direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C22.D8, C2.D8⋊6C6, C2.8(C6×D8), C22⋊C8⋊4C6, (C2×C6).26D8, C6.80(C2×D8), D4⋊C4⋊7C6, C4⋊D4.5C6, C22.4(C3×D8), (C2×C12).336D4, C23.50(C3×D4), C22.101(C6×D4), (C22×C6).167D4, C12.317(C4○D4), (C2×C24).190C22, (C2×C12).936C23, (C6×D4).195C22, C6.142(C8.C22), (C22×C12).428C22, C6.95(C22.D4), (C2×C4⋊C4)⋊11C6, (C6×C4⋊C4)⋊38C2, (C2×C8).9(C2×C6), C4⋊C4.57(C2×C6), (C3×C2.D8)⋊21C2, C4.29(C3×C4○D4), (C2×C4).37(C3×D4), (C3×C22⋊C8)⋊14C2, (C2×D4).18(C2×C6), (C2×C6).657(C2×D4), (C3×D4⋊C4)⋊18C2, (C3×C4⋊D4).15C2, (C22×C4).51(C2×C6), C2.17(C3×C8.C22), (C3×C4⋊C4).380C22, (C2×C4).111(C22×C6), C2.11(C3×C22.D4), SmallGroup(192,913)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.D8
G = < a,b,c,d,e | a3=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 226 in 114 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22×C12, C6×D4, C6×D4, C22.D8, C3×C22⋊C8, C3×D4⋊C4, C3×C2.D8, C6×C4⋊C4, C3×C4⋊D4, C3×C22.D8
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, D8, C2×D4, C4○D4, C3×D4, C22×C6, C22.D4, C2×D8, C8.C22, C3×D8, C6×D4, C3×C4○D4, C22.D8, C3×C22.D4, C6×D8, C3×C8.C22, C3×C22.D8
►On 96 points
Generators in S96
(1 34 16)(2 35 9)(3 36 10)(4 37 11)(5 38 12)(6 39 13)(7 40 14)(8 33 15)(17 53 43)(18 54 44)(19 55 45)(20 56 46)(21 49 47)(22 50 48)(23 51 41)(24 52 42)(25 92 70)(26 93 71)(27 94 72)(28 95 65)(29 96 66)(30 89 67)(31 90 68)(32 91 69)(57 80 87)(58 73 88)(59 74 81)(60 75 82)(61 76 83)(62 77 84)(63 78 85)(64 79 86)
(1 67)(2 62)(3 69)(4 64)(5 71)(6 58)(7 65)(8 60)(9 84)(10 91)(11 86)(12 93)(13 88)(14 95)(15 82)(16 89)(17 59)(18 66)(19 61)(20 68)(21 63)(22 70)(23 57)(24 72)(25 50)(26 38)(27 52)(28 40)(29 54)(30 34)(31 56)(32 36)(33 75)(35 77)(37 79)(39 73)(41 87)(42 94)(43 81)(44 96)(45 83)(46 90)(47 85)(48 92)(49 78)(51 80)(53 74)(55 76)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(25 79)(26 80)(27 73)(28 74)(29 75)(30 76)(31 77)(32 78)(33 54)(34 55)(35 56)(36 49)(37 50)(38 51)(39 52)(40 53)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)(81 95)(82 96)(83 89)(84 90)(85 91)(86 92)(87 93)(88 94)
(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 18)(3 7)(4 24)(6 22)(8 20)(9 44)(10 14)(11 42)(13 48)(15 46)(17 21)(25 27)(26 80)(28 78)(29 31)(30 76)(32 74)(33 56)(35 54)(36 40)(37 52)(39 50)(43 47)(49 53)(57 71)(58 64)(59 69)(60 62)(61 67)(63 65)(66 68)(70 72)(73 79)(75 77)(81 91)(82 84)(83 89)(85 95)(86 88)(87 93)(90 96)(92 94)
G:=sub<Sym(96)| (1,34,16)(2,35,9)(3,36,10)(4,37,11)(5,38,12)(6,39,13)(7,40,14)(8,33,15)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,92,70)(26,93,71)(27,94,72)(28,95,65)(29,96,66)(30,89,67)(31,90,68)(32,91,69)(57,80,87)(58,73,88)(59,74,81)(60,75,82)(61,76,83)(62,77,84)(63,78,85)(64,79,86), (1,67)(2,62)(3,69)(4,64)(5,71)(6,58)(7,65)(8,60)(9,84)(10,91)(11,86)(12,93)(13,88)(14,95)(15,82)(16,89)(17,59)(18,66)(19,61)(20,68)(21,63)(22,70)(23,57)(24,72)(25,50)(26,38)(27,52)(28,40)(29,54)(30,34)(31,56)(32,36)(33,75)(35,77)(37,79)(39,73)(41,87)(42,94)(43,81)(44,96)(45,83)(46,90)(47,85)(48,92)(49,78)(51,80)(53,74)(55,76), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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), (2,18)(3,7)(4,24)(6,22)(8,20)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,27)(26,80)(28,78)(29,31)(30,76)(32,74)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,71)(58,64)(59,69)(60,62)(61,67)(63,65)(66,68)(70,72)(73,79)(75,77)(81,91)(82,84)(83,89)(85,95)(86,88)(87,93)(90,96)(92,94)>;
G:=Group( (1,34,16)(2,35,9)(3,36,10)(4,37,11)(5,38,12)(6,39,13)(7,40,14)(8,33,15)(17,53,43)(18,54,44)(19,55,45)(20,56,46)(21,49,47)(22,50,48)(23,51,41)(24,52,42)(25,92,70)(26,93,71)(27,94,72)(28,95,65)(29,96,66)(30,89,67)(31,90,68)(32,91,69)(57,80,87)(58,73,88)(59,74,81)(60,75,82)(61,76,83)(62,77,84)(63,78,85)(64,79,86), (1,67)(2,62)(3,69)(4,64)(5,71)(6,58)(7,65)(8,60)(9,84)(10,91)(11,86)(12,93)(13,88)(14,95)(15,82)(16,89)(17,59)(18,66)(19,61)(20,68)(21,63)(22,70)(23,57)(24,72)(25,50)(26,38)(27,52)(28,40)(29,54)(30,34)(31,56)(32,36)(33,75)(35,77)(37,79)(39,73)(41,87)(42,94)(43,81)(44,96)(45,83)(46,90)(47,85)(48,92)(49,78)(51,80)(53,74)(55,76), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,46)(10,47)(11,48)(12,41)(13,42)(14,43)(15,44)(16,45)(25,79)(26,80)(27,73)(28,74)(29,75)(30,76)(31,77)(32,78)(33,54)(34,55)(35,56)(36,49)(37,50)(38,51)(39,52)(40,53)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70)(81,95)(82,96)(83,89)(84,90)(85,91)(86,92)(87,93)(88,94), (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), (2,18)(3,7)(4,24)(6,22)(8,20)(9,44)(10,14)(11,42)(13,48)(15,46)(17,21)(25,27)(26,80)(28,78)(29,31)(30,76)(32,74)(33,56)(35,54)(36,40)(37,52)(39,50)(43,47)(49,53)(57,71)(58,64)(59,69)(60,62)(61,67)(63,65)(66,68)(70,72)(73,79)(75,77)(81,91)(82,84)(83,89)(85,95)(86,88)(87,93)(90,96)(92,94) );
G=PermutationGroup([[(1,34,16),(2,35,9),(3,36,10),(4,37,11),(5,38,12),(6,39,13),(7,40,14),(8,33,15),(17,53,43),(18,54,44),(19,55,45),(20,56,46),(21,49,47),(22,50,48),(23,51,41),(24,52,42),(25,92,70),(26,93,71),(27,94,72),(28,95,65),(29,96,66),(30,89,67),(31,90,68),(32,91,69),(57,80,87),(58,73,88),(59,74,81),(60,75,82),(61,76,83),(62,77,84),(63,78,85),(64,79,86)], [(1,67),(2,62),(3,69),(4,64),(5,71),(6,58),(7,65),(8,60),(9,84),(10,91),(11,86),(12,93),(13,88),(14,95),(15,82),(16,89),(17,59),(18,66),(19,61),(20,68),(21,63),(22,70),(23,57),(24,72),(25,50),(26,38),(27,52),(28,40),(29,54),(30,34),(31,56),(32,36),(33,75),(35,77),(37,79),(39,73),(41,87),(42,94),(43,81),(44,96),(45,83),(46,90),(47,85),(48,92),(49,78),(51,80),(53,74),(55,76)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,46),(10,47),(11,48),(12,41),(13,42),(14,43),(15,44),(16,45),(25,79),(26,80),(27,73),(28,74),(29,75),(30,76),(31,77),(32,78),(33,54),(34,55),(35,56),(36,49),(37,50),(38,51),(39,52),(40,53),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70),(81,95),(82,96),(83,89),(84,90),(85,91),(86,92),(87,93),(88,94)], [(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)], [(2,18),(3,7),(4,24),(6,22),(8,20),(9,44),(10,14),(11,42),(13,48),(15,46),(17,21),(25,27),(26,80),(28,78),(29,31),(30,76),(32,74),(33,56),(35,54),(36,40),(37,52),(39,50),(43,47),(49,53),(57,71),(58,64),(59,69),(60,62),(61,67),(63,65),(66,68),(70,72),(73,79),(75,77),(81,91),(82,84),(83,89),(85,95),(86,88),(87,93),(90,96),(92,94)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | ||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | C4○D4 | D8 | C3×D4 | C3×D4 | C3×C4○D4 | C3×D8 | C8.C22 | C3×C8.C22 |
| kernel | C3×C22.D8 | C3×C22⋊C8 | C3×D4⋊C4 | C3×C2.D8 | C6×C4⋊C4 | C3×C4⋊D4 | C22.D8 | C22⋊C8 | D4⋊C4 | C2.D8 | C2×C4⋊C4 | C4⋊D4 | C2×C12 | C22×C6 | C12 | C2×C6 | C2×C4 | C23 | C4 | C22 | C6 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 1 | 2 |
Matrix representation of C3×C22.D8 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 57 | 32 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 72 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[46,0,0,0,0,27,0,0,0,0,0,57,0,0,32,32],[1,0,0,0,0,72,0,0,0,0,1,1,0,0,0,72] >;
C3×C22.D8 in GAP, Magma, Sage, TeX
C_3\times C_2^2.D_8
% in TeX
G:=Group("C3xC2^2.D8"); // GroupNames label
G:=SmallGroup(192,913);
// by ID
G=gap.SmallGroup(192,913);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,142,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations