direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C2×C6×C22⋊C4, C24⋊8C12, C25.6C6, (C23×C4)⋊6C6, (C23×C6)⋊6C4, (C23×C12)⋊5C2, C23⋊6(C2×C12), (C24×C6).2C2, (C2×C12)⋊13C23, C6.53(C23×C4), C24.34(C2×C6), C2.1(C23×C12), C22.57(C6×D4), C23.63(C3×D4), (C2×C6).332C24, C22⋊3(C22×C12), (C22×C6).219D4, C6.177(C22×D4), C22.5(C23×C6), (C22×C12)⋊57C22, C23.69(C22×C6), (C23×C6).88C22, (C22×C6).251C23, C2.1(D4×C2×C6), (C2×C6)⋊8(C22×C4), (C2×C4)⋊3(C22×C6), (C22×C4)⋊17(C2×C6), (C22×C6)⋊16(C2×C4), (C2×C6).679(C2×D4), SmallGroup(192,1401)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C6×C22⋊C4
G = < a,b,c,d,e | a2=b6=c2=d2=e4=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >
Subgroups: 1010 in 674 conjugacy classes, 338 normal (12 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C12, C2×C12, C22×C6, C22×C6, C2×C22⋊C4, C23×C4, C25, C3×C22⋊C4, C22×C12, C22×C12, C23×C6, C23×C6, C23×C6, C22×C22⋊C4, C6×C22⋊C4, C23×C12, C24×C6, C2×C6×C22⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C12, C2×C6, C22⋊C4, C22×C4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C2×C22⋊C4, C23×C4, C22×D4, C3×C22⋊C4, C22×C12, C6×D4, C23×C6, C22×C22⋊C4, C6×C22⋊C4, C23×C12, D4×C2×C6, C2×C6×C22⋊C4
►On 96 points
Generators in S96
(1 50)(2 51)(3 52)(4 53)(5 54)(6 49)(7 87)(8 88)(9 89)(10 90)(11 85)(12 86)(13 68)(14 69)(15 70)(16 71)(17 72)(18 67)(19 81)(20 82)(21 83)(22 84)(23 79)(24 80)(25 45)(26 46)(27 47)(28 48)(29 43)(30 44)(31 66)(32 61)(33 62)(34 63)(35 64)(36 65)(37 56)(38 57)(39 58)(40 59)(41 60)(42 55)(73 93)(74 94)(75 95)(76 96)(77 91)(78 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 32)(2 33)(3 34)(4 35)(5 36)(6 31)(7 20)(8 21)(9 22)(10 23)(11 24)(12 19)(13 92)(14 93)(15 94)(16 95)(17 96)(18 91)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)(43 59)(44 60)(45 55)(46 56)(47 57)(48 58)(49 66)(50 61)(51 62)(52 63)(53 64)(54 65)(67 77)(68 78)(69 73)(70 74)(71 75)(72 76)(79 90)(80 85)(81 86)(82 87)(83 88)(84 89)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 95)(20 96)(21 91)(22 92)(23 93)(24 94)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(43 64)(44 65)(45 66)(46 61)(47 62)(48 63)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(67 88)(68 89)(69 90)(70 85)(71 86)(72 87)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)
(1 8 35 94)(2 9 36 95)(3 10 31 96)(4 11 32 91)(5 12 33 92)(6 7 34 93)(13 30 19 38)(14 25 20 39)(15 26 21 40)(16 27 22 41)(17 28 23 42)(18 29 24 37)(43 80 56 67)(44 81 57 68)(45 82 58 69)(46 83 59 70)(47 84 60 71)(48 79 55 72)(49 87 63 73)(50 88 64 74)(51 89 65 75)(52 90 66 76)(53 85 61 77)(54 86 62 78)
G:=sub<Sym(96)| (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,68)(14,69)(15,70)(16,71)(17,72)(18,67)(19,81)(20,82)(21,83)(22,84)(23,79)(24,80)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(31,66)(32,61)(33,62)(34,63)(35,64)(36,65)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(73,93)(74,94)(75,95)(76,96)(77,91)(78,92), (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), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,92)(14,93)(15,94)(16,95)(17,96)(18,91)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,59)(44,60)(45,55)(46,56)(47,57)(48,58)(49,66)(50,61)(51,62)(52,63)(53,64)(54,65)(67,77)(68,78)(69,73)(70,74)(71,75)(72,76)(79,90)(80,85)(81,86)(82,87)(83,88)(84,89), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,95)(20,96)(21,91)(22,92)(23,93)(24,94)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(67,88)(68,89)(69,90)(70,85)(71,86)(72,87)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,8,35,94)(2,9,36,95)(3,10,31,96)(4,11,32,91)(5,12,33,92)(6,7,34,93)(13,30,19,38)(14,25,20,39)(15,26,21,40)(16,27,22,41)(17,28,23,42)(18,29,24,37)(43,80,56,67)(44,81,57,68)(45,82,58,69)(46,83,59,70)(47,84,60,71)(48,79,55,72)(49,87,63,73)(50,88,64,74)(51,89,65,75)(52,90,66,76)(53,85,61,77)(54,86,62,78)>;
G:=Group( (1,50)(2,51)(3,52)(4,53)(5,54)(6,49)(7,87)(8,88)(9,89)(10,90)(11,85)(12,86)(13,68)(14,69)(15,70)(16,71)(17,72)(18,67)(19,81)(20,82)(21,83)(22,84)(23,79)(24,80)(25,45)(26,46)(27,47)(28,48)(29,43)(30,44)(31,66)(32,61)(33,62)(34,63)(35,64)(36,65)(37,56)(38,57)(39,58)(40,59)(41,60)(42,55)(73,93)(74,94)(75,95)(76,96)(77,91)(78,92), (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), (1,32)(2,33)(3,34)(4,35)(5,36)(6,31)(7,20)(8,21)(9,22)(10,23)(11,24)(12,19)(13,92)(14,93)(15,94)(16,95)(17,96)(18,91)(25,42)(26,37)(27,38)(28,39)(29,40)(30,41)(43,59)(44,60)(45,55)(46,56)(47,57)(48,58)(49,66)(50,61)(51,62)(52,63)(53,64)(54,65)(67,77)(68,78)(69,73)(70,74)(71,75)(72,76)(79,90)(80,85)(81,86)(82,87)(83,88)(84,89), (1,37)(2,38)(3,39)(4,40)(5,41)(6,42)(7,17)(8,18)(9,13)(10,14)(11,15)(12,16)(19,95)(20,96)(21,91)(22,92)(23,93)(24,94)(25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(43,64)(44,65)(45,66)(46,61)(47,62)(48,63)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(67,88)(68,89)(69,90)(70,85)(71,86)(72,87)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84), (1,8,35,94)(2,9,36,95)(3,10,31,96)(4,11,32,91)(5,12,33,92)(6,7,34,93)(13,30,19,38)(14,25,20,39)(15,26,21,40)(16,27,22,41)(17,28,23,42)(18,29,24,37)(43,80,56,67)(44,81,57,68)(45,82,58,69)(46,83,59,70)(47,84,60,71)(48,79,55,72)(49,87,63,73)(50,88,64,74)(51,89,65,75)(52,90,66,76)(53,85,61,77)(54,86,62,78) );
G=PermutationGroup([[(1,50),(2,51),(3,52),(4,53),(5,54),(6,49),(7,87),(8,88),(9,89),(10,90),(11,85),(12,86),(13,68),(14,69),(15,70),(16,71),(17,72),(18,67),(19,81),(20,82),(21,83),(22,84),(23,79),(24,80),(25,45),(26,46),(27,47),(28,48),(29,43),(30,44),(31,66),(32,61),(33,62),(34,63),(35,64),(36,65),(37,56),(38,57),(39,58),(40,59),(41,60),(42,55),(73,93),(74,94),(75,95),(76,96),(77,91),(78,92)], [(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)], [(1,32),(2,33),(3,34),(4,35),(5,36),(6,31),(7,20),(8,21),(9,22),(10,23),(11,24),(12,19),(13,92),(14,93),(15,94),(16,95),(17,96),(18,91),(25,42),(26,37),(27,38),(28,39),(29,40),(30,41),(43,59),(44,60),(45,55),(46,56),(47,57),(48,58),(49,66),(50,61),(51,62),(52,63),(53,64),(54,65),(67,77),(68,78),(69,73),(70,74),(71,75),(72,76),(79,90),(80,85),(81,86),(82,87),(83,88),(84,89)], [(1,37),(2,38),(3,39),(4,40),(5,41),(6,42),(7,17),(8,18),(9,13),(10,14),(11,15),(12,16),(19,95),(20,96),(21,91),(22,92),(23,93),(24,94),(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(43,64),(44,65),(45,66),(46,61),(47,62),(48,63),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(67,88),(68,89),(69,90),(70,85),(71,86),(72,87),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84)], [(1,8,35,94),(2,9,36,95),(3,10,31,96),(4,11,32,91),(5,12,33,92),(6,7,34,93),(13,30,19,38),(14,25,20,39),(15,26,21,40),(16,27,22,41),(17,28,23,42),(18,29,24,37),(43,80,56,67),(44,81,57,68),(45,82,58,69),(46,83,59,70),(47,84,60,71),(48,79,55,72),(49,87,63,73),(50,88,64,74),(51,89,65,75),(52,90,66,76),(53,85,61,77),(54,86,62,78)]])
120 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 3A | 3B | 4A | ··· | 4P | 6A | ··· | 6AD | 6AE | ··· | 6AT | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
120 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | C3×D4 |
| kernel | C2×C6×C22⋊C4 | C6×C22⋊C4 | C23×C12 | C24×C6 | C22×C22⋊C4 | C23×C6 | C2×C22⋊C4 | C23×C4 | C25 | C24 | C22×C6 | C23 |
| # reps | 1 | 12 | 2 | 1 | 2 | 16 | 24 | 4 | 2 | 32 | 8 | 16 |
Matrix representation of C2×C6×C22⋊C4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 3 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,3,0,0,0,0,0,3],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,5,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,12,0] >;
C2×C6×C22⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_6\times C_2^2\rtimes C_4
% in TeX
G:=Group("C2xC6xC2^2:C4"); // GroupNames label
G:=SmallGroup(192,1401);
// by ID
G=gap.SmallGroup(192,1401);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-2,672,701]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations