direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C23.48D4, C2.D8⋊8C6, C2.8(C6×Q16), Q8⋊C4⋊7C6, C22⋊C8.4C6, C6.55(C2×Q16), (C2×C6).12Q16, C22⋊Q8.6C6, (C2×C12).339D4, C23.53(C3×D4), C22.3(C3×Q16), (C22×C6).170D4, C22.105(C6×D4), C12.321(C4○D4), C6.144(C8⋊C22), (C2×C24).191C22, (C2×C12).940C23, (C6×Q8).170C22, (C22×C12).432C22, C6.99(C22.D4), (C2×C4⋊C4).17C6, (C6×C4⋊C4).46C2, C4⋊C4.61(C2×C6), (C2×C8).10(C2×C6), (C3×C2.D8)⋊23C2, C4.33(C3×C4○D4), (C2×C4).40(C3×D4), (C2×C6).661(C2×D4), C2.19(C3×C8⋊C22), (C2×Q8).15(C2×C6), (C3×Q8⋊C4)⋊18C2, (C3×C22⋊C8).10C2, (C22×C4).55(C2×C6), (C3×C22⋊Q8).16C2, (C3×C4⋊C4).384C22, (C2×C4).115(C22×C6), C2.15(C3×C22.D4), SmallGroup(192,917)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C23.48D4
G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e4=f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=cde3 >
Subgroups: 178 in 104 conjugacy classes, 54 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C24, C2×C12, C2×C12, C3×Q8, C22×C6, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C22×C12, C22×C12, C6×Q8, C23.48D4, C3×C22⋊C8, C3×Q8⋊C4, C3×C2.D8, C6×C4⋊C4, C3×C22⋊Q8, C3×C23.48D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, Q16, C2×D4, C4○D4, C3×D4, C22×C6, C22.D4, C2×Q16, C8⋊C22, C3×Q16, C6×D4, C3×C4○D4, C23.48D4, C3×C22.D4, C6×Q16, C3×C8⋊C22, C3×C23.48D4
►On 96 points
Generators in S96
(1 74 91)(2 75 92)(3 76 93)(4 77 94)(5 78 95)(6 79 96)(7 80 89)(8 73 90)(9 65 81)(10 66 82)(11 67 83)(12 68 84)(13 69 85)(14 70 86)(15 71 87)(16 72 88)(17 55 29)(18 56 30)(19 49 31)(20 50 32)(21 51 25)(22 52 26)(23 53 27)(24 54 28)(33 41 62)(34 42 63)(35 43 64)(36 44 57)(37 45 58)(38 46 59)(39 47 60)(40 48 61)
(1 5)(2 52)(3 7)(4 54)(6 56)(8 50)(9 13)(10 57)(11 15)(12 59)(14 61)(16 63)(17 21)(18 96)(19 23)(20 90)(22 92)(24 94)(25 29)(26 75)(27 31)(28 77)(30 79)(32 73)(33 37)(34 72)(35 39)(36 66)(38 68)(40 70)(41 45)(42 88)(43 47)(44 82)(46 84)(48 86)(49 53)(51 55)(58 62)(60 64)(65 69)(67 71)(74 78)(76 80)(81 85)(83 87)(89 93)(91 95)
(1 55)(2 56)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 91)(18 92)(19 93)(20 94)(21 95)(22 96)(23 89)(24 90)(25 78)(26 79)(27 80)(28 73)(29 74)(30 75)(31 76)(32 77)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 65)(40 66)(41 83)(42 84)(43 85)(44 86)(45 87)(46 88)(47 81)(48 82)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 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)
(1 66 5 70)(2 39 6 35)(3 72 7 68)(4 37 8 33)(9 22 13 18)(10 95 14 91)(11 20 15 24)(12 93 16 89)(17 61 21 57)(19 59 23 63)(25 44 29 48)(26 85 30 81)(27 42 31 46)(28 83 32 87)(34 49 38 53)(36 55 40 51)(41 77 45 73)(43 75 47 79)(50 71 54 67)(52 69 56 65)(58 90 62 94)(60 96 64 92)(74 82 78 86)(76 88 80 84)
G:=sub<Sym(96)| (1,74,91)(2,75,92)(3,76,93)(4,77,94)(5,78,95)(6,79,96)(7,80,89)(8,73,90)(9,65,81)(10,66,82)(11,67,83)(12,68,84)(13,69,85)(14,70,86)(15,71,87)(16,72,88)(17,55,29)(18,56,30)(19,49,31)(20,50,32)(21,51,25)(22,52,26)(23,53,27)(24,54,28)(33,41,62)(34,42,63)(35,43,64)(36,44,57)(37,45,58)(38,46,59)(39,47,60)(40,48,61), (1,5)(2,52)(3,7)(4,54)(6,56)(8,50)(9,13)(10,57)(11,15)(12,59)(14,61)(16,63)(17,21)(18,96)(19,23)(20,90)(22,92)(24,94)(25,29)(26,75)(27,31)(28,77)(30,79)(32,73)(33,37)(34,72)(35,39)(36,66)(38,68)(40,70)(41,45)(42,88)(43,47)(44,82)(46,84)(48,86)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,89)(24,90)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,66,5,70)(2,39,6,35)(3,72,7,68)(4,37,8,33)(9,22,13,18)(10,95,14,91)(11,20,15,24)(12,93,16,89)(17,61,21,57)(19,59,23,63)(25,44,29,48)(26,85,30,81)(27,42,31,46)(28,83,32,87)(34,49,38,53)(36,55,40,51)(41,77,45,73)(43,75,47,79)(50,71,54,67)(52,69,56,65)(58,90,62,94)(60,96,64,92)(74,82,78,86)(76,88,80,84)>;
G:=Group( (1,74,91)(2,75,92)(3,76,93)(4,77,94)(5,78,95)(6,79,96)(7,80,89)(8,73,90)(9,65,81)(10,66,82)(11,67,83)(12,68,84)(13,69,85)(14,70,86)(15,71,87)(16,72,88)(17,55,29)(18,56,30)(19,49,31)(20,50,32)(21,51,25)(22,52,26)(23,53,27)(24,54,28)(33,41,62)(34,42,63)(35,43,64)(36,44,57)(37,45,58)(38,46,59)(39,47,60)(40,48,61), (1,5)(2,52)(3,7)(4,54)(6,56)(8,50)(9,13)(10,57)(11,15)(12,59)(14,61)(16,63)(17,21)(18,96)(19,23)(20,90)(22,92)(24,94)(25,29)(26,75)(27,31)(28,77)(30,79)(32,73)(33,37)(34,72)(35,39)(36,66)(38,68)(40,70)(41,45)(42,88)(43,47)(44,82)(46,84)(48,86)(49,53)(51,55)(58,62)(60,64)(65,69)(67,71)(74,78)(76,80)(81,85)(83,87)(89,93)(91,95), (1,55)(2,56)(3,49)(4,50)(5,51)(6,52)(7,53)(8,54)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,91)(18,92)(19,93)(20,94)(21,95)(22,96)(23,89)(24,90)(25,78)(26,79)(27,80)(28,73)(29,74)(30,75)(31,76)(32,77)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,65)(40,66)(41,83)(42,84)(43,85)(44,86)(45,87)(46,88)(47,81)(48,82), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96), (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), (1,66,5,70)(2,39,6,35)(3,72,7,68)(4,37,8,33)(9,22,13,18)(10,95,14,91)(11,20,15,24)(12,93,16,89)(17,61,21,57)(19,59,23,63)(25,44,29,48)(26,85,30,81)(27,42,31,46)(28,83,32,87)(34,49,38,53)(36,55,40,51)(41,77,45,73)(43,75,47,79)(50,71,54,67)(52,69,56,65)(58,90,62,94)(60,96,64,92)(74,82,78,86)(76,88,80,84) );
G=PermutationGroup([[(1,74,91),(2,75,92),(3,76,93),(4,77,94),(5,78,95),(6,79,96),(7,80,89),(8,73,90),(9,65,81),(10,66,82),(11,67,83),(12,68,84),(13,69,85),(14,70,86),(15,71,87),(16,72,88),(17,55,29),(18,56,30),(19,49,31),(20,50,32),(21,51,25),(22,52,26),(23,53,27),(24,54,28),(33,41,62),(34,42,63),(35,43,64),(36,44,57),(37,45,58),(38,46,59),(39,47,60),(40,48,61)], [(1,5),(2,52),(3,7),(4,54),(6,56),(8,50),(9,13),(10,57),(11,15),(12,59),(14,61),(16,63),(17,21),(18,96),(19,23),(20,90),(22,92),(24,94),(25,29),(26,75),(27,31),(28,77),(30,79),(32,73),(33,37),(34,72),(35,39),(36,66),(38,68),(40,70),(41,45),(42,88),(43,47),(44,82),(46,84),(48,86),(49,53),(51,55),(58,62),(60,64),(65,69),(67,71),(74,78),(76,80),(81,85),(83,87),(89,93),(91,95)], [(1,55),(2,56),(3,49),(4,50),(5,51),(6,52),(7,53),(8,54),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,91),(18,92),(19,93),(20,94),(21,95),(22,96),(23,89),(24,90),(25,78),(26,79),(27,80),(28,73),(29,74),(30,75),(31,76),(32,77),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,65),(40,66),(41,83),(42,84),(43,85),(44,86),(45,87),(46,88),(47,81),(48,82)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96)], [(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)], [(1,66,5,70),(2,39,6,35),(3,72,7,68),(4,37,8,33),(9,22,13,18),(10,95,14,91),(11,20,15,24),(12,93,16,89),(17,61,21,57),(19,59,23,63),(25,44,29,48),(26,85,30,81),(27,42,31,46),(28,83,32,87),(34,49,38,53),(36,55,40,51),(41,77,45,73),(43,75,47,79),(50,71,54,67),(52,69,56,65),(58,90,62,94),(60,96,64,92),(74,82,78,86),(76,88,80,84)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N | 12O | 12P | 12Q | 12R | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 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 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 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 | Q16 | C3×D4 | C3×D4 | C3×C4○D4 | C3×Q16 | C8⋊C22 | C3×C8⋊C22 |
| kernel | C3×C23.48D4 | C3×C22⋊C8 | C3×Q8⋊C4 | C3×C2.D8 | C6×C4⋊C4 | C3×C22⋊Q8 | C23.48D4 | C22⋊C8 | Q8⋊C4 | C2.D8 | C2×C4⋊C4 | C22⋊Q8 | 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×C23.48D4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 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 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 27 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 36 | 32 |
| 0 | 0 | 53 | 37 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[0,46,0,0,27,0,0,0,0,0,32,57,0,0,32,0],[0,1,0,0,1,0,0,0,0,0,36,53,0,0,32,37] >;
C3×C23.48D4 in GAP, Magma, Sage, TeX
C_3\times C_2^3._{48}D_4 % in TeX
G:=Group("C3xC2^3.48D4"); // GroupNames label
G:=SmallGroup(192,917);
// by ID
G=gap.SmallGroup(192,917);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,672,365,1094,646,6053,1531,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=1,e^4=f^2=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,e*b*e^-1=f*b*f^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;
// generators/relations