direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C4⋊C4⋊7S3, C4⋊C4⋊51D6, C6.12(C23×C4), (C2×C6).46C24, C6⋊3(C42⋊C2), C12.88(C22×C4), C4⋊Dic3⋊70C22, D6.18(C22×C4), (C22×C4).375D6, (C2×C12).578C23, D6⋊C4.115C22, (C4×Dic3)⋊75C22, C22.22(S3×C23), (S3×C23).97C22, C23.335(C22×S3), (C22×C6).395C23, Dic3.23(C22×C4), C22.73(D4⋊2S3), (C22×S3).153C23, (C22×C12).358C22, C22.33(Q8⋊3S3), (C2×Dic3).185C23, (C22×Dic3).210C22, (S3×C2×C4)⋊6C4, (C6×C4⋊C4)⋊8C2, C4.91(S3×C2×C4), (C2×C4⋊C4)⋊25S3, (C4×S3)⋊12(C2×C4), C3⋊3(C2×C42⋊C2), (C2×C4×Dic3)⋊32C2, C6.71(C2×C4○D4), (S3×C22×C4).4C2, C2.14(S3×C22×C4), C22.72(S3×C2×C4), (C3×C4⋊C4)⋊43C22, (C2×C4).161(C4×S3), (C2×D6⋊C4).23C2, (C2×C4⋊Dic3)⋊38C2, C2.4(C2×D4⋊2S3), C2.1(C2×Q8⋊3S3), (C2×C12).128(C2×C4), (S3×C2×C4).243C22, (C2×C6).171(C4○D4), (C22×S3).65(C2×C4), (C2×C4).265(C22×S3), (C2×C6).151(C22×C4), (C2×Dic3).111(C2×C4), SmallGroup(192,1061)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C4⋊C4⋊7S3
G = < a,b,c,d,e | a2=b4=c4=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b2c, ede=d-1 >
Subgroups: 696 in 330 conjugacy classes, 167 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C23×C4, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, C22×Dic3, C22×C12, C22×C12, S3×C23, C2×C42⋊C2, C4⋊C4⋊7S3, C2×C4×Dic3, C2×C4⋊Dic3, C2×D6⋊C4, C6×C4⋊C4, S3×C22×C4, C2×C4⋊C4⋊7S3
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4○D4, C24, C4×S3, C22×S3, C42⋊C2, C23×C4, C2×C4○D4, S3×C2×C4, D4⋊2S3, Q8⋊3S3, S3×C23, C2×C42⋊C2, C4⋊C4⋊7S3, S3×C22×C4, C2×D4⋊2S3, C2×Q8⋊3S3, C2×C4⋊C4⋊7S3
►On 96 points
(1 59)(2 60)(3 57)(4 58)(5 96)(6 93)(7 94)(8 95)(9 18)(10 19)(11 20)(12 17)(13 73)(14 74)(15 75)(16 76)(21 82)(22 83)(23 84)(24 81)(25 38)(26 39)(27 40)(28 37)(29 42)(30 43)(31 44)(32 41)(33 46)(34 47)(35 48)(36 45)(49 62)(50 63)(51 64)(52 61)(53 66)(54 67)(55 68)(56 65)(69 90)(70 91)(71 92)(72 89)(77 85)(78 86)(79 87)(80 88)
(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 5 89 66)(2 8 90 65)(3 7 91 68)(4 6 92 67)(9 25 48 83)(10 28 45 82)(11 27 46 81)(12 26 47 84)(13 43 51 86)(14 42 52 85)(15 41 49 88)(16 44 50 87)(17 39 34 23)(18 38 35 22)(19 37 36 21)(20 40 33 24)(29 61 77 74)(30 64 78 73)(31 63 79 76)(32 62 80 75)(53 59 96 72)(54 58 93 71)(55 57 94 70)(56 60 95 69)
(1 16 17)(2 13 18)(3 14 19)(4 15 20)(5 44 39)(6 41 40)(7 42 37)(8 43 38)(9 60 73)(10 57 74)(11 58 75)(12 59 76)(21 68 85)(22 65 86)(23 66 87)(24 67 88)(25 95 30)(26 96 31)(27 93 32)(28 94 29)(33 92 49)(34 89 50)(35 90 51)(36 91 52)(45 70 61)(46 71 62)(47 72 63)(48 69 64)(53 79 84)(54 80 81)(55 77 82)(56 78 83)
(1 57)(2 58)(3 59)(4 60)(5 96)(6 93)(7 94)(8 95)(9 15)(10 16)(11 13)(12 14)(17 74)(18 75)(19 76)(20 73)(21 77)(22 78)(23 79)(24 80)(25 43)(26 44)(27 41)(28 42)(29 37)(30 38)(31 39)(32 40)(33 64)(34 61)(35 62)(36 63)(45 50)(46 51)(47 52)(48 49)(53 66)(54 67)(55 68)(56 65)(69 92)(70 89)(71 90)(72 91)(81 88)(82 85)(83 86)(84 87)
G:=sub<Sym(96)| (1,59)(2,60)(3,57)(4,58)(5,96)(6,93)(7,94)(8,95)(9,18)(10,19)(11,20)(12,17)(13,73)(14,74)(15,75)(16,76)(21,82)(22,83)(23,84)(24,81)(25,38)(26,39)(27,40)(28,37)(29,42)(30,43)(31,44)(32,41)(33,46)(34,47)(35,48)(36,45)(49,62)(50,63)(51,64)(52,61)(53,66)(54,67)(55,68)(56,65)(69,90)(70,91)(71,92)(72,89)(77,85)(78,86)(79,87)(80,88), (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), (1,5,89,66)(2,8,90,65)(3,7,91,68)(4,6,92,67)(9,25,48,83)(10,28,45,82)(11,27,46,81)(12,26,47,84)(13,43,51,86)(14,42,52,85)(15,41,49,88)(16,44,50,87)(17,39,34,23)(18,38,35,22)(19,37,36,21)(20,40,33,24)(29,61,77,74)(30,64,78,73)(31,63,79,76)(32,62,80,75)(53,59,96,72)(54,58,93,71)(55,57,94,70)(56,60,95,69), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,44,39)(6,41,40)(7,42,37)(8,43,38)(9,60,73)(10,57,74)(11,58,75)(12,59,76)(21,68,85)(22,65,86)(23,66,87)(24,67,88)(25,95,30)(26,96,31)(27,93,32)(28,94,29)(33,92,49)(34,89,50)(35,90,51)(36,91,52)(45,70,61)(46,71,62)(47,72,63)(48,69,64)(53,79,84)(54,80,81)(55,77,82)(56,78,83), (1,57)(2,58)(3,59)(4,60)(5,96)(6,93)(7,94)(8,95)(9,15)(10,16)(11,13)(12,14)(17,74)(18,75)(19,76)(20,73)(21,77)(22,78)(23,79)(24,80)(25,43)(26,44)(27,41)(28,42)(29,37)(30,38)(31,39)(32,40)(33,64)(34,61)(35,62)(36,63)(45,50)(46,51)(47,52)(48,49)(53,66)(54,67)(55,68)(56,65)(69,92)(70,89)(71,90)(72,91)(81,88)(82,85)(83,86)(84,87)>;
G:=Group( (1,59)(2,60)(3,57)(4,58)(5,96)(6,93)(7,94)(8,95)(9,18)(10,19)(11,20)(12,17)(13,73)(14,74)(15,75)(16,76)(21,82)(22,83)(23,84)(24,81)(25,38)(26,39)(27,40)(28,37)(29,42)(30,43)(31,44)(32,41)(33,46)(34,47)(35,48)(36,45)(49,62)(50,63)(51,64)(52,61)(53,66)(54,67)(55,68)(56,65)(69,90)(70,91)(71,92)(72,89)(77,85)(78,86)(79,87)(80,88), (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), (1,5,89,66)(2,8,90,65)(3,7,91,68)(4,6,92,67)(9,25,48,83)(10,28,45,82)(11,27,46,81)(12,26,47,84)(13,43,51,86)(14,42,52,85)(15,41,49,88)(16,44,50,87)(17,39,34,23)(18,38,35,22)(19,37,36,21)(20,40,33,24)(29,61,77,74)(30,64,78,73)(31,63,79,76)(32,62,80,75)(53,59,96,72)(54,58,93,71)(55,57,94,70)(56,60,95,69), (1,16,17)(2,13,18)(3,14,19)(4,15,20)(5,44,39)(6,41,40)(7,42,37)(8,43,38)(9,60,73)(10,57,74)(11,58,75)(12,59,76)(21,68,85)(22,65,86)(23,66,87)(24,67,88)(25,95,30)(26,96,31)(27,93,32)(28,94,29)(33,92,49)(34,89,50)(35,90,51)(36,91,52)(45,70,61)(46,71,62)(47,72,63)(48,69,64)(53,79,84)(54,80,81)(55,77,82)(56,78,83), (1,57)(2,58)(3,59)(4,60)(5,96)(6,93)(7,94)(8,95)(9,15)(10,16)(11,13)(12,14)(17,74)(18,75)(19,76)(20,73)(21,77)(22,78)(23,79)(24,80)(25,43)(26,44)(27,41)(28,42)(29,37)(30,38)(31,39)(32,40)(33,64)(34,61)(35,62)(36,63)(45,50)(46,51)(47,52)(48,49)(53,66)(54,67)(55,68)(56,65)(69,92)(70,89)(71,90)(72,91)(81,88)(82,85)(83,86)(84,87) );
G=PermutationGroup([[(1,59),(2,60),(3,57),(4,58),(5,96),(6,93),(7,94),(8,95),(9,18),(10,19),(11,20),(12,17),(13,73),(14,74),(15,75),(16,76),(21,82),(22,83),(23,84),(24,81),(25,38),(26,39),(27,40),(28,37),(29,42),(30,43),(31,44),(32,41),(33,46),(34,47),(35,48),(36,45),(49,62),(50,63),(51,64),(52,61),(53,66),(54,67),(55,68),(56,65),(69,90),(70,91),(71,92),(72,89),(77,85),(78,86),(79,87),(80,88)], [(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)], [(1,5,89,66),(2,8,90,65),(3,7,91,68),(4,6,92,67),(9,25,48,83),(10,28,45,82),(11,27,46,81),(12,26,47,84),(13,43,51,86),(14,42,52,85),(15,41,49,88),(16,44,50,87),(17,39,34,23),(18,38,35,22),(19,37,36,21),(20,40,33,24),(29,61,77,74),(30,64,78,73),(31,63,79,76),(32,62,80,75),(53,59,96,72),(54,58,93,71),(55,57,94,70),(56,60,95,69)], [(1,16,17),(2,13,18),(3,14,19),(4,15,20),(5,44,39),(6,41,40),(7,42,37),(8,43,38),(9,60,73),(10,57,74),(11,58,75),(12,59,76),(21,68,85),(22,65,86),(23,66,87),(24,67,88),(25,95,30),(26,96,31),(27,93,32),(28,94,29),(33,92,49),(34,89,50),(35,90,51),(36,91,52),(45,70,61),(46,71,62),(47,72,63),(48,69,64),(53,79,84),(54,80,81),(55,77,82),(56,78,83)], [(1,57),(2,58),(3,59),(4,60),(5,96),(6,93),(7,94),(8,95),(9,15),(10,16),(11,13),(12,14),(17,74),(18,75),(19,76),(20,73),(21,77),(22,78),(23,79),(24,80),(25,43),(26,44),(27,41),(28,42),(29,37),(30,38),(31,39),(32,40),(33,64),(34,61),(35,62),(36,63),(45,50),(46,51),(47,52),(48,49),(53,66),(54,67),(55,68),(56,65),(69,92),(70,89),(71,90),(72,91),(81,88),(82,85),(83,86),(84,87)]])
60 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4L | 4M | ··· | 4T | 4U | ··· | 4AB | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D6 | D6 | C4○D4 | C4×S3 | D4⋊2S3 | Q8⋊3S3 |
| kernel | C2×C4⋊C4⋊7S3 | C4⋊C4⋊7S3 | C2×C4×Dic3 | C2×C4⋊Dic3 | C2×D6⋊C4 | C6×C4⋊C4 | S3×C22×C4 | S3×C2×C4 | C2×C4⋊C4 | C4⋊C4 | C22×C4 | C2×C6 | C2×C4 | C22 | C22 |
| # reps | 1 | 8 | 2 | 1 | 2 | 1 | 1 | 16 | 1 | 4 | 3 | 8 | 8 | 2 | 2 |
Matrix representation of C2×C4⋊C4⋊7S3 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 3 | 5 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,8,0,0,0,0,5,5,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,8,3,0,0,0,0,5,0,0,0,0,0,8,0,0,0,0,0,8],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,1,0],[1,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2×C4⋊C4⋊7S3 in GAP, Magma, Sage, TeX
C_2\times C_4\rtimes C_4\rtimes_7S_3
% in TeX
G:=Group("C2xC4:C4:7S3"); // GroupNames label
G:=SmallGroup(192,1061);
// by ID
G=gap.SmallGroup(192,1061);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,1123,297,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations