direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C42.7C22, (C4×C8)⋊3C6, C4⋊C8⋊14C6, (C4×C24)⋊8C2, C8⋊C4⋊8C6, C4⋊C4.6C12, C22⋊C8.8C6, C6.47(C8○D4), C22⋊C4.3C12, C42.61(C2×C6), C23.11(C2×C12), C42⋊C2.8C6, C12.352(C4○D4), (C2×C24).327C22, (C4×C12).247C22, (C2×C12).989C23, C6.61(C42⋊C2), C22.46(C22×C12), (C22×C12).416C22, (C3×C4⋊C8)⋊33C2, C2.6(C3×C8○D4), (C3×C4⋊C4).18C4, (C2×C8).51(C2×C6), (C3×C8⋊C4)⋊22C2, C4.50(C3×C4○D4), (C2×C4).27(C2×C12), (C2×C12).200(C2×C4), (C3×C22⋊C8).17C2, (C3×C22⋊C4).10C4, (C22×C4).40(C2×C6), (C22×C6).22(C2×C4), (C2×C4).157(C22×C6), (C2×C6).239(C22×C4), C2.12(C3×C42⋊C2), (C3×C42⋊C2).22C2, SmallGroup(192,866)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.7C22
G = < a,b,c,d,e | a3=b4=c4=e2=1, d2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1c2, ebe=bc2, cd=dc, ce=ec, ede=b2c2d >
Subgroups: 130 in 96 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C42⋊C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C42.7C22, C4×C24, C3×C8⋊C4, C3×C22⋊C8, C3×C4⋊C8, C3×C42⋊C2, C3×C42.7C22
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, C12, C2×C6, C22×C4, C4○D4, C2×C12, C22×C6, C42⋊C2, C8○D4, C22×C12, C3×C4○D4, C42.7C22, C3×C42⋊C2, C3×C8○D4, C3×C42.7C22
►On 96 points
Generators in S96
(1 71 23)(2 72 24)(3 65 17)(4 66 18)(5 67 19)(6 68 20)(7 69 21)(8 70 22)(9 58 26)(10 59 27)(11 60 28)(12 61 29)(13 62 30)(14 63 31)(15 64 32)(16 57 25)(33 73 81)(34 74 82)(35 75 83)(36 76 84)(37 77 85)(38 78 86)(39 79 87)(40 80 88)(41 53 89)(42 54 90)(43 55 91)(44 56 92)(45 49 93)(46 50 94)(47 51 95)(48 52 96)
(1 47 75 27)(2 32 76 44)(3 41 77 29)(4 26 78 46)(5 43 79 31)(6 28 80 48)(7 45 73 25)(8 30 74 42)(9 86 50 66)(10 71 51 83)(11 88 52 68)(12 65 53 85)(13 82 54 70)(14 67 55 87)(15 84 56 72)(16 69 49 81)(17 89 37 61)(18 58 38 94)(19 91 39 63)(20 60 40 96)(21 93 33 57)(22 62 34 90)(23 95 35 59)(24 64 36 92)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 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)
(2 80)(4 74)(6 76)(8 78)(9 50)(10 14)(11 52)(12 16)(13 54)(15 56)(18 34)(20 36)(22 38)(24 40)(25 29)(26 46)(27 31)(28 48)(30 42)(32 44)(41 45)(43 47)(49 53)(51 55)(57 61)(58 94)(59 63)(60 96)(62 90)(64 92)(66 82)(68 84)(70 86)(72 88)(89 93)(91 95)
G:=sub<Sym(96)| (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,47,75,27)(2,32,76,44)(3,41,77,29)(4,26,78,46)(5,43,79,31)(6,28,80,48)(7,45,73,25)(8,30,74,42)(9,86,50,66)(10,71,51,83)(11,88,52,68)(12,65,53,85)(13,82,54,70)(14,67,55,87)(15,84,56,72)(16,69,49,81)(17,89,37,61)(18,58,38,94)(19,91,39,63)(20,60,40,96)(21,93,33,57)(22,62,34,90)(23,95,35,59)(24,64,36,92), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (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), (2,80)(4,74)(6,76)(8,78)(9,50)(10,14)(11,52)(12,16)(13,54)(15,56)(18,34)(20,36)(22,38)(24,40)(25,29)(26,46)(27,31)(28,48)(30,42)(32,44)(41,45)(43,47)(49,53)(51,55)(57,61)(58,94)(59,63)(60,96)(62,90)(64,92)(66,82)(68,84)(70,86)(72,88)(89,93)(91,95)>;
G:=Group( (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,53,89)(42,54,90)(43,55,91)(44,56,92)(45,49,93)(46,50,94)(47,51,95)(48,52,96), (1,47,75,27)(2,32,76,44)(3,41,77,29)(4,26,78,46)(5,43,79,31)(6,28,80,48)(7,45,73,25)(8,30,74,42)(9,86,50,66)(10,71,51,83)(11,88,52,68)(12,65,53,85)(13,82,54,70)(14,67,55,87)(15,84,56,72)(16,69,49,81)(17,89,37,61)(18,58,38,94)(19,91,39,63)(20,60,40,96)(21,93,33,57)(22,62,34,90)(23,95,35,59)(24,64,36,92), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32)(33,35,37,39)(34,36,38,40)(41,43,45,47)(42,44,46,48)(49,51,53,55)(50,52,54,56)(57,59,61,63)(58,60,62,64)(65,67,69,71)(66,68,70,72)(73,75,77,79)(74,76,78,80)(81,83,85,87)(82,84,86,88)(89,91,93,95)(90,92,94,96), (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), (2,80)(4,74)(6,76)(8,78)(9,50)(10,14)(11,52)(12,16)(13,54)(15,56)(18,34)(20,36)(22,38)(24,40)(25,29)(26,46)(27,31)(28,48)(30,42)(32,44)(41,45)(43,47)(49,53)(51,55)(57,61)(58,94)(59,63)(60,96)(62,90)(64,92)(66,82)(68,84)(70,86)(72,88)(89,93)(91,95) );
G=PermutationGroup([[(1,71,23),(2,72,24),(3,65,17),(4,66,18),(5,67,19),(6,68,20),(7,69,21),(8,70,22),(9,58,26),(10,59,27),(11,60,28),(12,61,29),(13,62,30),(14,63,31),(15,64,32),(16,57,25),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,53,89),(42,54,90),(43,55,91),(44,56,92),(45,49,93),(46,50,94),(47,51,95),(48,52,96)], [(1,47,75,27),(2,32,76,44),(3,41,77,29),(4,26,78,46),(5,43,79,31),(6,28,80,48),(7,45,73,25),(8,30,74,42),(9,86,50,66),(10,71,51,83),(11,88,52,68),(12,65,53,85),(13,82,54,70),(14,67,55,87),(15,84,56,72),(16,69,49,81),(17,89,37,61),(18,58,38,94),(19,91,39,63),(20,60,40,96),(21,93,33,57),(22,62,34,90),(23,95,35,59),(24,64,36,92)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32),(33,35,37,39),(34,36,38,40),(41,43,45,47),(42,44,46,48),(49,51,53,55),(50,52,54,56),(57,59,61,63),(58,60,62,64),(65,67,69,71),(66,68,70,72),(73,75,77,79),(74,76,78,80),(81,83,85,87),(82,84,86,88),(89,91,93,95),(90,92,94,96)], [(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)], [(2,80),(4,74),(6,76),(8,78),(9,50),(10,14),(11,52),(12,16),(13,54),(15,56),(18,34),(20,36),(22,38),(24,40),(25,29),(26,46),(27,31),(28,48),(30,42),(32,44),(41,45),(43,47),(49,53),(51,55),(57,61),(58,94),(59,63),(60,96),(62,90),(64,92),(66,82),(68,84),(70,86),(72,88),(89,93),(91,95)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 6A | ··· | 6F | 6G | 6H | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12V | 24A | ··· | 24P | 24Q | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | C12 | C4○D4 | C8○D4 | C3×C4○D4 | C3×C8○D4 |
| kernel | C3×C42.7C22 | C4×C24 | C3×C8⋊C4 | C3×C22⋊C8 | C3×C4⋊C8 | C3×C42⋊C2 | C42.7C22 | C3×C22⋊C4 | C3×C4⋊C4 | C4×C8 | C8⋊C4 | C22⋊C8 | C4⋊C8 | C42⋊C2 | C22⋊C4 | C4⋊C4 | C12 | C6 | C4 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 2 | 8 | 8 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C42.7C22 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 1 | 0 | 0 |
| 71 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 72 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 63 | 0 | 0 | 0 |
| 0 | 63 | 0 | 0 |
| 0 | 0 | 0 | 22 |
| 0 | 0 | 51 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[1,71,0,0,1,72,0,0,0,0,0,72,0,0,72,0],[27,0,0,0,0,27,0,0,0,0,27,0,0,0,0,27],[63,0,0,0,0,63,0,0,0,0,0,51,0,0,22,0],[1,0,0,0,1,72,0,0,0,0,1,0,0,0,0,72] >;
C3×C42.7C22 in GAP, Magma, Sage, TeX
C_3\times C_4^2._7C_2^2
% in TeX
G:=Group("C3xC4^2.7C2^2"); // GroupNames label
G:=SmallGroup(192,866);
// by ID
G=gap.SmallGroup(192,866);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,1059,142,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=e^2=1,d^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=b^-1*c^2,e*b*e=b*c^2,c*d=d*c,c*e=e*c,e*d*e=b^2*c^2*d>;
// generators/relations