metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.230D6, C6.33(C2×Q16), (C2×C6).17Q16, (C2×C12).285D4, C6.Q16⋊26C2, C4.88(C4○D12), C6.SD16⋊25C2, C6.85(C8⋊C22), (C22×C6).188D4, (C22×C4).112D6, C12.176(C4○D4), (C2×C12).323C23, C2.7(D12⋊6C22), C12.48D4.9C2, C12.55D4.3C2, C3⋊4(C23.48D4), C23.86(C3⋊D4), C22.5(C3⋊Q16), C4⋊Dic3.132C22, (C2×Dic6).96C22, (C22×C12).138C22, C6.60(C22.D4), C2.10(C23.28D6), (C6×C4⋊C4).7C2, (C2×C4⋊C4).8S3, C2.5(C2×C3⋊Q16), (C2×C6).443(C2×D4), (C2×C3⋊C8).83C22, (C2×C4).33(C3⋊D4), (C3×C4⋊C4).261C22, (C2×C4).423(C22×S3), C22.132(C2×C3⋊D4), SmallGroup(192,529)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.230D6
G = < a,b,c,d | a4=b4=c6=1, d2=b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >
Subgroups: 248 in 104 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C23.48D4, C6.Q16, C6.SD16, C12.55D4, C12.48D4, C6×C4⋊C4, C4⋊C4.230D6
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×Q16, C8⋊C22, C3⋊Q16, C4○D12, C2×C3⋊D4, C23.48D4, C23.28D6, D12⋊6C22, C2×C3⋊Q16, C4⋊C4.230D6
►On 96 points
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 13 10 16)(8 14 11 17)(9 15 12 18)(25 46 28 43)(26 47 29 44)(27 48 30 45)(31 40 34 37)(32 41 35 38)(33 42 36 39)(49 75 66 67)(50 76 61 68)(51 77 62 69)(52 78 63 70)(53 73 64 71)(54 74 65 72)(55 83 92 85)(56 84 93 86)(57 79 94 87)(58 80 95 88)(59 81 96 89)(60 82 91 90)
(1 34 10 25)(2 35 11 26)(3 36 12 27)(4 31 7 28)(5 32 8 29)(6 33 9 30)(13 46 22 37)(14 47 23 38)(15 48 24 39)(16 43 19 40)(17 44 20 41)(18 45 21 42)(49 57 63 91)(50 58 64 92)(51 59 65 93)(52 60 66 94)(53 55 61 95)(54 56 62 96)(67 79 78 90)(68 80 73 85)(69 81 74 86)(70 82 75 87)(71 83 76 88)(72 84 77 89)
(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 63 10 49)(2 65 11 51)(3 61 12 53)(4 52 7 66)(5 54 8 62)(6 50 9 64)(13 75 22 70)(14 77 23 72)(15 73 24 68)(16 67 19 78)(17 69 20 74)(18 71 21 76)(25 87 34 82)(26 89 35 84)(27 85 36 80)(28 79 31 90)(29 81 32 86)(30 83 33 88)(37 60 46 94)(38 56 47 96)(39 58 48 92)(40 91 43 57)(41 93 44 59)(42 95 45 55)
G:=sub<Sym(96)| (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,13,10,16)(8,14,11,17)(9,15,12,18)(25,46,28,43)(26,47,29,44)(27,48,30,45)(31,40,34,37)(32,41,35,38)(33,42,36,39)(49,75,66,67)(50,76,61,68)(51,77,62,69)(52,78,63,70)(53,73,64,71)(54,74,65,72)(55,83,92,85)(56,84,93,86)(57,79,94,87)(58,80,95,88)(59,81,96,89)(60,82,91,90), (1,34,10,25)(2,35,11,26)(3,36,12,27)(4,31,7,28)(5,32,8,29)(6,33,9,30)(13,46,22,37)(14,47,23,38)(15,48,24,39)(16,43,19,40)(17,44,20,41)(18,45,21,42)(49,57,63,91)(50,58,64,92)(51,59,65,93)(52,60,66,94)(53,55,61,95)(54,56,62,96)(67,79,78,90)(68,80,73,85)(69,81,74,86)(70,82,75,87)(71,83,76,88)(72,84,77,89), (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), (1,63,10,49)(2,65,11,51)(3,61,12,53)(4,52,7,66)(5,54,8,62)(6,50,9,64)(13,75,22,70)(14,77,23,72)(15,73,24,68)(16,67,19,78)(17,69,20,74)(18,71,21,76)(25,87,34,82)(26,89,35,84)(27,85,36,80)(28,79,31,90)(29,81,32,86)(30,83,33,88)(37,60,46,94)(38,56,47,96)(39,58,48,92)(40,91,43,57)(41,93,44,59)(42,95,45,55)>;
G:=Group( (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,13,10,16)(8,14,11,17)(9,15,12,18)(25,46,28,43)(26,47,29,44)(27,48,30,45)(31,40,34,37)(32,41,35,38)(33,42,36,39)(49,75,66,67)(50,76,61,68)(51,77,62,69)(52,78,63,70)(53,73,64,71)(54,74,65,72)(55,83,92,85)(56,84,93,86)(57,79,94,87)(58,80,95,88)(59,81,96,89)(60,82,91,90), (1,34,10,25)(2,35,11,26)(3,36,12,27)(4,31,7,28)(5,32,8,29)(6,33,9,30)(13,46,22,37)(14,47,23,38)(15,48,24,39)(16,43,19,40)(17,44,20,41)(18,45,21,42)(49,57,63,91)(50,58,64,92)(51,59,65,93)(52,60,66,94)(53,55,61,95)(54,56,62,96)(67,79,78,90)(68,80,73,85)(69,81,74,86)(70,82,75,87)(71,83,76,88)(72,84,77,89), (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), (1,63,10,49)(2,65,11,51)(3,61,12,53)(4,52,7,66)(5,54,8,62)(6,50,9,64)(13,75,22,70)(14,77,23,72)(15,73,24,68)(16,67,19,78)(17,69,20,74)(18,71,21,76)(25,87,34,82)(26,89,35,84)(27,85,36,80)(28,79,31,90)(29,81,32,86)(30,83,33,88)(37,60,46,94)(38,56,47,96)(39,58,48,92)(40,91,43,57)(41,93,44,59)(42,95,45,55) );
G=PermutationGroup([[(1,19,4,22),(2,20,5,23),(3,21,6,24),(7,13,10,16),(8,14,11,17),(9,15,12,18),(25,46,28,43),(26,47,29,44),(27,48,30,45),(31,40,34,37),(32,41,35,38),(33,42,36,39),(49,75,66,67),(50,76,61,68),(51,77,62,69),(52,78,63,70),(53,73,64,71),(54,74,65,72),(55,83,92,85),(56,84,93,86),(57,79,94,87),(58,80,95,88),(59,81,96,89),(60,82,91,90)], [(1,34,10,25),(2,35,11,26),(3,36,12,27),(4,31,7,28),(5,32,8,29),(6,33,9,30),(13,46,22,37),(14,47,23,38),(15,48,24,39),(16,43,19,40),(17,44,20,41),(18,45,21,42),(49,57,63,91),(50,58,64,92),(51,59,65,93),(52,60,66,94),(53,55,61,95),(54,56,62,96),(67,79,78,90),(68,80,73,85),(69,81,74,86),(70,82,75,87),(71,83,76,88),(72,84,77,89)], [(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)], [(1,63,10,49),(2,65,11,51),(3,61,12,53),(4,52,7,66),(5,54,8,62),(6,50,9,64),(13,75,22,70),(14,77,23,72),(15,73,24,68),(16,67,19,78),(17,69,20,74),(18,71,21,76),(25,87,34,82),(26,89,35,84),(27,85,36,80),(28,79,31,90),(29,81,32,86),(30,83,33,88),(37,60,46,94),(38,56,47,96),(39,58,48,92),(40,91,43,57),(41,93,44,59),(42,95,45,55)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
39 irreducible representations
| dim | 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 | S3 | D4 | D4 | D6 | D6 | C4○D4 | Q16 | C3⋊D4 | C3⋊D4 | C4○D12 | C8⋊C22 | C3⋊Q16 | D12⋊6C22 |
| kernel | C4⋊C4.230D6 | C6.Q16 | C6.SD16 | C12.55D4 | C12.48D4 | C6×C4⋊C4 | C2×C4⋊C4 | C2×C12 | C22×C6 | C4⋊C4 | C22×C4 | C12 | C2×C6 | C2×C4 | C23 | C4 | C6 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 1 | 2 | 2 |
Matrix representation of C4⋊C4.230D6 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 0 | 1 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 0 | 72 | 61 |
| 0 | 0 | 61 | 1 |
| 8 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 64 | 0 | 0 |
| 65 | 0 | 0 | 0 |
| 0 | 0 | 62 | 30 |
| 0 | 0 | 30 | 11 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,72,0],[27,0,0,0,0,27,0,0,0,0,72,61,0,0,61,1],[8,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[0,65,0,0,64,0,0,0,0,0,62,30,0,0,30,11] >;
C4⋊C4.230D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{230}D_6 % in TeX
G:=Group("C4:C4.230D6"); // GroupNames label
G:=SmallGroup(192,529);
// by ID
G=gap.SmallGroup(192,529);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,268,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^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^-1>;
// generators/relations