direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C22.C42, M4(2)⋊3C12, C12.53(C4⋊C4), (C2×C6).9C42, (C2×C12).37Q8, (C2×C12).509D4, (C3×M4(2))⋊9C4, C22.2(C4×C12), (C22×C4).4C12, (C22×C12).6C4, C23.28(C2×C12), (C2×M4(2)).8C6, C6.15(C4.D4), (C6×M4(2)).20C2, C12.105(C22⋊C4), C6.13(C4.10D4), C6.27(C2.C42), (C22×C12).388C22, C4.4(C3×C4⋊C4), (C2×C4⋊C4).3C6, (C6×C4⋊C4).30C2, (C2×C4).2(C3×Q8), C22.5(C3×C4⋊C4), (C2×C6).22(C4⋊C4), (C2×C4).14(C2×C12), (C2×C4).114(C3×D4), C4.20(C3×C22⋊C4), C2.2(C3×C4.D4), (C2×C12).181(C2×C4), (C22×C4).23(C2×C6), C2.2(C3×C4.10D4), (C22×C6).108(C2×C4), C22.30(C3×C22⋊C4), C2.8(C3×C2.C42), (C2×C6).132(C22⋊C4), SmallGroup(192,149)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22.C42
G = < a,b,c,d,e | a3=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >
Subgroups: 154 in 98 conjugacy classes, 58 normal (26 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C2×M4(2), C3×C4⋊C4, C2×C24, C3×M4(2), C3×M4(2), C22×C12, C22×C12, C22.C42, C6×C4⋊C4, C6×M4(2), C3×C22.C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4.D4, C4.10D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22.C42, C3×C2.C42, C3×C4.D4, C3×C4.10D4, C3×C22.C42
►On 96 points
Generators in S96
(1 64 67)(2 57 68)(3 58 69)(4 59 70)(5 60 71)(6 61 72)(7 62 65)(8 63 66)(9 17 31)(10 18 32)(11 19 25)(12 20 26)(13 21 27)(14 22 28)(15 23 29)(16 24 30)(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 55 89)(42 56 90)(43 49 91)(44 50 92)(45 51 93)(46 52 94)(47 53 95)(48 54 96)
(1 79)(2 76)(3 73)(4 78)(5 75)(6 80)(7 77)(8 74)(9 47)(10 44)(11 41)(12 46)(13 43)(14 48)(15 45)(16 42)(17 53)(18 50)(19 55)(20 52)(21 49)(22 54)(23 51)(24 56)(25 89)(26 94)(27 91)(28 96)(29 93)(30 90)(31 95)(32 92)(33 69)(34 66)(35 71)(36 68)(37 65)(38 70)(39 67)(40 72)(57 84)(58 81)(59 86)(60 83)(61 88)(62 85)(63 82)(64 87)
(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 43 79 13)(2 10 80 48)(3 41 73 11)(4 16 74 46)(5 47 75 9)(6 14 76 44)(7 45 77 15)(8 12 78 42)(17 60 53 83)(18 88 54 57)(19 58 55 81)(20 86 56 63)(21 64 49 87)(22 84 50 61)(23 62 51 85)(24 82 52 59)(25 69 89 33)(26 38 90 66)(27 67 91 39)(28 36 92 72)(29 65 93 37)(30 34 94 70)(31 71 95 35)(32 40 96 68)
G:=sub<Sym(96)| (1,64,67)(2,57,68)(3,58,69)(4,59,70)(5,60,71)(6,61,72)(7,62,65)(8,63,66)(9,17,31)(10,18,32)(11,19,25)(12,20,26)(13,21,27)(14,22,28)(15,23,29)(16,24,30)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (1,79)(2,76)(3,73)(4,78)(5,75)(6,80)(7,77)(8,74)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(17,53)(18,50)(19,55)(20,52)(21,49)(22,54)(23,51)(24,56)(25,89)(26,94)(27,91)(28,96)(29,93)(30,90)(31,95)(32,92)(33,69)(34,66)(35,71)(36,68)(37,65)(38,70)(39,67)(40,72)(57,84)(58,81)(59,86)(60,83)(61,88)(62,85)(63,82)(64,87), (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), (1,43,79,13)(2,10,80,48)(3,41,73,11)(4,16,74,46)(5,47,75,9)(6,14,76,44)(7,45,77,15)(8,12,78,42)(17,60,53,83)(18,88,54,57)(19,58,55,81)(20,86,56,63)(21,64,49,87)(22,84,50,61)(23,62,51,85)(24,82,52,59)(25,69,89,33)(26,38,90,66)(27,67,91,39)(28,36,92,72)(29,65,93,37)(30,34,94,70)(31,71,95,35)(32,40,96,68)>;
G:=Group( (1,64,67)(2,57,68)(3,58,69)(4,59,70)(5,60,71)(6,61,72)(7,62,65)(8,63,66)(9,17,31)(10,18,32)(11,19,25)(12,20,26)(13,21,27)(14,22,28)(15,23,29)(16,24,30)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,55,89)(42,56,90)(43,49,91)(44,50,92)(45,51,93)(46,52,94)(47,53,95)(48,54,96), (1,79)(2,76)(3,73)(4,78)(5,75)(6,80)(7,77)(8,74)(9,47)(10,44)(11,41)(12,46)(13,43)(14,48)(15,45)(16,42)(17,53)(18,50)(19,55)(20,52)(21,49)(22,54)(23,51)(24,56)(25,89)(26,94)(27,91)(28,96)(29,93)(30,90)(31,95)(32,92)(33,69)(34,66)(35,71)(36,68)(37,65)(38,70)(39,67)(40,72)(57,84)(58,81)(59,86)(60,83)(61,88)(62,85)(63,82)(64,87), (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), (1,43,79,13)(2,10,80,48)(3,41,73,11)(4,16,74,46)(5,47,75,9)(6,14,76,44)(7,45,77,15)(8,12,78,42)(17,60,53,83)(18,88,54,57)(19,58,55,81)(20,86,56,63)(21,64,49,87)(22,84,50,61)(23,62,51,85)(24,82,52,59)(25,69,89,33)(26,38,90,66)(27,67,91,39)(28,36,92,72)(29,65,93,37)(30,34,94,70)(31,71,95,35)(32,40,96,68) );
G=PermutationGroup([[(1,64,67),(2,57,68),(3,58,69),(4,59,70),(5,60,71),(6,61,72),(7,62,65),(8,63,66),(9,17,31),(10,18,32),(11,19,25),(12,20,26),(13,21,27),(14,22,28),(15,23,29),(16,24,30),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,55,89),(42,56,90),(43,49,91),(44,50,92),(45,51,93),(46,52,94),(47,53,95),(48,54,96)], [(1,79),(2,76),(3,73),(4,78),(5,75),(6,80),(7,77),(8,74),(9,47),(10,44),(11,41),(12,46),(13,43),(14,48),(15,45),(16,42),(17,53),(18,50),(19,55),(20,52),(21,49),(22,54),(23,51),(24,56),(25,89),(26,94),(27,91),(28,96),(29,93),(30,90),(31,95),(32,92),(33,69),(34,66),(35,71),(36,68),(37,65),(38,70),(39,67),(40,72),(57,84),(58,81),(59,86),(60,83),(61,88),(62,85),(63,82),(64,87)], [(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)], [(1,43,79,13),(2,10,80,48),(3,41,73,11),(4,16,74,46),(5,47,75,9),(6,14,76,44),(7,45,77,15),(8,12,78,42),(17,60,53,83),(18,88,54,57),(19,58,55,81),(20,86,56,63),(21,64,49,87),(22,84,50,61),(23,62,51,85),(24,82,52,59),(25,69,89,33),(26,38,90,66),(27,67,91,39),(28,36,92,72),(29,65,93,37),(30,34,94,70),(31,71,95,35),(32,40,96,68)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12P | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | |||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C4.D4 | C4.10D4 | C3×C4.D4 | C3×C4.10D4 |
| kernel | C3×C22.C42 | C6×C4⋊C4 | C6×M4(2) | C22.C42 | C3×M4(2) | C22×C12 | C2×C4⋊C4 | C2×M4(2) | M4(2) | C22×C4 | C2×C12 | C2×C12 | C2×C4 | C2×C4 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 8 | 4 | 2 | 4 | 16 | 8 | 3 | 1 | 6 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C3×C22.C42 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0],[46,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,27] >;
C3×C22.C42 in GAP, Magma, Sage, TeX
C_3\times C_2^2.C_4^2
% in TeX
G:=Group("C3xC2^2.C4^2"); // GroupNames label
G:=SmallGroup(192,149);
// by ID
G=gap.SmallGroup(192,149);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,3027,2111,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^2=e^4=1,d^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
// generators/relations