Copied to
clipboard

## G = C6×C42⋊2C2order 192 = 26·3

### Direct product of C6 and C42⋊2C2

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C6×C42⋊2C2
 Chief series C1 — C2 — C22 — C2×C6 — C22×C6 — C3×C22⋊C4 — C3×C42⋊2C2 — C6×C42⋊2C2
 Lower central C1 — C22 — C6×C42⋊2C2
 Upper central C1 — C22×C6 — C6×C42⋊2C2

Generators and relations for C6×C422C2
G = < a,b,c,d | a6=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, dcd=b2c-1 >

Subgroups: 354 in 246 conjugacy classes, 162 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C24, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C422C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C23×C6, C2×C422C2, C2×C4×C12, C6×C22⋊C4, C6×C4⋊C4, C3×C422C2, C6×C422C2
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C24, C22×C6, C422C2, C2×C4○D4, C3×C4○D4, C23×C6, C2×C422C2, C3×C422C2, C6×C4○D4, C6×C422C2

Smallest permutation representation of C6×C422C2
On 96 points
Generators in S96
(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 77 38 89)(2 78 39 90)(3 73 40 85)(4 74 41 86)(5 75 42 87)(6 76 37 88)(7 53 21 65)(8 54 22 66)(9 49 23 61)(10 50 24 62)(11 51 19 63)(12 52 20 64)(13 45 96 57)(14 46 91 58)(15 47 92 59)(16 48 93 60)(17 43 94 55)(18 44 95 56)(25 72 33 84)(26 67 34 79)(27 68 35 80)(28 69 36 81)(29 70 31 82)(30 71 32 83)
(1 47 32 65)(2 48 33 66)(3 43 34 61)(4 44 35 62)(5 45 36 63)(6 46 31 64)(7 77 92 83)(8 78 93 84)(9 73 94 79)(10 74 95 80)(11 75 96 81)(12 76 91 82)(13 69 19 87)(14 70 20 88)(15 71 21 89)(16 72 22 90)(17 67 23 85)(18 68 24 86)(25 54 39 60)(26 49 40 55)(27 50 41 56)(28 51 42 57)(29 52 37 58)(30 53 38 59)
(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)(13 96)(14 91)(15 92)(16 93)(17 94)(18 95)(43 49)(44 50)(45 51)(46 52)(47 53)(48 54)(55 61)(56 62)(57 63)(58 64)(59 65)(60 66)(67 85)(68 86)(69 87)(70 88)(71 89)(72 90)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)

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

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

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

84 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3A 3B 4A ··· 4L 4M ··· 4R 6A ··· 6N 6O 6P 6Q 6R 12A ··· 12X 12Y ··· 12AJ order 1 2 ··· 2 2 2 3 3 4 ··· 4 4 ··· 4 6 ··· 6 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 ··· 1 4 4 1 1 2 ··· 2 4 ··· 4 1 ··· 1 4 4 4 4 2 ··· 2 4 ··· 4

84 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 C6×C42⋊2C2 C2×C4×C12 C6×C22⋊C4 C6×C4⋊C4 C3×C42⋊2C2 C2×C42⋊2C2 C2×C42 C2×C22⋊C4 C2×C4⋊C4 C42⋊2C2 C2×C6 C22 # reps 1 1 3 3 8 2 2 6 6 16 12 24

Matrix representation of C6×C422C2 in GL5(𝔽13)

 12 0 0 0 0 0 10 0 0 0 0 0 10 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 12 0 0 0 2 1 0 0 0 0 0 5 0 0 0 0 0 5
,
 1 0 0 0 0 0 8 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 12 0
,
 12 0 0 0 0 0 1 1 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 12

G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,10,0,0,0,0,0,10,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,2,0,0,0,12,1,0,0,0,0,0,5,0,0,0,0,0,5],[1,0,0,0,0,0,8,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,12,0],[12,0,0,0,0,0,1,0,0,0,0,1,12,0,0,0,0,0,1,0,0,0,0,0,12] >;

C6×C422C2 in GAP, Magma, Sage, TeX

C_6\times C_4^2\rtimes_2C_2
% in TeX

G:=Group("C6xC4^2:2C2");
// GroupNames label

G:=SmallGroup(192,1417);
// by ID

G=gap.SmallGroup(192,1417);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,701,1016,2102,268]);
// Polycyclic

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

׿
×
𝔽