direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C42.78C22, (C4×C8)⋊6C6, (C4×C24)⋊11C2, Q8⋊C4⋊3C6, C42.C2⋊2C6, D4⋊C4.1C6, (C2×C12).366D4, C42.78(C2×C6), C4.4D4.5C6, C6.130(C4○D8), C22.109(C6×D4), C12.270(C4○D4), (C2×C12).944C23, (C2×C24).368C22, (C4×C12).362C22, C6.73(C4.4D4), (C6×D4).199C22, (C6×Q8).173C22, C4⋊C4.19(C2×C6), (C2×C8).70(C2×C6), C2.17(C3×C4○D8), C4.15(C3×C4○D4), (C2×C4).56(C3×D4), (C3×Q8⋊C4)⋊3C2, (C2×D4).22(C2×C6), (C2×C6).665(C2×D4), (C2×Q8).18(C2×C6), (C3×D4⋊C4).1C2, (C3×C42.C2)⋊19C2, C2.11(C3×C4.4D4), (C3×C4⋊C4).239C22, (C2×C4).119(C22×C6), (C3×C4.4D4).14C2, SmallGroup(192,921)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C42.78C22
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, be=eb, dcd=c-1, ce=ec, ede-1=b2cd >
Subgroups: 178 in 96 conjugacy classes, 50 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, 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, C24, C2×C12, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C4×C8, D4⋊C4, Q8⋊C4, C4.4D4, C42.C2, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×C24, C6×D4, C6×Q8, C42.78C22, C4×C24, C3×D4⋊C4, C3×Q8⋊C4, C3×C4.4D4, C3×C42.C2, C3×C42.78C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C4○D4, C3×D4, C22×C6, C4.4D4, C4○D8, C6×D4, C3×C4○D4, C42.78C22, C3×C4.4D4, C3×C4○D8, C3×C42.78C22
►On 96 points
Generators in S96
(1 71 23)(2 72 24)(3 65 17)(4 66 18)(5 67 19)(6 68 20)(7 69 21)(8 70 22)(9 58 26)(10 59 27)(11 60 28)(12 61 29)(13 62 30)(14 63 31)(15 64 32)(16 57 25)(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 52 89)(42 53 90)(43 54 91)(44 55 92)(45 56 93)(46 49 94)(47 50 95)(48 51 96)
(1 48 73 30)(2 41 74 31)(3 42 75 32)(4 43 76 25)(5 44 77 26)(6 45 78 27)(7 46 79 28)(8 47 80 29)(9 67 55 85)(10 68 56 86)(11 69 49 87)(12 70 50 88)(13 71 51 81)(14 72 52 82)(15 65 53 83)(16 66 54 84)(17 90 35 64)(18 91 36 57)(19 92 37 58)(20 93 38 59)(21 94 39 60)(22 95 40 61)(23 96 33 62)(24 89 34 63)
(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 80)(3 7)(4 78)(6 76)(8 74)(9 51)(10 12)(11 49)(13 55)(14 16)(15 53)(17 21)(18 38)(20 36)(22 34)(24 40)(25 31)(26 48)(27 29)(28 46)(30 44)(32 42)(35 39)(41 43)(45 47)(50 56)(52 54)(57 63)(58 96)(59 61)(60 94)(62 92)(64 90)(65 69)(66 86)(68 84)(70 82)(72 88)(75 79)(83 87)(89 91)(93 95)
(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)| (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,52,89)(42,53,90)(43,54,91)(44,55,92)(45,56,93)(46,49,94)(47,50,95)(48,51,96), (1,48,73,30)(2,41,74,31)(3,42,75,32)(4,43,76,25)(5,44,77,26)(6,45,78,27)(7,46,79,28)(8,47,80,29)(9,67,55,85)(10,68,56,86)(11,69,49,87)(12,70,50,88)(13,71,51,81)(14,72,52,82)(15,65,53,83)(16,66,54,84)(17,90,35,64)(18,91,36,57)(19,92,37,58)(20,93,38,59)(21,94,39,60)(22,95,40,61)(23,96,33,62)(24,89,34,63), (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), (2,80)(3,7)(4,78)(6,76)(8,74)(9,51)(10,12)(11,49)(13,55)(14,16)(15,53)(17,21)(18,38)(20,36)(22,34)(24,40)(25,31)(26,48)(27,29)(28,46)(30,44)(32,42)(35,39)(41,43)(45,47)(50,56)(52,54)(57,63)(58,96)(59,61)(60,94)(62,92)(64,90)(65,69)(66,86)(68,84)(70,82)(72,88)(75,79)(83,87)(89,91)(93,95), (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( (1,71,23)(2,72,24)(3,65,17)(4,66,18)(5,67,19)(6,68,20)(7,69,21)(8,70,22)(9,58,26)(10,59,27)(11,60,28)(12,61,29)(13,62,30)(14,63,31)(15,64,32)(16,57,25)(33,73,81)(34,74,82)(35,75,83)(36,76,84)(37,77,85)(38,78,86)(39,79,87)(40,80,88)(41,52,89)(42,53,90)(43,54,91)(44,55,92)(45,56,93)(46,49,94)(47,50,95)(48,51,96), (1,48,73,30)(2,41,74,31)(3,42,75,32)(4,43,76,25)(5,44,77,26)(6,45,78,27)(7,46,79,28)(8,47,80,29)(9,67,55,85)(10,68,56,86)(11,69,49,87)(12,70,50,88)(13,71,51,81)(14,72,52,82)(15,65,53,83)(16,66,54,84)(17,90,35,64)(18,91,36,57)(19,92,37,58)(20,93,38,59)(21,94,39,60)(22,95,40,61)(23,96,33,62)(24,89,34,63), (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), (2,80)(3,7)(4,78)(6,76)(8,74)(9,51)(10,12)(11,49)(13,55)(14,16)(15,53)(17,21)(18,38)(20,36)(22,34)(24,40)(25,31)(26,48)(27,29)(28,46)(30,44)(32,42)(35,39)(41,43)(45,47)(50,56)(52,54)(57,63)(58,96)(59,61)(60,94)(62,92)(64,90)(65,69)(66,86)(68,84)(70,82)(72,88)(75,79)(83,87)(89,91)(93,95), (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([[(1,71,23),(2,72,24),(3,65,17),(4,66,18),(5,67,19),(6,68,20),(7,69,21),(8,70,22),(9,58,26),(10,59,27),(11,60,28),(12,61,29),(13,62,30),(14,63,31),(15,64,32),(16,57,25),(33,73,81),(34,74,82),(35,75,83),(36,76,84),(37,77,85),(38,78,86),(39,79,87),(40,80,88),(41,52,89),(42,53,90),(43,54,91),(44,55,92),(45,56,93),(46,49,94),(47,50,95),(48,51,96)], [(1,48,73,30),(2,41,74,31),(3,42,75,32),(4,43,76,25),(5,44,77,26),(6,45,78,27),(7,46,79,28),(8,47,80,29),(9,67,55,85),(10,68,56,86),(11,69,49,87),(12,70,50,88),(13,71,51,81),(14,72,52,82),(15,65,53,83),(16,66,54,84),(17,90,35,64),(18,91,36,57),(19,92,37,58),(20,93,38,59),(21,94,39,60),(22,95,40,61),(23,96,33,62),(24,89,34,63)], [(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)], [(2,80),(3,7),(4,78),(6,76),(8,74),(9,51),(10,12),(11,49),(13,55),(14,16),(15,53),(17,21),(18,38),(20,36),(22,34),(24,40),(25,31),(26,48),(27,29),(28,46),(30,44),(32,42),(35,39),(41,43),(45,47),(50,56),(52,54),(57,63),(58,96),(59,61),(60,94),(62,92),(64,90),(65,69),(66,86),(68,84),(70,82),(72,88),(75,79),(83,87),(89,91),(93,95)], [(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)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | ··· | 4F | 4G | 4H | 4I | 6A | ··· | 6F | 6G | 6H | 8A | ··· | 8H | 12A | ··· | 12L | 12M | ··· | 12R | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | ··· | 2 | 8 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 2 | ··· | 2 | 2 | ··· | 2 | 8 | ··· | 8 | 2 | ··· | 2 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | C4○D4 | C3×D4 | C4○D8 | C3×C4○D4 | C3×C4○D8 |
| kernel | C3×C42.78C22 | C4×C24 | C3×D4⋊C4 | C3×Q8⋊C4 | C3×C4.4D4 | C3×C42.C2 | C42.78C22 | C4×C8 | D4⋊C4 | Q8⋊C4 | C4.4D4 | C42.C2 | C2×C12 | C12 | C2×C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C42.78C22 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 72 | 2 | 0 | 0 |
| 72 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
| 12 | 61 | 0 | 0 |
| 6 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,0,46,0,0,46,0],[72,72,0,0,2,1,0,0,0,0,1,0,0,0,0,1],[1,1,0,0,0,72,0,0,0,0,1,0,0,0,0,72],[12,6,0,0,61,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C42.78C22 in GAP, Magma, Sage, TeX
C_3\times C_4^2._{78}C_2^2 % in TeX
G:=Group("C3xC4^2.78C2^2"); // GroupNames label
G:=SmallGroup(192,921);
// by ID
G=gap.SmallGroup(192,921);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1016,1094,142,4204,172,6053,124]);
// 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,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c*d>;
// generators/relations