metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.92D6, C6.502- (1+4), C4⋊C4.269D6, C12⋊2Q8⋊7C2, C4.72(C2×D12), (C2×C4).56D12, C42⋊7S3⋊4C2, C4.D12⋊11C2, C12.288(C2×D4), (C2×C12).202D4, (C2×C6).70C24, (C4×C12).8C22, D6⋊C4.2C22, C2.8(Q8○D12), C22⋊C4.94D6, C6.14(C22×D4), C42⋊C2⋊10S3, C22.21(C2×D12), C2.16(C22×D12), (C22×C4).207D6, (C2×C12).145C23, (C22×Dic6)⋊15C2, C22.D12⋊4C2, C4⋊Dic3.33C22, C22.99(S3×C23), (C2×D12).206C22, (C22×S3).20C23, C23.168(C22×S3), (C22×C6).140C23, (C2×Dic3).24C23, (C22×C12).230C22, C3⋊1(C23.38C23), (C2×Dic6).285C22, (C22×Dic3).87C22, (C2×C6).51(C2×D4), (S3×C2×C4).59C22, (C2×C4○D12).19C2, (C3×C42⋊C2)⋊12C2, (C3×C4⋊C4).307C22, (C2×C4).576(C22×S3), (C2×C3⋊D4).101C22, (C3×C22⋊C4).102C22, SmallGroup(192,1085)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 680 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], S3 [×2], C6, C6 [×2], C6 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×14], D4 [×6], Q8 [×10], C23, C23 [×2], Dic3 [×6], C12 [×4], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×8], C22×C4, C22×C4 [×4], C2×D4 [×3], C2×Q8 [×9], C4○D4 [×4], Dic6 [×10], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×8], C22×S3 [×2], C22×C6, C42⋊C2, C22⋊Q8 [×4], C22.D4 [×4], C4.4D4 [×2], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C4⋊Dic3 [×8], D6⋊C4 [×8], C4×C12 [×2], C3×C22⋊C4 [×2], C3×C4⋊C4 [×2], C2×Dic6, C2×Dic6 [×4], C2×Dic6 [×4], S3×C2×C4 [×2], C2×D12, C4○D12 [×4], C22×Dic3 [×2], C2×C3⋊D4 [×2], C22×C12, C23.38C23, C12⋊2Q8 [×2], C42⋊7S3 [×2], C22.D12 [×4], C4.D12 [×4], C3×C42⋊C2, C22×Dic6, C2×C4○D12, C42.92D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, 2- (1+4) [×2], C2×D12 [×6], S3×C23, C23.38C23, C22×D12, Q8○D12 [×2], C42.92D6
Generators and relations
G = < a,b,c,d | a4=b4=1, c6=d2=a2, ab=ba, ac=ca, dad-1=a-1, cbc-1=a2b, dbd-1=b-1, dcd-1=c5 >
►On 96 points
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)(25 67 31 61)(26 68 32 62)(27 69 33 63)(28 70 34 64)(29 71 35 65)(30 72 36 66)(37 74 43 80)(38 75 44 81)(39 76 45 82)(40 77 46 83)(41 78 47 84)(42 79 48 73)(49 89 55 95)(50 90 56 96)(51 91 57 85)(52 92 58 86)(53 93 59 87)(54 94 60 88)
(1 65 76 53)(2 72 77 60)(3 67 78 55)(4 62 79 50)(5 69 80 57)(6 64 81 52)(7 71 82 59)(8 66 83 54)(9 61 84 49)(10 68 73 56)(11 63 74 51)(12 70 75 58)(13 31 47 95)(14 26 48 90)(15 33 37 85)(16 28 38 92)(17 35 39 87)(18 30 40 94)(19 25 41 89)(20 32 42 96)(21 27 43 91)(22 34 44 86)(23 29 45 93)(24 36 46 88)
(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 22 7 16)(2 15 8 21)(3 20 9 14)(4 13 10 19)(5 18 11 24)(6 23 12 17)(25 50 31 56)(26 55 32 49)(27 60 33 54)(28 53 34 59)(29 58 35 52)(30 51 36 57)(37 83 43 77)(38 76 44 82)(39 81 45 75)(40 74 46 80)(41 79 47 73)(42 84 48 78)(61 90 67 96)(62 95 68 89)(63 88 69 94)(64 93 70 87)(65 86 71 92)(66 91 72 85)
G:=sub<Sym(96)| (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(37,74,43,80)(38,75,44,81)(39,76,45,82)(40,77,46,83)(41,78,47,84)(42,79,48,73)(49,89,55,95)(50,90,56,96)(51,91,57,85)(52,92,58,86)(53,93,59,87)(54,94,60,88), (1,65,76,53)(2,72,77,60)(3,67,78,55)(4,62,79,50)(5,69,80,57)(6,64,81,52)(7,71,82,59)(8,66,83,54)(9,61,84,49)(10,68,73,56)(11,63,74,51)(12,70,75,58)(13,31,47,95)(14,26,48,90)(15,33,37,85)(16,28,38,92)(17,35,39,87)(18,30,40,94)(19,25,41,89)(20,32,42,96)(21,27,43,91)(22,34,44,86)(23,29,45,93)(24,36,46,88), (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), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,83,43,77)(38,76,44,82)(39,81,45,75)(40,74,46,80)(41,79,47,73)(42,84,48,78)(61,90,67,96)(62,95,68,89)(63,88,69,94)(64,93,70,87)(65,86,71,92)(66,91,72,85)>;
G:=Group( (1,23,7,17)(2,24,8,18)(3,13,9,19)(4,14,10,20)(5,15,11,21)(6,16,12,22)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(37,74,43,80)(38,75,44,81)(39,76,45,82)(40,77,46,83)(41,78,47,84)(42,79,48,73)(49,89,55,95)(50,90,56,96)(51,91,57,85)(52,92,58,86)(53,93,59,87)(54,94,60,88), (1,65,76,53)(2,72,77,60)(3,67,78,55)(4,62,79,50)(5,69,80,57)(6,64,81,52)(7,71,82,59)(8,66,83,54)(9,61,84,49)(10,68,73,56)(11,63,74,51)(12,70,75,58)(13,31,47,95)(14,26,48,90)(15,33,37,85)(16,28,38,92)(17,35,39,87)(18,30,40,94)(19,25,41,89)(20,32,42,96)(21,27,43,91)(22,34,44,86)(23,29,45,93)(24,36,46,88), (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), (1,22,7,16)(2,15,8,21)(3,20,9,14)(4,13,10,19)(5,18,11,24)(6,23,12,17)(25,50,31,56)(26,55,32,49)(27,60,33,54)(28,53,34,59)(29,58,35,52)(30,51,36,57)(37,83,43,77)(38,76,44,82)(39,81,45,75)(40,74,46,80)(41,79,47,73)(42,84,48,78)(61,90,67,96)(62,95,68,89)(63,88,69,94)(64,93,70,87)(65,86,71,92)(66,91,72,85) );
G=PermutationGroup([(1,23,7,17),(2,24,8,18),(3,13,9,19),(4,14,10,20),(5,15,11,21),(6,16,12,22),(25,67,31,61),(26,68,32,62),(27,69,33,63),(28,70,34,64),(29,71,35,65),(30,72,36,66),(37,74,43,80),(38,75,44,81),(39,76,45,82),(40,77,46,83),(41,78,47,84),(42,79,48,73),(49,89,55,95),(50,90,56,96),(51,91,57,85),(52,92,58,86),(53,93,59,87),(54,94,60,88)], [(1,65,76,53),(2,72,77,60),(3,67,78,55),(4,62,79,50),(5,69,80,57),(6,64,81,52),(7,71,82,59),(8,66,83,54),(9,61,84,49),(10,68,73,56),(11,63,74,51),(12,70,75,58),(13,31,47,95),(14,26,48,90),(15,33,37,85),(16,28,38,92),(17,35,39,87),(18,30,40,94),(19,25,41,89),(20,32,42,96),(21,27,43,91),(22,34,44,86),(23,29,45,93),(24,36,46,88)], [(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)], [(1,22,7,16),(2,15,8,21),(3,20,9,14),(4,13,10,19),(5,18,11,24),(6,23,12,17),(25,50,31,56),(26,55,32,49),(27,60,33,54),(28,53,34,59),(29,58,35,52),(30,51,36,57),(37,83,43,77),(38,76,44,82),(39,81,45,75),(40,74,46,80),(41,79,47,73),(42,84,48,78),(61,90,67,96),(62,95,68,89),(63,88,69,94),(64,93,70,87),(65,86,71,92),(66,91,72,85)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 6 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 7 |
| 0 | 0 | 0 | 0 | 6 | 3 |
| 1 | 8 | 0 | 0 | 0 | 0 |
| 3 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 4 | 2 | 0 |
| 0 | 0 | 9 | 11 | 0 | 2 |
| 0 | 0 | 12 | 0 | 11 | 9 |
| 0 | 0 | 0 | 12 | 4 | 2 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 11 | 8 | 8 |
| 0 | 0 | 2 | 10 | 5 | 0 |
| 0 | 0 | 6 | 7 | 5 | 2 |
| 0 | 0 | 6 | 12 | 11 | 3 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 10 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 11 | 2 |
| 0 | 0 | 8 | 10 | 4 | 2 |
| 0 | 0 | 12 | 1 | 11 | 3 |
| 0 | 0 | 2 | 1 | 5 | 2 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,10,6,0,0,0,0,7,3,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[1,3,0,0,0,0,8,12,0,0,0,0,0,0,2,9,12,0,0,0,4,11,0,12,0,0,2,0,11,4,0,0,0,2,9,2],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,2,6,6,0,0,11,10,7,12,0,0,8,5,5,11,0,0,8,0,2,3],[12,10,0,0,0,0,0,1,0,0,0,0,0,0,3,8,12,2,0,0,11,10,1,1,0,0,11,4,11,5,0,0,2,2,3,2] >;
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | ··· | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | D6 | D12 | 2- (1+4) | Q8○D12 |
| kernel | C42.92D6 | C12⋊2Q8 | C42⋊7S3 | C22.D12 | C4.D12 | C3×C42⋊C2 | C22×Dic6 | C2×C4○D12 | C42⋊C2 | C2×C12 | C42 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×C4 | C6 | C2 |
| # reps | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 8 | 2 | 4 |
In GAP, Magma, Sage, TeX
C_4^2._{92}D_6 % in TeX
G:=Group("C4^2.92D6"); // GroupNames label
G:=SmallGroup(192,1085);
// by ID
G=gap.SmallGroup(192,1085);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,297,192,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=1,c^6=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations