metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.150D6, Q8⋊3S3⋊2C4, D12.3(C2×C4), (C4×S3).40D4, (C2×C8).209D6, C4.165(S3×D4), Q8.15(C4×S3), Q8⋊C4⋊22S3, C6.47(C4○D8), C12.119(C2×D4), Q8⋊2Dic3⋊6C2, C6.D8⋊10C2, C2.D24⋊28C2, (C2×Q8).130D6, C22.79(S3×D4), C12.15(C22×C4), D6.7(C22⋊C4), (C22×S3).49D4, (C6×Q8).27C22, C2.2(D24⋊C2), (C2×C24).242C22, (C2×C12).244C23, (C2×Dic3).204D4, C2.4(Q8.7D6), (C2×D12).61C22, C3⋊2(C23.24D4), C4⋊Dic3.92C22, Dic3.19(C22⋊C4), (S3×C2×C8)⋊20C2, C4.15(S3×C2×C4), C4⋊C4⋊7S3⋊5C2, (C3×Q8).4(C2×C4), (C4×S3).15(C2×C4), (C2×C6).257(C2×D4), C2.24(S3×C22⋊C4), C6.23(C2×C22⋊C4), (C3×Q8⋊C4)⋊25C2, (C3×C4⋊C4).45C22, (C2×C3⋊C8).219C22, (S3×C2×C4).227C22, (C2×Q8⋊3S3).3C2, (C2×C4).351(C22×S3), SmallGroup(192,363)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.150D6
G = < a,b,c,d | a4=b4=d2=1, c6=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b-1, dbd=ab-1, dcd=a2c5 >
Subgroups: 456 in 158 conjugacy classes, 55 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, D4⋊C4, Q8⋊C4, Q8⋊C4, C42⋊C2, C22×C8, C2×C4○D4, S3×C8, C2×C3⋊C8, C4×Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, Q8⋊3S3, C6×Q8, C23.24D4, C6.D8, C2.D24, Q8⋊2Dic3, C3×Q8⋊C4, C4⋊C4⋊7S3, S3×C2×C8, C2×Q8⋊3S3, C4⋊C4.150D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C4○D8, S3×C2×C4, S3×D4, C23.24D4, S3×C22⋊C4, Q8.7D6, D24⋊C2, C4⋊C4.150D6
►On 96 points
(1 82 7 76)(2 77 8 83)(3 84 9 78)(4 79 10 73)(5 74 11 80)(6 81 12 75)(13 29 19 35)(14 36 20 30)(15 31 21 25)(16 26 22 32)(17 33 23 27)(18 28 24 34)(37 58 43 52)(38 53 44 59)(39 60 45 54)(40 55 46 49)(41 50 47 56)(42 57 48 51)(61 88 67 94)(62 95 68 89)(63 90 69 96)(64 85 70 91)(65 92 71 86)(66 87 72 93)
(1 39 22 86)(2 66 23 49)(3 41 24 88)(4 68 13 51)(5 43 14 90)(6 70 15 53)(7 45 16 92)(8 72 17 55)(9 47 18 94)(10 62 19 57)(11 37 20 96)(12 64 21 59)(25 44 75 91)(26 65 76 60)(27 46 77 93)(28 67 78 50)(29 48 79 95)(30 69 80 52)(31 38 81 85)(32 71 82 54)(33 40 83 87)(34 61 84 56)(35 42 73 89)(36 63 74 58)
(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 9)(2 8)(3 7)(4 6)(10 12)(13 15)(16 24)(17 23)(18 22)(19 21)(25 29)(26 28)(30 36)(31 35)(32 34)(37 69)(38 68)(39 67)(40 66)(41 65)(42 64)(43 63)(44 62)(45 61)(46 72)(47 71)(48 70)(49 87)(50 86)(51 85)(52 96)(53 95)(54 94)(55 93)(56 92)(57 91)(58 90)(59 89)(60 88)(73 81)(74 80)(75 79)(76 78)(82 84)
G:=sub<Sym(96)| (1,82,7,76)(2,77,8,83)(3,84,9,78)(4,79,10,73)(5,74,11,80)(6,81,12,75)(13,29,19,35)(14,36,20,30)(15,31,21,25)(16,26,22,32)(17,33,23,27)(18,28,24,34)(37,58,43,52)(38,53,44,59)(39,60,45,54)(40,55,46,49)(41,50,47,56)(42,57,48,51)(61,88,67,94)(62,95,68,89)(63,90,69,96)(64,85,70,91)(65,92,71,86)(66,87,72,93), (1,39,22,86)(2,66,23,49)(3,41,24,88)(4,68,13,51)(5,43,14,90)(6,70,15,53)(7,45,16,92)(8,72,17,55)(9,47,18,94)(10,62,19,57)(11,37,20,96)(12,64,21,59)(25,44,75,91)(26,65,76,60)(27,46,77,93)(28,67,78,50)(29,48,79,95)(30,69,80,52)(31,38,81,85)(32,71,82,54)(33,40,83,87)(34,61,84,56)(35,42,73,89)(36,63,74,58), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,72)(47,71)(48,70)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(73,81)(74,80)(75,79)(76,78)(82,84)>;
G:=Group( (1,82,7,76)(2,77,8,83)(3,84,9,78)(4,79,10,73)(5,74,11,80)(6,81,12,75)(13,29,19,35)(14,36,20,30)(15,31,21,25)(16,26,22,32)(17,33,23,27)(18,28,24,34)(37,58,43,52)(38,53,44,59)(39,60,45,54)(40,55,46,49)(41,50,47,56)(42,57,48,51)(61,88,67,94)(62,95,68,89)(63,90,69,96)(64,85,70,91)(65,92,71,86)(66,87,72,93), (1,39,22,86)(2,66,23,49)(3,41,24,88)(4,68,13,51)(5,43,14,90)(6,70,15,53)(7,45,16,92)(8,72,17,55)(9,47,18,94)(10,62,19,57)(11,37,20,96)(12,64,21,59)(25,44,75,91)(26,65,76,60)(27,46,77,93)(28,67,78,50)(29,48,79,95)(30,69,80,52)(31,38,81,85)(32,71,82,54)(33,40,83,87)(34,61,84,56)(35,42,73,89)(36,63,74,58), (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), (1,9)(2,8)(3,7)(4,6)(10,12)(13,15)(16,24)(17,23)(18,22)(19,21)(25,29)(26,28)(30,36)(31,35)(32,34)(37,69)(38,68)(39,67)(40,66)(41,65)(42,64)(43,63)(44,62)(45,61)(46,72)(47,71)(48,70)(49,87)(50,86)(51,85)(52,96)(53,95)(54,94)(55,93)(56,92)(57,91)(58,90)(59,89)(60,88)(73,81)(74,80)(75,79)(76,78)(82,84) );
G=PermutationGroup([[(1,82,7,76),(2,77,8,83),(3,84,9,78),(4,79,10,73),(5,74,11,80),(6,81,12,75),(13,29,19,35),(14,36,20,30),(15,31,21,25),(16,26,22,32),(17,33,23,27),(18,28,24,34),(37,58,43,52),(38,53,44,59),(39,60,45,54),(40,55,46,49),(41,50,47,56),(42,57,48,51),(61,88,67,94),(62,95,68,89),(63,90,69,96),(64,85,70,91),(65,92,71,86),(66,87,72,93)], [(1,39,22,86),(2,66,23,49),(3,41,24,88),(4,68,13,51),(5,43,14,90),(6,70,15,53),(7,45,16,92),(8,72,17,55),(9,47,18,94),(10,62,19,57),(11,37,20,96),(12,64,21,59),(25,44,75,91),(26,65,76,60),(27,46,77,93),(28,67,78,50),(29,48,79,95),(30,69,80,52),(31,38,81,85),(32,71,82,54),(33,40,83,87),(34,61,84,56),(35,42,73,89),(36,63,74,58)], [(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)], [(1,9),(2,8),(3,7),(4,6),(10,12),(13,15),(16,24),(17,23),(18,22),(19,21),(25,29),(26,28),(30,36),(31,35),(32,34),(37,69),(38,68),(39,67),(40,66),(41,65),(42,64),(43,63),(44,62),(45,61),(46,72),(47,71),(48,70),(49,87),(50,86),(51,85),(52,96),(53,95),(54,94),(55,93),(56,92),(57,91),(58,90),(59,89),(60,88),(73,81),(74,80),(75,79),(76,78),(82,84)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4×S3 | C4○D8 | S3×D4 | S3×D4 | Q8.7D6 | D24⋊C2 |
| kernel | C4⋊C4.150D6 | C6.D8 | C2.D24 | Q8⋊2Dic3 | C3×Q8⋊C4 | C4⋊C4⋊7S3 | S3×C2×C8 | C2×Q8⋊3S3 | Q8⋊3S3 | Q8⋊C4 | C4×S3 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 4 | 8 | 1 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.150D6 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 16 | 16 |
| 0 | 0 | 16 | 57 |
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 46 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,0,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,16,16,0,0,16,57],[1,72,0,0,1,0,0,0,0,0,0,46,0,0,46,0],[72,0,0,0,72,1,0,0,0,0,72,0,0,0,0,1] >;
C4⋊C4.150D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{150}D_6 % in TeX
G:=Group("C4:C4.150D6"); // GroupNames label
G:=SmallGroup(192,363);
// by ID
G=gap.SmallGroup(192,363);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,232,219,58,570,136,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^6=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b^-1,d*b*d=a*b^-1,d*c*d=a^2*c^5>;
// generators/relations