metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4:C4.237D6, C4.64(C2xD12), (C2xC4).47D12, C4.24(D6:C4), (C2xDic6):11C4, (C2xC12).473D4, C12.144(C2xD4), (C22xC6).77D4, C6.SD16:28C2, C42:C2.9S3, C12.66(C22xC4), Dic6.27(C2xC4), (C22xC4).128D6, C12.48(C22:C4), (C2xC12).331C23, C22.24(D6:C4), C2.2(Q8.14D6), C23.62(C3:D4), C3:2(C23.38D4), C6.106(C8.C22), (C22xDic6).12C2, (C22xC12).153C22, (C2xDic6).263C22, C4.53(S3xC2xC4), (C2xC4).45(C4xS3), C2.19(C2xD6:C4), (C2xC12).93(C2xC4), (C2xC6).460(C2xD4), C6.46(C2xC22:C4), (C2xC3:C8).88C22, C22.74(C2xC3:D4), (C2xC4).242(C3:D4), (C3xC4:C4).268C22, (C2xC6).16(C22:C4), (C2xC4).431(C22xS3), (C2xC4.Dic3).19C2, (C3xC42:C2).10C2, SmallGroup(192,563)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4:C4.237D6
G = < a,b,c,d | a4=b4=c6=1, d2=a2, bab-1=dad-1=a-1, ac=ca, cbc-1=a2b, dbd-1=ab-1, dcd-1=c-1 >
Subgroups: 360 in 150 conjugacy classes, 63 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C4, C22, C22, C22, C6, C6, C6, C8, C2xC4, C2xC4, C2xC4, Q8, C23, Dic3, C12, C12, C12, C2xC6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C2xQ8, C3:C8, Dic6, Dic6, C2xDic3, C2xC12, C2xC12, C2xC12, C22xC6, Q8:C4, C42:C2, C2xM4(2), C22xQ8, C2xC3:C8, C4.Dic3, C4xC12, C3xC22:C4, C3xC4:C4, C2xDic6, C2xDic6, C22xDic3, C22xC12, C23.38D4, C6.SD16, C2xC4.Dic3, C3xC42:C2, C22xDic6, C4:C4.237D6
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, C22xC4, C2xD4, C4xS3, D12, C3:D4, C22xS3, C2xC22:C4, C8.C22, D6:C4, S3xC2xC4, C2xD12, C2xC3:D4, C23.38D4, C2xD6:C4, Q8.14D6, C4:C4.237D6
►On 96 points
(1 19 4 22)(2 20 5 23)(3 21 6 24)(7 16 10 13)(8 17 11 14)(9 18 12 15)(25 33 41 35)(26 31 42 36)(27 32 40 34)(28 39 44 48)(29 37 45 46)(30 38 43 47)(49 94 52 91)(50 95 53 92)(51 96 54 93)(55 89 58 86)(56 90 59 87)(57 85 60 88)(61 69 64 72)(62 70 65 67)(63 71 66 68)(73 82 76 79)(74 83 77 80)(75 84 78 81)
(1 92 16 70)(2 96 17 68)(3 94 18 72)(4 95 13 67)(5 93 14 71)(6 91 15 69)(7 65 22 50)(8 63 23 54)(9 61 24 52)(10 62 19 53)(11 66 20 51)(12 64 21 49)(25 56 46 81)(26 60 47 79)(27 58 48 83)(28 74 32 89)(29 78 33 87)(30 76 31 85)(34 86 44 77)(35 90 45 75)(36 88 43 73)(37 84 41 59)(38 82 42 57)(39 80 40 55)
(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 36 4 31)(2 35 5 33)(3 34 6 32)(7 47 10 38)(8 46 11 37)(9 48 12 39)(13 30 16 43)(14 29 17 45)(15 28 18 44)(19 42 22 26)(20 41 23 25)(21 40 24 27)(49 77 52 74)(50 76 53 73)(51 75 54 78)(55 72 58 69)(56 71 59 68)(57 70 60 67)(61 89 64 86)(62 88 65 85)(63 87 66 90)(79 95 82 92)(80 94 83 91)(81 93 84 96)
G:=sub<Sym(96)| (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,33,41,35)(26,31,42,36)(27,32,40,34)(28,39,44,48)(29,37,45,46)(30,38,43,47)(49,94,52,91)(50,95,53,92)(51,96,54,93)(55,89,58,86)(56,90,59,87)(57,85,60,88)(61,69,64,72)(62,70,65,67)(63,71,66,68)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,92,16,70)(2,96,17,68)(3,94,18,72)(4,95,13,67)(5,93,14,71)(6,91,15,69)(7,65,22,50)(8,63,23,54)(9,61,24,52)(10,62,19,53)(11,66,20,51)(12,64,21,49)(25,56,46,81)(26,60,47,79)(27,58,48,83)(28,74,32,89)(29,78,33,87)(30,76,31,85)(34,86,44,77)(35,90,45,75)(36,88,43,73)(37,84,41,59)(38,82,42,57)(39,80,40,55), (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), (1,36,4,31)(2,35,5,33)(3,34,6,32)(7,47,10,38)(8,46,11,37)(9,48,12,39)(13,30,16,43)(14,29,17,45)(15,28,18,44)(19,42,22,26)(20,41,23,25)(21,40,24,27)(49,77,52,74)(50,76,53,73)(51,75,54,78)(55,72,58,69)(56,71,59,68)(57,70,60,67)(61,89,64,86)(62,88,65,85)(63,87,66,90)(79,95,82,92)(80,94,83,91)(81,93,84,96)>;
G:=Group( (1,19,4,22)(2,20,5,23)(3,21,6,24)(7,16,10,13)(8,17,11,14)(9,18,12,15)(25,33,41,35)(26,31,42,36)(27,32,40,34)(28,39,44,48)(29,37,45,46)(30,38,43,47)(49,94,52,91)(50,95,53,92)(51,96,54,93)(55,89,58,86)(56,90,59,87)(57,85,60,88)(61,69,64,72)(62,70,65,67)(63,71,66,68)(73,82,76,79)(74,83,77,80)(75,84,78,81), (1,92,16,70)(2,96,17,68)(3,94,18,72)(4,95,13,67)(5,93,14,71)(6,91,15,69)(7,65,22,50)(8,63,23,54)(9,61,24,52)(10,62,19,53)(11,66,20,51)(12,64,21,49)(25,56,46,81)(26,60,47,79)(27,58,48,83)(28,74,32,89)(29,78,33,87)(30,76,31,85)(34,86,44,77)(35,90,45,75)(36,88,43,73)(37,84,41,59)(38,82,42,57)(39,80,40,55), (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), (1,36,4,31)(2,35,5,33)(3,34,6,32)(7,47,10,38)(8,46,11,37)(9,48,12,39)(13,30,16,43)(14,29,17,45)(15,28,18,44)(19,42,22,26)(20,41,23,25)(21,40,24,27)(49,77,52,74)(50,76,53,73)(51,75,54,78)(55,72,58,69)(56,71,59,68)(57,70,60,67)(61,89,64,86)(62,88,65,85)(63,87,66,90)(79,95,82,92)(80,94,83,91)(81,93,84,96) );
G=PermutationGroup([[(1,19,4,22),(2,20,5,23),(3,21,6,24),(7,16,10,13),(8,17,11,14),(9,18,12,15),(25,33,41,35),(26,31,42,36),(27,32,40,34),(28,39,44,48),(29,37,45,46),(30,38,43,47),(49,94,52,91),(50,95,53,92),(51,96,54,93),(55,89,58,86),(56,90,59,87),(57,85,60,88),(61,69,64,72),(62,70,65,67),(63,71,66,68),(73,82,76,79),(74,83,77,80),(75,84,78,81)], [(1,92,16,70),(2,96,17,68),(3,94,18,72),(4,95,13,67),(5,93,14,71),(6,91,15,69),(7,65,22,50),(8,63,23,54),(9,61,24,52),(10,62,19,53),(11,66,20,51),(12,64,21,49),(25,56,46,81),(26,60,47,79),(27,58,48,83),(28,74,32,89),(29,78,33,87),(30,76,31,85),(34,86,44,77),(35,90,45,75),(36,88,43,73),(37,84,41,59),(38,82,42,57),(39,80,40,55)], [(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)], [(1,36,4,31),(2,35,5,33),(3,34,6,32),(7,47,10,38),(8,46,11,37),(9,48,12,39),(13,30,16,43),(14,29,17,45),(15,28,18,44),(19,42,22,26),(20,41,23,25),(21,40,24,27),(49,77,52,74),(50,76,53,73),(51,75,54,78),(55,72,58,69),(56,71,59,68),(57,70,60,67),(61,89,64,86),(62,88,65,85),(63,87,66,90),(79,95,82,92),(80,94,83,91),(81,93,84,96)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D4 | D6 | D6 | C4xS3 | D12 | C3:D4 | C3:D4 | C8.C22 | Q8.14D6 |
| kernel | C4:C4.237D6 | C6.SD16 | C2xC4.Dic3 | C3xC42:C2 | C22xDic6 | C2xDic6 | C42:C2 | C2xC12 | C22xC6 | C4:C4 | C22xC4 | C2xC4 | C2xC4 | C2xC4 | C23 | C6 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 3 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 |
Matrix representation of C4:C4.237D6 ►in GL6(F73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 59 | 66 | 0 | 0 |
| 0 | 0 | 57 | 12 | 7 | 59 |
| 0 | 0 | 45 | 57 | 14 | 66 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 50 | 48 | 48 |
| 0 | 0 | 23 | 34 | 50 | 48 |
| 0 | 0 | 8 | 9 | 39 | 23 |
| 0 | 0 | 72 | 8 | 50 | 62 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 28 | 12 | 0 | 1 |
| 0 | 0 | 16 | 28 | 72 | 1 |
| 3 | 28 | 0 | 0 | 0 | 0 |
| 31 | 70 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 43 | 0 | 0 |
| 0 | 0 | 13 | 43 | 0 | 0 |
| 0 | 0 | 67 | 26 | 46 | 27 |
| 0 | 0 | 26 | 32 | 0 | 27 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,7,59,57,45,0,0,14,66,12,57,0,0,0,0,7,14,0,0,0,0,59,66],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,11,23,8,72,0,0,50,34,9,8,0,0,48,50,39,50,0,0,48,48,23,62],[0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,1,28,16,0,0,72,0,12,28,0,0,0,0,0,72,0,0,0,0,1,1],[3,31,0,0,0,0,28,70,0,0,0,0,0,0,30,13,67,26,0,0,43,43,26,32,0,0,0,0,46,0,0,0,0,0,27,27] >;
C4:C4.237D6 in GAP, Magma, Sage, TeX
C_4\rtimes C_4._{237}D_6 % in TeX
G:=Group("C4:C4.237D6"); // GroupNames label
G:=SmallGroup(192,563);
// by ID
G=gap.SmallGroup(192,563);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,387,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,c*b*c^-1=a^2*b,d*b*d^-1=a*b^-1,d*c*d^-1=c^-1>;
// generators/relations