metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.61D6, Dic6.23D4, C4.49(S3×D4), C12⋊C8⋊28C2, (C2×D4).44D6, C12.23(C2×D4), (C2×Q8).58D6, (C4×Dic6)⋊21C2, (C2×C12).269D4, C3⋊5(Q8.D4), C4.4D4.4S3, C6.103(C4○D8), C12.65(C4○D4), C4.1(D4⋊2S3), Q8⋊2Dic3⋊19C2, (C6×D4).60C22, (C6×Q8).52C22, C2.10(D6⋊3D4), C6.101(C4⋊D4), (C4×C12).103C22, (C2×C12).372C23, D4⋊Dic3.11C2, C2.17(Q8.14D6), C2.22(Q8.13D6), C6.118(C8.C22), C4⋊Dic3.340C22, (C2×Dic6).272C22, (C2×C3⋊Q16)⋊12C2, (C2×C6).503(C2×D4), (C2×D4.S3).6C2, (C2×C4).59(C3⋊D4), (C2×C3⋊C8).119C22, (C3×C4.4D4).2C2, (C2×C4).472(C22×S3), C22.178(C2×C3⋊D4), SmallGroup(192,613)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42.61D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2b, ab=ba, cac-1=dad-1=a-1b2, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >
Subgroups: 288 in 112 conjugacy classes, 41 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, C3⋊C8, Dic6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C3⋊Q16, C4×C12, C3×C22⋊C4, C2×Dic6, C6×D4, C6×Q8, Q8.D4, C12⋊C8, D4⋊Dic3, Q8⋊2Dic3, C4×Dic6, C2×D4.S3, C2×C3⋊Q16, C3×C4.4D4, C42.61D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C4○D8, C8.C22, S3×D4, D4⋊2S3, C2×C3⋊D4, Q8.D4, D6⋊3D4, Q8.13D6, Q8.14D6, C42.61D6
►On 96 points
(1 30 75 54)(2 51 76 27)(3 32 77 56)(4 53 78 29)(5 26 79 50)(6 55 80 31)(7 28 73 52)(8 49 74 25)(9 57 21 40)(10 37 22 62)(11 59 23 34)(12 39 24 64)(13 61 17 36)(14 33 18 58)(15 63 19 38)(16 35 20 60)(41 67 92 82)(42 87 93 72)(43 69 94 84)(44 81 95 66)(45 71 96 86)(46 83 89 68)(47 65 90 88)(48 85 91 70)
(1 77 5 73)(2 78 6 74)(3 79 7 75)(4 80 8 76)(9 23 13 19)(10 24 14 20)(11 17 15 21)(12 18 16 22)(25 51 29 55)(26 52 30 56)(27 53 31 49)(28 54 32 50)(33 60 37 64)(34 61 38 57)(35 62 39 58)(36 63 40 59)(41 94 45 90)(42 95 46 91)(43 96 47 92)(44 89 48 93)(65 82 69 86)(66 83 70 87)(67 84 71 88)(68 85 72 81)
(1 95 58)(2 36 96 78 59 47)(3 93 60 7 89 64)(4 34 90 76 61 45)(5 91 62)(6 40 92 74 63 43)(8 38 94 80 57 41)(9 71 25 11 69 27)(10 54 70 18 26 81)(12 52 72 24 28 87)(13 67 29 15 65 31)(14 50 66 22 30 85)(16 56 68 20 32 83)(17 82 53 19 88 55)(21 86 49 23 84 51)(33 75 44)(35 73 46 39 77 42)(37 79 48)
(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)
G:=sub<Sym(96)| (1,30,75,54)(2,51,76,27)(3,32,77,56)(4,53,78,29)(5,26,79,50)(6,55,80,31)(7,28,73,52)(8,49,74,25)(9,57,21,40)(10,37,22,62)(11,59,23,34)(12,39,24,64)(13,61,17,36)(14,33,18,58)(15,63,19,38)(16,35,20,60)(41,67,92,82)(42,87,93,72)(43,69,94,84)(44,81,95,66)(45,71,96,86)(46,83,89,68)(47,65,90,88)(48,85,91,70), (1,77,5,73)(2,78,6,74)(3,79,7,75)(4,80,8,76)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59)(41,94,45,90)(42,95,46,91)(43,96,47,92)(44,89,48,93)(65,82,69,86)(66,83,70,87)(67,84,71,88)(68,85,72,81), (1,95,58)(2,36,96,78,59,47)(3,93,60,7,89,64)(4,34,90,76,61,45)(5,91,62)(6,40,92,74,63,43)(8,38,94,80,57,41)(9,71,25,11,69,27)(10,54,70,18,26,81)(12,52,72,24,28,87)(13,67,29,15,65,31)(14,50,66,22,30,85)(16,56,68,20,32,83)(17,82,53,19,88,55)(21,86,49,23,84,51)(33,75,44)(35,73,46,39,77,42)(37,79,48), (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)>;
G:=Group( (1,30,75,54)(2,51,76,27)(3,32,77,56)(4,53,78,29)(5,26,79,50)(6,55,80,31)(7,28,73,52)(8,49,74,25)(9,57,21,40)(10,37,22,62)(11,59,23,34)(12,39,24,64)(13,61,17,36)(14,33,18,58)(15,63,19,38)(16,35,20,60)(41,67,92,82)(42,87,93,72)(43,69,94,84)(44,81,95,66)(45,71,96,86)(46,83,89,68)(47,65,90,88)(48,85,91,70), (1,77,5,73)(2,78,6,74)(3,79,7,75)(4,80,8,76)(9,23,13,19)(10,24,14,20)(11,17,15,21)(12,18,16,22)(25,51,29,55)(26,52,30,56)(27,53,31,49)(28,54,32,50)(33,60,37,64)(34,61,38,57)(35,62,39,58)(36,63,40,59)(41,94,45,90)(42,95,46,91)(43,96,47,92)(44,89,48,93)(65,82,69,86)(66,83,70,87)(67,84,71,88)(68,85,72,81), (1,95,58)(2,36,96,78,59,47)(3,93,60,7,89,64)(4,34,90,76,61,45)(5,91,62)(6,40,92,74,63,43)(8,38,94,80,57,41)(9,71,25,11,69,27)(10,54,70,18,26,81)(12,52,72,24,28,87)(13,67,29,15,65,31)(14,50,66,22,30,85)(16,56,68,20,32,83)(17,82,53,19,88,55)(21,86,49,23,84,51)(33,75,44)(35,73,46,39,77,42)(37,79,48), (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) );
G=PermutationGroup([[(1,30,75,54),(2,51,76,27),(3,32,77,56),(4,53,78,29),(5,26,79,50),(6,55,80,31),(7,28,73,52),(8,49,74,25),(9,57,21,40),(10,37,22,62),(11,59,23,34),(12,39,24,64),(13,61,17,36),(14,33,18,58),(15,63,19,38),(16,35,20,60),(41,67,92,82),(42,87,93,72),(43,69,94,84),(44,81,95,66),(45,71,96,86),(46,83,89,68),(47,65,90,88),(48,85,91,70)], [(1,77,5,73),(2,78,6,74),(3,79,7,75),(4,80,8,76),(9,23,13,19),(10,24,14,20),(11,17,15,21),(12,18,16,22),(25,51,29,55),(26,52,30,56),(27,53,31,49),(28,54,32,50),(33,60,37,64),(34,61,38,57),(35,62,39,58),(36,63,40,59),(41,94,45,90),(42,95,46,91),(43,96,47,92),(44,89,48,93),(65,82,69,86),(66,83,70,87),(67,84,71,88),(68,85,72,81)], [(1,95,58),(2,36,96,78,59,47),(3,93,60,7,89,64),(4,34,90,76,61,45),(5,91,62),(6,40,92,74,63,43),(8,38,94,80,57,41),(9,71,25,11,69,27),(10,54,70,18,26,81),(12,52,72,24,28,87),(13,67,29,15,65,31),(14,50,66,22,30,85),(16,56,68,20,32,83),(17,82,53,19,88,55),(21,86,49,23,84,51),(33,75,44),(35,73,46,39,77,42),(37,79,48)], [(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)]])
33 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 | ··· | 12F | 12G | 12H |
| 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 |
| size | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 8 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 8 | 8 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | C4○D8 | C8.C22 | S3×D4 | D4⋊2S3 | Q8.13D6 | Q8.14D6 |
| kernel | C42.61D6 | C12⋊C8 | D4⋊Dic3 | Q8⋊2Dic3 | C4×Dic6 | C2×D4.S3 | C2×C3⋊Q16 | C3×C4.4D4 | C4.4D4 | Dic6 | C2×C12 | C42 | C2×D4 | C2×Q8 | C12 | C2×C4 | C6 | C6 | C4 | C4 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C42.61D6 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 71 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 11 | 71 | 0 | 0 | 0 | 0 |
| 60 | 62 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 57 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,71,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72],[11,60,0,0,0,0,71,62,0,0,0,0,0,0,0,57,0,0,0,0,32,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C42.61D6 in GAP, Magma, Sage, TeX
C_4^2._{61}D_6 % in TeX
G:=Group("C4^2.61D6"); // GroupNames label
G:=SmallGroup(192,613);
// by ID
G=gap.SmallGroup(192,613);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,344,254,219,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations