73 13 72)(6 74 14 67)(7 42 95 35)(8 37 96 36)(9 38 91 31)(10 39 92 32)(11 40 93 33)(12 41 94 34)(19 62 26 55)(20 63 27 56)(21 64 28 57)(22 65 29 58)(23 66 30 59)(24 61 25 60)(43 86 50 79)(44 87 51 80)(45 88 52 81)(46 89 53 82)(47 90 54 83)(48 85 49 84)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 67)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(43 79)(44 80)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 89)(54 90)

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

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

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

120 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2S 3A 3B 4A ··· 4H 4I ··· 4T 6A ··· 6N 6O ··· 6AL 12A ··· 12P 12Q ··· 12AN order 1 2 ··· 2 2 ··· 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 2 ··· 2 1 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2

120 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C3 C6 C6 C6 C6 C4○D4 C3×C4○D4 kernel C2×C6×C4○D4 C23×C12 D4×C2×C6 Q8×C2×C6 C6×C4○D4 C22×C4○D4 C23×C4 C22×D4 C22×Q8 C2×C4○D4 C2×C6 C22 # reps 1 3 3 1 24 2 6 6 2 48 8 16

Matrix representation of C2×C6×C4○D4 in GL4(𝔽13) generated by

 1 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 12 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 12 0 0 0 0 12 0 0 0 0 5 0 0 0 0 5
,
 1 0 0 0 0 12 0 0 0 0 12 2 0 0 12 1
,
 12 0 0 0 0 1 0 0 0 0 1 0 0 0 1 12
G:=sub<GL(4,GF(13))| [1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,5,0,0,0,0,5],[1,0,0,0,0,12,0,0,0,0,12,12,0,0,2,1],[12,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;

C2×C6×C4○D4 in GAP, Magma, Sage, TeX

C_2\times C_6\times C_4\circ D_4
% in TeX

G:=Group("C2xC6xC4oD4");
// GroupNames label

G:=SmallGroup(192,1533);
// by ID

G=gap.SmallGroup(192,1533);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-2,1373,520]);
// Polycyclic

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

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