metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.277D6, (C4×D12)⋊4C2, (C2×C42)⋊13S3, (C4×Dic6)⋊4C2, (C2×C6).23C24, C42⋊3S3⋊21C2, C42⋊2S3⋊27C2, C42⋊7S3⋊34C2, C12⋊7D4.20C2, D6⋊C4.82C22, C12.6Q8⋊32C2, (C22×C4).426D6, C4.118(C4○D12), C12.234(C4○D4), C12.48D4⋊50C2, (C4×C12).316C22, (C2×C12).696C23, (C22×S3).5C23, C22.66(S3×C23), (C2×Dic3).7C23, (C2×D12).204C22, C22.22(C4○D12), C23.28D6⋊32C2, Dic3⋊C4.96C22, C4⋊Dic3.290C22, C23.229(C22×S3), (C22×C6).385C23, (C22×C12).565C22, C3⋊1(C23.36C23), (C4×Dic3).191C22, (C2×Dic6).225C22, C6.D4.81C22, (C2×C4×C12)⋊14C2, (C4×C3⋊D4)⋊32C2, C6.10(C2×C4○D4), C2.12(C2×C4○D12), (C2×C6).99(C4○D4), (S3×C2×C4).188C22, (C2×C4).651(C22×S3), (C2×C3⋊D4).86C22, SmallGroup(192,1038)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 536 in 234 conjugacy classes, 103 normal (51 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×4], C4 [×10], C22, C22 [×2], C22 [×8], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×6], Q8 [×2], C23, C23 [×2], Dic3 [×6], C12 [×4], C12 [×4], D6 [×6], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×2], C2×D4 [×3], C2×Q8, Dic6 [×2], C4×S3 [×4], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×6], C2×C12 [×6], C22×S3 [×2], C22×C6, C2×C42, C42⋊C2 [×2], C4×D4 [×3], C4×Q8, C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2 [×2], C4×Dic3 [×2], Dic3⋊C4 [×8], C4⋊Dic3 [×2], D6⋊C4 [×8], C6.D4 [×2], C4×C12 [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C2×C3⋊D4 [×2], C22×C12 [×3], C23.36C23, C4×Dic6, C12.6Q8, C42⋊2S3 [×2], C4×D12, C42⋊7S3, C42⋊3S3 [×2], C12.48D4, C4×C3⋊D4 [×2], C23.28D6 [×2], C12⋊7D4, C2×C4×C12, C42.277D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×6], C24, C22×S3 [×7], C2×C4○D4 [×3], C4○D12 [×6], S3×C23, C23.36C23, C2×C4○D12 [×3], C42.277D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=b2, ab=ba, ac=ca, dad-1=ab2, bc=cb, dbd-1=a2b, dcd-1=b2c-1 >
►On 96 points
(1 72 57 22)(2 67 58 23)(3 68 59 24)(4 69 60 19)(5 70 55 20)(6 71 56 21)(7 66 25 16)(8 61 26 17)(9 62 27 18)(10 63 28 13)(11 64 29 14)(12 65 30 15)(31 90 75 46)(32 85 76 47)(33 86 77 48)(34 87 78 43)(35 88 73 44)(36 89 74 45)(37 96 81 50)(38 91 82 51)(39 92 83 52)(40 93 84 53)(41 94 79 54)(42 95 80 49)
(1 79 73 13)(2 80 74 14)(3 81 75 15)(4 82 76 16)(5 83 77 17)(6 84 78 18)(7 69 51 47)(8 70 52 48)(9 71 53 43)(10 72 54 44)(11 67 49 45)(12 68 50 46)(19 91 85 25)(20 92 86 26)(21 93 87 27)(22 94 88 28)(23 95 89 29)(24 96 90 30)(31 65 59 37)(32 66 60 38)(33 61 55 39)(34 62 56 40)(35 63 57 41)(36 64 58 42)
(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 73 78)(2 77 74 5)(3 4 75 76)(7 30 51 96)(8 95 52 29)(9 28 53 94)(10 93 54 27)(11 26 49 92)(12 91 50 25)(13 62 79 40)(14 39 80 61)(15 66 81 38)(16 37 82 65)(17 64 83 42)(18 41 84 63)(19 24 85 90)(20 89 86 23)(21 22 87 88)(31 32 59 60)(33 36 55 58)(34 57 56 35)(43 44 71 72)(45 48 67 70)(46 69 68 47)
G:=sub<Sym(96)| (1,72,57,22)(2,67,58,23)(3,68,59,24)(4,69,60,19)(5,70,55,20)(6,71,56,21)(7,66,25,16)(8,61,26,17)(9,62,27,18)(10,63,28,13)(11,64,29,14)(12,65,30,15)(31,90,75,46)(32,85,76,47)(33,86,77,48)(34,87,78,43)(35,88,73,44)(36,89,74,45)(37,96,81,50)(38,91,82,51)(39,92,83,52)(40,93,84,53)(41,94,79,54)(42,95,80,49), (1,79,73,13)(2,80,74,14)(3,81,75,15)(4,82,76,16)(5,83,77,17)(6,84,78,18)(7,69,51,47)(8,70,52,48)(9,71,53,43)(10,72,54,44)(11,67,49,45)(12,68,50,46)(19,91,85,25)(20,92,86,26)(21,93,87,27)(22,94,88,28)(23,95,89,29)(24,96,90,30)(31,65,59,37)(32,66,60,38)(33,61,55,39)(34,62,56,40)(35,63,57,41)(36,64,58,42), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,30,51,96)(8,95,52,29)(9,28,53,94)(10,93,54,27)(11,26,49,92)(12,91,50,25)(13,62,79,40)(14,39,80,61)(15,66,81,38)(16,37,82,65)(17,64,83,42)(18,41,84,63)(19,24,85,90)(20,89,86,23)(21,22,87,88)(31,32,59,60)(33,36,55,58)(34,57,56,35)(43,44,71,72)(45,48,67,70)(46,69,68,47)>;
G:=Group( (1,72,57,22)(2,67,58,23)(3,68,59,24)(4,69,60,19)(5,70,55,20)(6,71,56,21)(7,66,25,16)(8,61,26,17)(9,62,27,18)(10,63,28,13)(11,64,29,14)(12,65,30,15)(31,90,75,46)(32,85,76,47)(33,86,77,48)(34,87,78,43)(35,88,73,44)(36,89,74,45)(37,96,81,50)(38,91,82,51)(39,92,83,52)(40,93,84,53)(41,94,79,54)(42,95,80,49), (1,79,73,13)(2,80,74,14)(3,81,75,15)(4,82,76,16)(5,83,77,17)(6,84,78,18)(7,69,51,47)(8,70,52,48)(9,71,53,43)(10,72,54,44)(11,67,49,45)(12,68,50,46)(19,91,85,25)(20,92,86,26)(21,93,87,27)(22,94,88,28)(23,95,89,29)(24,96,90,30)(31,65,59,37)(32,66,60,38)(33,61,55,39)(34,62,56,40)(35,63,57,41)(36,64,58,42), (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), (1,6,73,78)(2,77,74,5)(3,4,75,76)(7,30,51,96)(8,95,52,29)(9,28,53,94)(10,93,54,27)(11,26,49,92)(12,91,50,25)(13,62,79,40)(14,39,80,61)(15,66,81,38)(16,37,82,65)(17,64,83,42)(18,41,84,63)(19,24,85,90)(20,89,86,23)(21,22,87,88)(31,32,59,60)(33,36,55,58)(34,57,56,35)(43,44,71,72)(45,48,67,70)(46,69,68,47) );
G=PermutationGroup([(1,72,57,22),(2,67,58,23),(3,68,59,24),(4,69,60,19),(5,70,55,20),(6,71,56,21),(7,66,25,16),(8,61,26,17),(9,62,27,18),(10,63,28,13),(11,64,29,14),(12,65,30,15),(31,90,75,46),(32,85,76,47),(33,86,77,48),(34,87,78,43),(35,88,73,44),(36,89,74,45),(37,96,81,50),(38,91,82,51),(39,92,83,52),(40,93,84,53),(41,94,79,54),(42,95,80,49)], [(1,79,73,13),(2,80,74,14),(3,81,75,15),(4,82,76,16),(5,83,77,17),(6,84,78,18),(7,69,51,47),(8,70,52,48),(9,71,53,43),(10,72,54,44),(11,67,49,45),(12,68,50,46),(19,91,85,25),(20,92,86,26),(21,93,87,27),(22,94,88,28),(23,95,89,29),(24,96,90,30),(31,65,59,37),(32,66,60,38),(33,61,55,39),(34,62,56,40),(35,63,57,41),(36,64,58,42)], [(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)], [(1,6,73,78),(2,77,74,5),(3,4,75,76),(7,30,51,96),(8,95,52,29),(9,28,53,94),(10,93,54,27),(11,26,49,92),(12,91,50,25),(13,62,79,40),(14,39,80,61),(15,66,81,38),(16,37,82,65),(17,64,83,42),(18,41,84,63),(19,24,85,90),(20,89,86,23),(21,22,87,88),(31,32,59,60),(33,36,55,58),(34,57,56,35),(43,44,71,72),(45,48,67,70),(46,69,68,47)])
Matrix representation ►G ⊆ GL4(𝔽13) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 2 | 4 |
| 0 | 0 | 9 | 11 |
| 1 | 11 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 2 | 11 |
| 0 | 0 | 2 | 4 |
| 12 | 0 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 11 | 9 |
| 0 | 0 | 11 | 2 |
G:=sub<GL(4,GF(13))| [8,0,0,0,0,8,0,0,0,0,2,9,0,0,4,11],[1,0,0,0,11,12,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,2,2,0,0,11,4],[12,12,0,0,0,1,0,0,0,0,11,11,0,0,9,2] >;
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T | 6A | ··· | 6G | 12A | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | C4○D4 | C4○D4 | C4○D12 | C4○D12 |
| kernel | C42.277D6 | C4×Dic6 | C12.6Q8 | C42⋊2S3 | C4×D12 | C42⋊7S3 | C42⋊3S3 | C12.48D4 | C4×C3⋊D4 | C23.28D6 | C12⋊7D4 | C2×C4×C12 | C2×C42 | C42 | C22×C4 | C12 | C2×C6 | C4 | C22 |
| # reps | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 3 | 8 | 4 | 16 | 8 |
In GAP, Magma, Sage, TeX
C_4^2._{277}D_6 % in TeX
G:=Group("C4^2.277D6"); // GroupNames label
G:=SmallGroup(192,1038);
// by ID
G=gap.SmallGroup(192,1038);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,675,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=b^2*c^-1>;
// generators/relations