metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.228D6, (C2×C12).284D4, C6.D8⋊26C2, (C2×C6).41SD16, C6.67(C2×SD16), C4.87(C4○D12), C12.55D4⋊5C2, C12⋊7D4.10C2, C6.84(C8⋊C22), C12.Q8⋊25C2, (C22×C4).111D6, (C22×C6).186D4, C12.175(C4○D4), (C2×C12).321C23, C2.6(D12⋊6C22), (C2×D12).91C22, C3⋊4(C23.46D4), C23.85(C3⋊D4), C4⋊Dic3.131C22, C22.8(Q8⋊2S3), (C22×C12).136C22, C6.59(C22.D4), C2.9(C23.28D6), (C6×C4⋊C4)⋊4C2, (C2×C4⋊C4)⋊4S3, (C2×C6).441(C2×D4), (C2×C3⋊C8).82C22, C2.5(C2×Q8⋊2S3), (C2×C4).32(C3⋊D4), (C3×C4⋊C4).259C22, (C2×C4).421(C22×S3), C22.131(C2×C3⋊D4), SmallGroup(192,527)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4.228D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=a2b2c-1 >
Subgroups: 344 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C3⋊C8, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C22×C12, C23.46D4, C12.Q8, C6.D8, C12.55D4, C12⋊7D4, C6×C4⋊C4, C4⋊C4.228D6
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C22.D4, C2×SD16, C8⋊C22, Q8⋊2S3, C4○D12, C2×C3⋊D4, C23.46D4, C23.28D6, D12⋊6C22, C2×Q8⋊2S3, C4⋊C4.228D6
►On 96 points
(1 61 73 41)(2 62 74 42)(3 63 75 37)(4 64 76 38)(5 65 77 39)(6 66 78 40)(7 23 51 85)(8 24 52 86)(9 19 53 87)(10 20 54 88)(11 21 49 89)(12 22 50 90)(13 31 81 57)(14 32 82 58)(15 33 83 59)(16 34 84 60)(17 35 79 55)(18 36 80 56)(25 69 93 43)(26 70 94 44)(27 71 95 45)(28 72 96 46)(29 67 91 47)(30 68 92 48)
(1 44 35 20)(2 45 36 21)(3 46 31 22)(4 47 32 23)(5 48 33 24)(6 43 34 19)(7 64 91 82)(8 65 92 83)(9 66 93 84)(10 61 94 79)(11 62 95 80)(12 63 96 81)(13 50 37 28)(14 51 38 29)(15 52 39 30)(16 53 40 25)(17 54 41 26)(18 49 42 27)(55 88 73 70)(56 89 74 71)(57 90 75 72)(58 85 76 67)(59 86 77 68)(60 87 78 69)
(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 6 55 60)(2 59 56 5)(3 4 57 58)(7 72 29 22)(8 21 30 71)(9 70 25 20)(10 19 26 69)(11 68 27 24)(12 23 28 67)(13 82 63 38)(14 37 64 81)(15 80 65 42)(16 41 66 79)(17 84 61 40)(18 39 62 83)(31 32 75 76)(33 36 77 74)(34 73 78 35)(43 54 87 94)(44 93 88 53)(45 52 89 92)(46 91 90 51)(47 50 85 96)(48 95 86 49)
G:=sub<Sym(96)| (1,61,73,41)(2,62,74,42)(3,63,75,37)(4,64,76,38)(5,65,77,39)(6,66,78,40)(7,23,51,85)(8,24,52,86)(9,19,53,87)(10,20,54,88)(11,21,49,89)(12,22,50,90)(13,31,81,57)(14,32,82,58)(15,33,83,59)(16,34,84,60)(17,35,79,55)(18,36,80,56)(25,69,93,43)(26,70,94,44)(27,71,95,45)(28,72,96,46)(29,67,91,47)(30,68,92,48), (1,44,35,20)(2,45,36,21)(3,46,31,22)(4,47,32,23)(5,48,33,24)(6,43,34,19)(7,64,91,82)(8,65,92,83)(9,66,93,84)(10,61,94,79)(11,62,95,80)(12,63,96,81)(13,50,37,28)(14,51,38,29)(15,52,39,30)(16,53,40,25)(17,54,41,26)(18,49,42,27)(55,88,73,70)(56,89,74,71)(57,90,75,72)(58,85,76,67)(59,86,77,68)(60,87,78,69), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,72,29,22)(8,21,30,71)(9,70,25,20)(10,19,26,69)(11,68,27,24)(12,23,28,67)(13,82,63,38)(14,37,64,81)(15,80,65,42)(16,41,66,79)(17,84,61,40)(18,39,62,83)(31,32,75,76)(33,36,77,74)(34,73,78,35)(43,54,87,94)(44,93,88,53)(45,52,89,92)(46,91,90,51)(47,50,85,96)(48,95,86,49)>;
G:=Group( (1,61,73,41)(2,62,74,42)(3,63,75,37)(4,64,76,38)(5,65,77,39)(6,66,78,40)(7,23,51,85)(8,24,52,86)(9,19,53,87)(10,20,54,88)(11,21,49,89)(12,22,50,90)(13,31,81,57)(14,32,82,58)(15,33,83,59)(16,34,84,60)(17,35,79,55)(18,36,80,56)(25,69,93,43)(26,70,94,44)(27,71,95,45)(28,72,96,46)(29,67,91,47)(30,68,92,48), (1,44,35,20)(2,45,36,21)(3,46,31,22)(4,47,32,23)(5,48,33,24)(6,43,34,19)(7,64,91,82)(8,65,92,83)(9,66,93,84)(10,61,94,79)(11,62,95,80)(12,63,96,81)(13,50,37,28)(14,51,38,29)(15,52,39,30)(16,53,40,25)(17,54,41,26)(18,49,42,27)(55,88,73,70)(56,89,74,71)(57,90,75,72)(58,85,76,67)(59,86,77,68)(60,87,78,69), (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), (1,6,55,60)(2,59,56,5)(3,4,57,58)(7,72,29,22)(8,21,30,71)(9,70,25,20)(10,19,26,69)(11,68,27,24)(12,23,28,67)(13,82,63,38)(14,37,64,81)(15,80,65,42)(16,41,66,79)(17,84,61,40)(18,39,62,83)(31,32,75,76)(33,36,77,74)(34,73,78,35)(43,54,87,94)(44,93,88,53)(45,52,89,92)(46,91,90,51)(47,50,85,96)(48,95,86,49) );
G=PermutationGroup([[(1,61,73,41),(2,62,74,42),(3,63,75,37),(4,64,76,38),(5,65,77,39),(6,66,78,40),(7,23,51,85),(8,24,52,86),(9,19,53,87),(10,20,54,88),(11,21,49,89),(12,22,50,90),(13,31,81,57),(14,32,82,58),(15,33,83,59),(16,34,84,60),(17,35,79,55),(18,36,80,56),(25,69,93,43),(26,70,94,44),(27,71,95,45),(28,72,96,46),(29,67,91,47),(30,68,92,48)], [(1,44,35,20),(2,45,36,21),(3,46,31,22),(4,47,32,23),(5,48,33,24),(6,43,34,19),(7,64,91,82),(8,65,92,83),(9,66,93,84),(10,61,94,79),(11,62,95,80),(12,63,96,81),(13,50,37,28),(14,51,38,29),(15,52,39,30),(16,53,40,25),(17,54,41,26),(18,49,42,27),(55,88,73,70),(56,89,74,71),(57,90,75,72),(58,85,76,67),(59,86,77,68),(60,87,78,69)], [(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)], [(1,6,55,60),(2,59,56,5),(3,4,57,58),(7,72,29,22),(8,21,30,71),(9,70,25,20),(10,19,26,69),(11,68,27,24),(12,23,28,67),(13,82,63,38),(14,37,64,81),(15,80,65,42),(16,41,66,79),(17,84,61,40),(18,39,62,83),(31,32,75,76),(33,36,77,74),(34,73,78,35),(43,54,87,94),(44,93,88,53),(45,52,89,92),(46,91,90,51),(47,50,85,96),(48,95,86,49)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 6A | ··· | 6G | 8A | 8B | 8C | 8D | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 24 | 2 | 2 | 2 | 4 | ··· | 4 | 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 | SD16 | C3⋊D4 | C3⋊D4 | C4○D12 | C8⋊C22 | Q8⋊2S3 | D12⋊6C22 |
| kernel | C4⋊C4.228D6 | C12.Q8 | C6.D8 | C12.55D4 | C12⋊7D4 | 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⋊C4.228D6 ►in GL6(𝔽73)
| 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 | 2 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 61 | 12 |
| 0 | 0 | 0 | 0 | 67 | 12 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,61,67,0,0,0,0,12,12],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;
C4⋊C4.228D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{228}D_6 % in TeX
G:=Group("C4:C4.228D6"); // GroupNames label
G:=SmallGroup(192,527);
// by ID
G=gap.SmallGroup(192,527);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,268,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^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