metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.241D6, C4⋊Q8⋊20S3, (C4×S3)⋊5Q8, D6.5(C2×Q8), C4.40(S3×Q8), C4⋊C4.219D6, C12.54(C2×Q8), C12⋊2Q8⋊36C2, (Q8×Dic3)⋊22C2, (C2×Q8).171D6, C6.48(C22×Q8), (S3×C42).10C2, (C2×C6).272C24, D6⋊C4.51C22, C4.D12.13C2, D6⋊3Q8.11C2, C4.Dic6⋊42C2, Dic3.17(C2×Q8), Dic6⋊C4⋊42C2, C12.136(C4○D4), C4.41(D4⋊2S3), (C2×C12).105C23, (C4×C12).213C22, C4.22(Q8⋊3S3), (C6×Q8).139C22, Dic3⋊C4.61C22, C4⋊Dic3.251C22, C22.293(S3×C23), (C22×S3).233C23, C3⋊6(C23.37C23), (C2×Dic3).143C23, (C2×Dic6).190C22, (C4×Dic3).161C22, C2.31(C2×S3×Q8), (C3×C4⋊Q8)⋊14C2, C6.100(C2×C4○D4), C4⋊C4⋊7S3.14C2, C2.64(C2×D4⋊2S3), (S3×C2×C4).252C22, C2.29(C2×Q8⋊3S3), (C3×C4⋊C4).215C22, (C2×C4).600(C22×S3), SmallGroup(192,1287)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.241D6
G = < a,b,c,d | a4=b4=1, c6=b2, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a2c5 >
Subgroups: 464 in 222 conjugacy classes, 111 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, C2×C4, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C2×Q8, Dic6, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C2×C42, C42⋊C2, C4×Q8, C22⋊Q8, C42.C2, C4⋊Q8, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C4×C12, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C6×Q8, C23.37C23, C12⋊2Q8, S3×C42, Dic6⋊C4, C4.Dic6, C4⋊C4⋊7S3, C4.D12, Q8×Dic3, D6⋊3Q8, C3×C4⋊Q8, C42.241D6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C24, C22×S3, C22×Q8, C2×C4○D4, D4⋊2S3, S3×Q8, Q8⋊3S3, S3×C23, C23.37C23, C2×D4⋊2S3, C2×S3×Q8, C2×Q8⋊3S3, C42.241D6
►On 96 points
(1 44 77 51)(2 52 78 45)(3 46 79 53)(4 54 80 47)(5 48 81 55)(6 56 82 37)(7 38 83 57)(8 58 84 39)(9 40 73 59)(10 60 74 41)(11 42 75 49)(12 50 76 43)(13 63 34 92)(14 93 35 64)(15 65 36 94)(16 95 25 66)(17 67 26 96)(18 85 27 68)(19 69 28 86)(20 87 29 70)(21 71 30 88)(22 89 31 72)(23 61 32 90)(24 91 33 62)
(1 31 7 25)(2 26 8 32)(3 33 9 27)(4 28 10 34)(5 35 11 29)(6 30 12 36)(13 80 19 74)(14 75 20 81)(15 82 21 76)(16 77 22 83)(17 84 23 78)(18 79 24 73)(37 71 43 65)(38 66 44 72)(39 61 45 67)(40 68 46 62)(41 63 47 69)(42 70 48 64)(49 87 55 93)(50 94 56 88)(51 89 57 95)(52 96 58 90)(53 91 59 85)(54 86 60 92)
(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 83 76)(2 75 84 5)(3 4 73 74)(7 12 77 82)(8 81 78 11)(9 10 79 80)(13 27 28 24)(14 23 29 26)(15 25 30 22)(16 21 31 36)(17 35 32 20)(18 19 33 34)(37 57 50 44)(38 43 51 56)(39 55 52 42)(40 41 53 54)(45 49 58 48)(46 47 59 60)(61 87 96 64)(62 63 85 86)(65 95 88 72)(66 71 89 94)(67 93 90 70)(68 69 91 92)
G:=sub<Sym(96)| (1,44,77,51)(2,52,78,45)(3,46,79,53)(4,54,80,47)(5,48,81,55)(6,56,82,37)(7,38,83,57)(8,58,84,39)(9,40,73,59)(10,60,74,41)(11,42,75,49)(12,50,76,43)(13,63,34,92)(14,93,35,64)(15,65,36,94)(16,95,25,66)(17,67,26,96)(18,85,27,68)(19,69,28,86)(20,87,29,70)(21,71,30,88)(22,89,31,72)(23,61,32,90)(24,91,33,62), (1,31,7,25)(2,26,8,32)(3,33,9,27)(4,28,10,34)(5,35,11,29)(6,30,12,36)(13,80,19,74)(14,75,20,81)(15,82,21,76)(16,77,22,83)(17,84,23,78)(18,79,24,73)(37,71,43,65)(38,66,44,72)(39,61,45,67)(40,68,46,62)(41,63,47,69)(42,70,48,64)(49,87,55,93)(50,94,56,88)(51,89,57,95)(52,96,58,90)(53,91,59,85)(54,86,60,92), (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), (1,6,83,76)(2,75,84,5)(3,4,73,74)(7,12,77,82)(8,81,78,11)(9,10,79,80)(13,27,28,24)(14,23,29,26)(15,25,30,22)(16,21,31,36)(17,35,32,20)(18,19,33,34)(37,57,50,44)(38,43,51,56)(39,55,52,42)(40,41,53,54)(45,49,58,48)(46,47,59,60)(61,87,96,64)(62,63,85,86)(65,95,88,72)(66,71,89,94)(67,93,90,70)(68,69,91,92)>;
G:=Group( (1,44,77,51)(2,52,78,45)(3,46,79,53)(4,54,80,47)(5,48,81,55)(6,56,82,37)(7,38,83,57)(8,58,84,39)(9,40,73,59)(10,60,74,41)(11,42,75,49)(12,50,76,43)(13,63,34,92)(14,93,35,64)(15,65,36,94)(16,95,25,66)(17,67,26,96)(18,85,27,68)(19,69,28,86)(20,87,29,70)(21,71,30,88)(22,89,31,72)(23,61,32,90)(24,91,33,62), (1,31,7,25)(2,26,8,32)(3,33,9,27)(4,28,10,34)(5,35,11,29)(6,30,12,36)(13,80,19,74)(14,75,20,81)(15,82,21,76)(16,77,22,83)(17,84,23,78)(18,79,24,73)(37,71,43,65)(38,66,44,72)(39,61,45,67)(40,68,46,62)(41,63,47,69)(42,70,48,64)(49,87,55,93)(50,94,56,88)(51,89,57,95)(52,96,58,90)(53,91,59,85)(54,86,60,92), (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), (1,6,83,76)(2,75,84,5)(3,4,73,74)(7,12,77,82)(8,81,78,11)(9,10,79,80)(13,27,28,24)(14,23,29,26)(15,25,30,22)(16,21,31,36)(17,35,32,20)(18,19,33,34)(37,57,50,44)(38,43,51,56)(39,55,52,42)(40,41,53,54)(45,49,58,48)(46,47,59,60)(61,87,96,64)(62,63,85,86)(65,95,88,72)(66,71,89,94)(67,93,90,70)(68,69,91,92) );
G=PermutationGroup([[(1,44,77,51),(2,52,78,45),(3,46,79,53),(4,54,80,47),(5,48,81,55),(6,56,82,37),(7,38,83,57),(8,58,84,39),(9,40,73,59),(10,60,74,41),(11,42,75,49),(12,50,76,43),(13,63,34,92),(14,93,35,64),(15,65,36,94),(16,95,25,66),(17,67,26,96),(18,85,27,68),(19,69,28,86),(20,87,29,70),(21,71,30,88),(22,89,31,72),(23,61,32,90),(24,91,33,62)], [(1,31,7,25),(2,26,8,32),(3,33,9,27),(4,28,10,34),(5,35,11,29),(6,30,12,36),(13,80,19,74),(14,75,20,81),(15,82,21,76),(16,77,22,83),(17,84,23,78),(18,79,24,73),(37,71,43,65),(38,66,44,72),(39,61,45,67),(40,68,46,62),(41,63,47,69),(42,70,48,64),(49,87,55,93),(50,94,56,88),(51,89,57,95),(52,96,58,90),(53,91,59,85),(54,86,60,92)], [(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)], [(1,6,83,76),(2,75,84,5),(3,4,73,74),(7,12,77,82),(8,81,78,11),(9,10,79,80),(13,27,28,24),(14,23,29,26),(15,25,30,22),(16,21,31,36),(17,35,32,20),(18,19,33,34),(37,57,50,44),(38,43,51,56),(39,55,52,42),(40,41,53,54),(45,49,58,48),(46,47,59,60),(61,87,96,64),(62,63,85,86),(65,95,88,72),(66,71,89,94),(67,93,90,70),(68,69,91,92)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 4Q | 4R | 4S | 4T | 4U | 4V | 6A | 6B | 6C | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | C4○D4 | D4⋊2S3 | S3×Q8 | Q8⋊3S3 |
| kernel | C42.241D6 | C12⋊2Q8 | S3×C42 | Dic6⋊C4 | C4.Dic6 | C4⋊C4⋊7S3 | C4.D12 | Q8×Dic3 | D6⋊3Q8 | C3×C4⋊Q8 | C4⋊Q8 | C4×S3 | C42 | C4⋊C4 | C2×Q8 | C12 | C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 2 | 2 | 2 |
Matrix representation of C42.241D6 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 5 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 11 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 0 | 12 |
G:=sub<GL(6,GF(13))| [5,5,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[5,5,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,1,0,0,0,0,11,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,2,1,0,0,0,0,0,0,0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12] >;
C42.241D6 in GAP, Magma, Sage, TeX
C_4^2._{241}D_6 % in TeX
G:=Group("C4^2.241D6"); // GroupNames label
G:=SmallGroup(192,1287);
// by ID
G=gap.SmallGroup(192,1287);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,100,1123,570,185,80,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=b^2,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^2*c^5>;
// generators/relations