direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C42.28C22, C4⋊Q8⋊6C6, C8⋊C4⋊10C6, Q8⋊C4⋊19C6, D4⋊C4.7C6, (C2×C12).340D4, C42.26(C2×C6), C4.4D4.6C6, C22.110(C6×D4), C12.271(C4○D4), C6.145(C8⋊C22), (C2×C24).335C22, (C4×C12).268C22, (C2×C12).945C23, C6.74(C4.4D4), (C6×D4).200C22, (C6×Q8).174C22, C6.145(C8.C22), (C3×C4⋊Q8)⋊27C2, C4⋊C4.20(C2×C6), (C2×C8).56(C2×C6), (C3×C8⋊C4)⋊24C2, C4.16(C3×C4○D4), (C2×C4).41(C3×D4), (C2×D4).23(C2×C6), (C2×C6).666(C2×D4), C2.20(C3×C8⋊C22), (C2×Q8).19(C2×C6), (C3×Q8⋊C4)⋊42C2, C2.12(C3×C4.4D4), C2.20(C3×C8.C22), (C3×D4⋊C4).16C2, (C3×C4⋊C4).240C22, (C2×C4).120(C22×C6), (C3×C4.4D4).15C2, SmallGroup(192,922)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.28C22
G = < a,b,c,d,e | a3=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >
Subgroups: 194 in 100 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×D4, C2×Q8, C2×Q8, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C8⋊C4, D4⋊C4, Q8⋊C4, C4.4D4, C4⋊Q8, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×D4, C6×Q8, C6×Q8, C42.28C22, C3×C8⋊C4, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.4D4, C3×C4⋊Q8, C3×C42.28C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C8⋊C22, C8.C22, C6×D4, C3×C4○D4, C42.28C22, C3×C4.4D4, C3×C8⋊C22, C3×C8.C22, C3×C42.28C22
►On 96 points
Generators in S96
(1 65 17)(2 66 18)(3 67 19)(4 68 20)(5 69 21)(6 70 22)(7 71 23)(8 72 24)(9 32 61)(10 25 62)(11 26 63)(12 27 64)(13 28 57)(14 29 58)(15 30 59)(16 31 60)(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 48 73 32)(2 45 74 29)(3 42 75 26)(4 47 76 31)(5 44 77 28)(6 41 78 25)(7 46 79 30)(8 43 80 27)(9 17 96 33)(10 22 89 38)(11 19 90 35)(12 24 91 40)(13 21 92 37)(14 18 93 34)(15 23 94 39)(16 20 95 36)(49 82 58 66)(50 87 59 71)(51 84 60 68)(52 81 61 65)(53 86 62 70)(54 83 63 67)(55 88 64 72)(56 85 57 69)
(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)
(2 76)(3 7)(4 74)(6 80)(8 78)(9 92)(10 16)(11 90)(12 14)(13 96)(15 94)(18 36)(19 23)(20 34)(22 40)(24 38)(25 31)(26 42)(27 29)(28 48)(30 46)(32 44)(35 39)(41 47)(43 45)(49 55)(50 59)(51 53)(52 57)(54 63)(56 61)(58 64)(60 62)(66 84)(67 71)(68 82)(70 88)(72 86)(75 79)(83 87)(89 95)(91 93)
(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)
G:=sub<Sym(96)| (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,61)(10,25,62)(11,26,63)(12,27,64)(13,28,57)(14,29,58)(15,30,59)(16,31,60)(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), (1,48,73,32)(2,45,74,29)(3,42,75,26)(4,47,76,31)(5,44,77,28)(6,41,78,25)(7,46,79,30)(8,43,80,27)(9,17,96,33)(10,22,89,38)(11,19,90,35)(12,24,91,40)(13,21,92,37)(14,18,93,34)(15,23,94,39)(16,20,95,36)(49,82,58,66)(50,87,59,71)(51,84,60,68)(52,81,61,65)(53,86,62,70)(54,83,63,67)(55,88,64,72)(56,85,57,69), (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), (2,76)(3,7)(4,74)(6,80)(8,78)(9,92)(10,16)(11,90)(12,14)(13,96)(15,94)(18,36)(19,23)(20,34)(22,40)(24,38)(25,31)(26,42)(27,29)(28,48)(30,46)(32,44)(35,39)(41,47)(43,45)(49,55)(50,59)(51,53)(52,57)(54,63)(56,61)(58,64)(60,62)(66,84)(67,71)(68,82)(70,88)(72,86)(75,79)(83,87)(89,95)(91,93), (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)>;
G:=Group( (1,65,17)(2,66,18)(3,67,19)(4,68,20)(5,69,21)(6,70,22)(7,71,23)(8,72,24)(9,32,61)(10,25,62)(11,26,63)(12,27,64)(13,28,57)(14,29,58)(15,30,59)(16,31,60)(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), (1,48,73,32)(2,45,74,29)(3,42,75,26)(4,47,76,31)(5,44,77,28)(6,41,78,25)(7,46,79,30)(8,43,80,27)(9,17,96,33)(10,22,89,38)(11,19,90,35)(12,24,91,40)(13,21,92,37)(14,18,93,34)(15,23,94,39)(16,20,95,36)(49,82,58,66)(50,87,59,71)(51,84,60,68)(52,81,61,65)(53,86,62,70)(54,83,63,67)(55,88,64,72)(56,85,57,69), (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), (2,76)(3,7)(4,74)(6,80)(8,78)(9,92)(10,16)(11,90)(12,14)(13,96)(15,94)(18,36)(19,23)(20,34)(22,40)(24,38)(25,31)(26,42)(27,29)(28,48)(30,46)(32,44)(35,39)(41,47)(43,45)(49,55)(50,59)(51,53)(52,57)(54,63)(56,61)(58,64)(60,62)(66,84)(67,71)(68,82)(70,88)(72,86)(75,79)(83,87)(89,95)(91,93), (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) );
G=PermutationGroup([[(1,65,17),(2,66,18),(3,67,19),(4,68,20),(5,69,21),(6,70,22),(7,71,23),(8,72,24),(9,32,61),(10,25,62),(11,26,63),(12,27,64),(13,28,57),(14,29,58),(15,30,59),(16,31,60),(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)], [(1,48,73,32),(2,45,74,29),(3,42,75,26),(4,47,76,31),(5,44,77,28),(6,41,78,25),(7,46,79,30),(8,43,80,27),(9,17,96,33),(10,22,89,38),(11,19,90,35),(12,24,91,40),(13,21,92,37),(14,18,93,34),(15,23,94,39),(16,20,95,36),(49,82,58,66),(50,87,59,71),(51,84,60,68),(52,81,61,65),(53,86,62,70),(54,83,63,67),(55,88,64,72),(56,85,57,69)], [(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)], [(2,76),(3,7),(4,74),(6,80),(8,78),(9,92),(10,16),(11,90),(12,14),(13,96),(15,94),(18,36),(19,23),(20,34),(22,40),(24,38),(25,31),(26,42),(27,29),(28,48),(30,46),(32,44),(35,39),(41,47),(43,45),(49,55),(50,59),(51,53),(52,57),(54,63),(56,61),(58,64),(60,62),(66,84),(67,71),(68,82),(70,88),(72,86),(75,79),(83,87),(89,95),(91,93)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | ··· | 6F | 6G | 6H | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | ··· | 12N | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 8 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C3×C4○D4 | C8⋊C22 | C8.C22 | C3×C8⋊C22 | C3×C8.C22 |
| kernel | C3×C42.28C22 | C3×C8⋊C4 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C4.4D4 | C3×C4⋊Q8 | C42.28C22 | C8⋊C4 | D4⋊C4 | Q8⋊C4 | C4.4D4 | C4⋊Q8 | C2×C12 | C12 | C2×C4 | C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C42.28C22 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 61 | 48 | 0 | 0 | 0 | 0 |
| 35 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 67 | 0 | 0 | 1 |
| 0 | 0 | 0 | 67 | 36 | 72 |
| 0 | 0 | 1 | 1 | 6 | 0 |
| 0 | 0 | 36 | 0 | 0 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 57 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 41 | 55 | 0 | 0 | 0 | 0 |
| 69 | 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 67 | 61 |
| 0 | 0 | 37 | 1 | 6 | 6 |
| 0 | 0 | 67 | 0 | 0 | 1 |
| 0 | 0 | 0 | 67 | 36 | 72 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[61,35,0,0,0,0,48,12,0,0,0,0,0,0,67,0,1,36,0,0,0,67,1,0,0,0,0,36,6,0,0,0,1,72,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,71,1,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[1,0,0,0,0,0,57,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,72],[41,69,0,0,0,0,55,32,0,0,0,0,0,0,0,37,67,0,0,0,72,1,0,67,0,0,67,6,0,36,0,0,61,6,1,72] >;
C3×C42.28C22 in GAP, Magma, Sage, TeX
C_3\times C_4^2._{28}C_2^2 % in TeX
G:=Group("C3xC4^2.28C2^2"); // GroupNames label
G:=SmallGroup(192,922);
// by ID
G=gap.SmallGroup(192,922);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,344,1094,1059,142,4204,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^4=d^2=1,e^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=b^-1*c^2,e*b*e^-1=b*c^2,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;
// generators/relations