direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×M4(2)⋊C4, M4(2)⋊1C12, C8⋊1(C2×C12), C4.Q8⋊2C6, C24⋊16(C2×C4), C4.4(C6×Q8), C2.D8⋊10C6, C12.64(C4⋊C4), (C2×C12).43Q8, C12.93(C2×Q8), (C2×C12).522D4, (C3×M4(2))⋊2C4, C22.50(C6×D4), C23.45(C3×D4), C42⋊C2.7C6, C4.27(C22×C12), (C22×C6).162D4, (C2×M4(2)).1C6, (C6×M4(2)).4C2, C6.128(C8⋊C22), C12.185(C22×C4), (C2×C24).197C22, (C2×C12).901C23, C6.128(C8.C22), (C22×C12).414C22, C2.14(C6×C4⋊C4), C4.15(C3×C4⋊C4), C6.70(C2×C4⋊C4), (C2×C4⋊C4).15C6, (C6×C4⋊C4).44C2, (C2×C4).6(C3×Q8), C4⋊C4.44(C2×C6), (C2×C8).16(C2×C6), (C3×C4.Q8)⋊11C2, (C3×C2.D8)⋊25C2, C2.3(C3×C8⋊C22), (C2×C6).27(C4⋊C4), (C2×C4).25(C2×C12), (C2×C6).626(C2×D4), (C2×C4).125(C3×D4), C22.10(C3×C4⋊C4), C2.3(C3×C8.C22), (C2×C12).198(C2×C4), (C2×C4).76(C22×C6), (C22×C4).38(C2×C6), (C3×C4⋊C4).365C22, (C3×C42⋊C2).21C2, SmallGroup(192,861)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×M4(2)⋊C4
G = < a,b,c,d | a3=b8=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b5, dbd-1=b-1, cd=dc >
Subgroups: 178 in 118 conjugacy classes, 82 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, C23, C12, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C24, C2×C12, C2×C12, C2×C12, C22×C6, C4.Q8, C2.D8, C2×C4⋊C4, C42⋊C2, C2×M4(2), C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C3×M4(2), C22×C12, C22×C12, M4(2)⋊C4, C3×C4.Q8, C3×C2.D8, C6×C4⋊C4, C3×C42⋊C2, C6×M4(2), C3×M4(2)⋊C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C23, C12, C2×C6, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2×C12, C3×D4, C3×Q8, C22×C6, C2×C4⋊C4, C8⋊C22, C8.C22, C3×C4⋊C4, C22×C12, C6×D4, C6×Q8, M4(2)⋊C4, C6×C4⋊C4, C3×C8⋊C22, C3×C8.C22, C3×M4(2)⋊C4
►On 96 points
(1 53 43)(2 54 44)(3 55 45)(4 56 46)(5 49 47)(6 50 48)(7 51 41)(8 52 42)(9 40 22)(10 33 23)(11 34 24)(12 35 17)(13 36 18)(14 37 19)(15 38 20)(16 39 21)(25 90 68)(26 91 69)(27 92 70)(28 93 71)(29 94 72)(30 95 65)(31 96 66)(32 89 67)(57 79 87)(58 80 88)(59 73 81)(60 74 82)(61 75 83)(62 76 84)(63 77 85)(64 78 86)
(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 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)(81 85)(83 87)(90 94)(92 96)
(1 78 39 71)(2 77 40 70)(3 76 33 69)(4 75 34 68)(5 74 35 67)(6 73 36 66)(7 80 37 65)(8 79 38 72)(9 92 44 63)(10 91 45 62)(11 90 46 61)(12 89 47 60)(13 96 48 59)(14 95 41 58)(15 94 42 57)(16 93 43 64)(17 32 49 82)(18 31 50 81)(19 30 51 88)(20 29 52 87)(21 28 53 86)(22 27 54 85)(23 26 55 84)(24 25 56 83)
G:=sub<Sym(96)| (1,53,43)(2,54,44)(3,55,45)(4,56,46)(5,49,47)(6,50,48)(7,51,41)(8,52,42)(9,40,22)(10,33,23)(11,34,24)(12,35,17)(13,36,18)(14,37,19)(15,38,20)(16,39,21)(25,90,68)(26,91,69)(27,92,70)(28,93,71)(29,94,72)(30,95,65)(31,96,66)(32,89,67)(57,79,87)(58,80,88)(59,73,81)(60,74,82)(61,75,83)(62,76,84)(63,77,85)(64,78,86), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79)(81,85)(83,87)(90,94)(92,96), (1,78,39,71)(2,77,40,70)(3,76,33,69)(4,75,34,68)(5,74,35,67)(6,73,36,66)(7,80,37,65)(8,79,38,72)(9,92,44,63)(10,91,45,62)(11,90,46,61)(12,89,47,60)(13,96,48,59)(14,95,41,58)(15,94,42,57)(16,93,43,64)(17,32,49,82)(18,31,50,81)(19,30,51,88)(20,29,52,87)(21,28,53,86)(22,27,54,85)(23,26,55,84)(24,25,56,83)>;
G:=Group( (1,53,43)(2,54,44)(3,55,45)(4,56,46)(5,49,47)(6,50,48)(7,51,41)(8,52,42)(9,40,22)(10,33,23)(11,34,24)(12,35,17)(13,36,18)(14,37,19)(15,38,20)(16,39,21)(25,90,68)(26,91,69)(27,92,70)(28,93,71)(29,94,72)(30,95,65)(31,96,66)(32,89,67)(57,79,87)(58,80,88)(59,73,81)(60,74,82)(61,75,83)(62,76,84)(63,77,85)(64,78,86), (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), (2,6)(4,8)(9,13)(11,15)(18,22)(20,24)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(73,77)(75,79)(81,85)(83,87)(90,94)(92,96), (1,78,39,71)(2,77,40,70)(3,76,33,69)(4,75,34,68)(5,74,35,67)(6,73,36,66)(7,80,37,65)(8,79,38,72)(9,92,44,63)(10,91,45,62)(11,90,46,61)(12,89,47,60)(13,96,48,59)(14,95,41,58)(15,94,42,57)(16,93,43,64)(17,32,49,82)(18,31,50,81)(19,30,51,88)(20,29,52,87)(21,28,53,86)(22,27,54,85)(23,26,55,84)(24,25,56,83) );
G=PermutationGroup([[(1,53,43),(2,54,44),(3,55,45),(4,56,46),(5,49,47),(6,50,48),(7,51,41),(8,52,42),(9,40,22),(10,33,23),(11,34,24),(12,35,17),(13,36,18),(14,37,19),(15,38,20),(16,39,21),(25,90,68),(26,91,69),(27,92,70),(28,93,71),(29,94,72),(30,95,65),(31,96,66),(32,89,67),(57,79,87),(58,80,88),(59,73,81),(60,74,82),(61,75,83),(62,76,84),(63,77,85),(64,78,86)], [(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)], [(2,6),(4,8),(9,13),(11,15),(18,22),(20,24),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(73,77),(75,79),(81,85),(83,87),(90,94),(92,96)], [(1,78,39,71),(2,77,40,70),(3,76,33,69),(4,75,34,68),(5,74,35,67),(6,73,36,66),(7,80,37,65),(8,79,38,72),(9,92,44,63),(10,91,45,62),(11,90,46,61),(12,89,47,60),(13,96,48,59),(14,95,41,58),(15,94,42,57),(16,93,43,64),(17,32,49,82),(18,31,50,81),(19,30,51,88),(20,29,52,87),(21,28,53,86),(22,27,54,85),(23,26,55,84),(24,25,56,83)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 8A | 8B | 8C | 8D | 12A | ··· | 12H | 12I | ··· | 12X | 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 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | Q8 | D4 | C3×D4 | C3×Q8 | C3×D4 | C8⋊C22 | C8.C22 | C3×C8⋊C22 | C3×C8.C22 |
| kernel | C3×M4(2)⋊C4 | C3×C4.Q8 | C3×C2.D8 | C6×C4⋊C4 | C3×C42⋊C2 | C6×M4(2) | M4(2)⋊C4 | C3×M4(2) | C4.Q8 | C2.D8 | C2×C4⋊C4 | C42⋊C2 | C2×M4(2) | M4(2) | C2×C12 | C2×C12 | C22×C6 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 8 | 4 | 4 | 2 | 2 | 2 | 16 | 1 | 2 | 1 | 2 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of C3×M4(2)⋊C4 ►in GL6(𝔽73)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 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 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 54 | 21 | 0 | 0 | 0 | 0 |
| 21 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 7 | 0 | 0 |
| 0 | 0 | 7 | 68 | 0 | 0 |
| 0 | 0 | 0 | 0 | 66 | 5 |
| 0 | 0 | 0 | 0 | 5 | 7 |
G:=sub<GL(6,GF(73))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[54,21,0,0,0,0,21,19,0,0,0,0,0,0,5,7,0,0,0,0,7,68,0,0,0,0,0,0,66,5,0,0,0,0,5,7] >;
C3×M4(2)⋊C4 in GAP, Magma, Sage, TeX
C_3\times M_4(2)\rtimes C_4
% in TeX
G:=Group("C3xM4(2):C4"); // GroupNames label
G:=SmallGroup(192,861);
// by ID
G=gap.SmallGroup(192,861);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,336,365,176,1059,4204,172]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^5,d*b*d^-1=b^-1,c*d=d*c>;
// generators/relations