metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.233D6, C6.83(C4○D8), (C2×C12).449D4, C6.Q16⋊28C2, (C22×C6).72D4, C4.90(C4○D12), C6.SD16⋊27C2, C42⋊C2.5S3, C12.Q8⋊28C2, (C22×C4).124D6, C12.178(C4○D4), (C2×C12).326C23, C2.6(Q8.14D6), C2.8(Q8.13D6), C12.55D4.5C2, C3⋊6(C23.20D4), C23.26(C3⋊D4), C12.48D4.11C2, C6.105(C8.C22), C4⋊Dic3.134C22, (C2×Dic6).98C22, (C22×C12).146C22, C6.65(C22.D4), C2.16(C23.28D6), (C2×C6).455(C2×D4), (C2×C3⋊C8).86C22, (C2×C4).214(C3⋊D4), (C3×C4⋊C4).264C22, (C3×C42⋊C2).5C2, (C2×C4).426(C22×S3), C22.144(C2×C3⋊D4), SmallGroup(192,555)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.233D6
G = < a,b,c,d | a4=b4=1, c6=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c5 >
Subgroups: 232 in 96 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C23.20D4, C6.Q16, C12.Q8, C6.SD16, C12.55D4, C12.48D4, C3×C42⋊C2, C4⋊C4.233D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C4○D8, C8.C22, C4○D12, C2×C3⋊D4, C23.20D4, C23.28D6, Q8.13D6, Q8.14D6, C4⋊C4.233D6
►On 96 points
Generators in S96
(1 29 7 35)(2 30 8 36)(3 31 9 25)(4 32 10 26)(5 33 11 27)(6 34 12 28)(13 22 19 16)(14 23 20 17)(15 24 21 18)(37 57 43 51)(38 58 44 52)(39 59 45 53)(40 60 46 54)(41 49 47 55)(42 50 48 56)(61 64 67 70)(62 65 68 71)(63 66 69 72)(73 76 79 82)(74 77 80 83)(75 78 81 84)(85 94 91 88)(86 95 92 89)(87 96 93 90)
(1 60 26 37)(2 49 27 38)(3 50 28 39)(4 51 29 40)(5 52 30 41)(6 53 31 42)(7 54 32 43)(8 55 33 44)(9 56 34 45)(10 57 35 46)(11 58 36 47)(12 59 25 48)(13 68 86 79)(14 69 87 80)(15 70 88 81)(16 71 89 82)(17 72 90 83)(18 61 91 84)(19 62 92 73)(20 63 93 74)(21 64 94 75)(22 65 95 76)(23 66 96 77)(24 67 85 78)
(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 67 7 61)(2 77 8 83)(3 65 9 71)(4 75 10 81)(5 63 11 69)(6 73 12 79)(13 45 19 39)(14 55 20 49)(15 43 21 37)(16 53 22 59)(17 41 23 47)(18 51 24 57)(25 68 31 62)(26 78 32 84)(27 66 33 72)(28 76 34 82)(29 64 35 70)(30 74 36 80)(38 87 44 93)(40 85 46 91)(42 95 48 89)(50 86 56 92)(52 96 58 90)(54 94 60 88)
G:=sub<Sym(96)| (1,29,7,35)(2,30,8,36)(3,31,9,25)(4,32,10,26)(5,33,11,27)(6,34,12,28)(13,22,19,16)(14,23,20,17)(15,24,21,18)(37,57,43,51)(38,58,44,52)(39,59,45,53)(40,60,46,54)(41,49,47,55)(42,50,48,56)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,94,91,88)(86,95,92,89)(87,96,93,90), (1,60,26,37)(2,49,27,38)(3,50,28,39)(4,51,29,40)(5,52,30,41)(6,53,31,42)(7,54,32,43)(8,55,33,44)(9,56,34,45)(10,57,35,46)(11,58,36,47)(12,59,25,48)(13,68,86,79)(14,69,87,80)(15,70,88,81)(16,71,89,82)(17,72,90,83)(18,61,91,84)(19,62,92,73)(20,63,93,74)(21,64,94,75)(22,65,95,76)(23,66,96,77)(24,67,85,78), (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), (1,67,7,61)(2,77,8,83)(3,65,9,71)(4,75,10,81)(5,63,11,69)(6,73,12,79)(13,45,19,39)(14,55,20,49)(15,43,21,37)(16,53,22,59)(17,41,23,47)(18,51,24,57)(25,68,31,62)(26,78,32,84)(27,66,33,72)(28,76,34,82)(29,64,35,70)(30,74,36,80)(38,87,44,93)(40,85,46,91)(42,95,48,89)(50,86,56,92)(52,96,58,90)(54,94,60,88)>;
G:=Group( (1,29,7,35)(2,30,8,36)(3,31,9,25)(4,32,10,26)(5,33,11,27)(6,34,12,28)(13,22,19,16)(14,23,20,17)(15,24,21,18)(37,57,43,51)(38,58,44,52)(39,59,45,53)(40,60,46,54)(41,49,47,55)(42,50,48,56)(61,64,67,70)(62,65,68,71)(63,66,69,72)(73,76,79,82)(74,77,80,83)(75,78,81,84)(85,94,91,88)(86,95,92,89)(87,96,93,90), (1,60,26,37)(2,49,27,38)(3,50,28,39)(4,51,29,40)(5,52,30,41)(6,53,31,42)(7,54,32,43)(8,55,33,44)(9,56,34,45)(10,57,35,46)(11,58,36,47)(12,59,25,48)(13,68,86,79)(14,69,87,80)(15,70,88,81)(16,71,89,82)(17,72,90,83)(18,61,91,84)(19,62,92,73)(20,63,93,74)(21,64,94,75)(22,65,95,76)(23,66,96,77)(24,67,85,78), (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), (1,67,7,61)(2,77,8,83)(3,65,9,71)(4,75,10,81)(5,63,11,69)(6,73,12,79)(13,45,19,39)(14,55,20,49)(15,43,21,37)(16,53,22,59)(17,41,23,47)(18,51,24,57)(25,68,31,62)(26,78,32,84)(27,66,33,72)(28,76,34,82)(29,64,35,70)(30,74,36,80)(38,87,44,93)(40,85,46,91)(42,95,48,89)(50,86,56,92)(52,96,58,90)(54,94,60,88) );
G=PermutationGroup([[(1,29,7,35),(2,30,8,36),(3,31,9,25),(4,32,10,26),(5,33,11,27),(6,34,12,28),(13,22,19,16),(14,23,20,17),(15,24,21,18),(37,57,43,51),(38,58,44,52),(39,59,45,53),(40,60,46,54),(41,49,47,55),(42,50,48,56),(61,64,67,70),(62,65,68,71),(63,66,69,72),(73,76,79,82),(74,77,80,83),(75,78,81,84),(85,94,91,88),(86,95,92,89),(87,96,93,90)], [(1,60,26,37),(2,49,27,38),(3,50,28,39),(4,51,29,40),(5,52,30,41),(6,53,31,42),(7,54,32,43),(8,55,33,44),(9,56,34,45),(10,57,35,46),(11,58,36,47),(12,59,25,48),(13,68,86,79),(14,69,87,80),(15,70,88,81),(16,71,89,82),(17,72,90,83),(18,61,91,84),(19,62,92,73),(20,63,93,74),(21,64,94,75),(22,65,95,76),(23,66,96,77),(24,67,85,78)], [(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)], [(1,67,7,61),(2,77,8,83),(3,65,9,71),(4,75,10,81),(5,63,11,69),(6,73,12,79),(13,45,19,39),(14,55,20,49),(15,43,21,37),(16,53,22,59),(17,41,23,47),(18,51,24,57),(25,68,31,62),(26,78,32,84),(27,66,33,72),(28,76,34,82),(29,64,35,70),(30,74,36,80),(38,87,44,93),(40,85,46,91),(42,95,48,89),(50,86,56,92),(52,96,58,90),(54,94,60,88)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | C3⋊D4 | C4○D8 | C4○D12 | C8.C22 | Q8.13D6 | Q8.14D6 |
| kernel | C4⋊C4.233D6 | C6.Q16 | C12.Q8 | C6.SD16 | C12.55D4 | C12.48D4 | C3×C42⋊C2 | C42⋊C2 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C12 | C2×C4 | C23 | C6 | C4 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.233D6 ►in GL6(𝔽73)
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 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 | 46 | 0 | 0 | 0 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 60 | 0 | 0 |
| 0 | 0 | 13 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 63 | 27 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 22 | 72 |
| 0 | 51 | 0 | 0 | 0 | 0 |
| 10 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 70 | 31 | 0 | 0 |
| 0 | 0 | 28 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 2 |
| 0 | 0 | 0 | 0 | 14 | 22 |
G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[0,46,0,0,0,0,46,0,0,0,0,0,0,0,43,13,0,0,0,0,60,30,0,0,0,0,0,0,46,63,0,0,0,0,0,27],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,1,22,0,0,0,0,0,72],[0,10,0,0,0,0,51,0,0,0,0,0,0,0,70,28,0,0,0,0,31,3,0,0,0,0,0,0,51,14,0,0,0,0,2,22] >;
C4⋊C4.233D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{233}D_6 % in TeX
G:=Group("C4:C4.233D6"); // GroupNames label
G:=SmallGroup(192,555);
// by ID
G=gap.SmallGroup(192,555);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,254,100,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=a^2*b^2*c^5>;
// generators/relations