G = C6.242- 1+4 order 192 = 26·3
24th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.242- 1+4, C6.1192+ 1+4, C12⋊Q8⋊28C2, C4⋊C4.99D6, C22⋊Q8⋊20S3, C4.D12⋊29C2, (Q8×Dic3)⋊17C2, (C2×Q8).157D6, C22⋊C4.22D6, Dic3⋊5D4⋊30C2, C23.9D6⋊27C2, C2.37(D4○D12), C12⋊7D4.16C2, (C2×C6).187C24, D6⋊C4.28C22, (C22×C4).265D6, C12.214(C4○D4), C4.73(D4⋊2S3), C12.23D4⋊16C2, (C2×C12).175C23, (C6×Q8).117C22, C23.11D6⋊27C2, (C2×D12).153C22, C23.26D6⋊31C2, Dic3⋊C4.35C22, (C22×S3).78C23, C4⋊Dic3.312C22, C22.208(S3×C23), C23.135(C22×S3), (C22×C6).215C23, (C2×Dic3).94C23, (C22×C12).262C22, C2.25(Q8.15D6), C3⋊6(C22.36C24), (C2×Dic6).164C22, (C4×Dic3).115C22, C6.D4.126C22, C4⋊C4⋊S3⋊22C2, C6.91(C2×C4○D4), (C3×C22⋊Q8)⋊23C2, C2.50(C2×D4⋊2S3), (S3×C2×C4).104C22, (C3×C4⋊C4).168C22, (C2×C4).593(C22×S3), (C2×C3⋊D4).39C22, (C3×C22⋊C4).42C22, SmallGroup(192,1202)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.242- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=b2, e2=a3b2, bab-1=cac=a-1, ad=da, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, cd=dc, ece-1=a3c, ede-1=b2d >
Subgroups: 544 in 216 conjugacy classes, 95 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C42⋊2C2, C4⋊Q8, C4×Dic3, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C6×Q8, C22.36C24, C23.9D6, C23.11D6, C12⋊Q8, Dic3⋊5D4, C4.D12, C4⋊C4⋊S3, C23.26D6, C12⋊7D4, Q8×Dic3, C12.23D4, C3×C22⋊Q8, C6.242- 1+4
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, 2+ 1+4, 2- 1+4, D4⋊2S3, S3×C23, C22.36C24, C2×D4⋊2S3, Q8.15D6, D4○D12, C6.242- 1+4
►On 96 points
Generators in S96
(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 24 17 29)(2 23 18 28)(3 22 13 27)(4 21 14 26)(5 20 15 25)(6 19 16 30)(7 80 94 85)(8 79 95 90)(9 84 96 89)(10 83 91 88)(11 82 92 87)(12 81 93 86)(31 52 42 47)(32 51 37 46)(33 50 38 45)(34 49 39 44)(35 54 40 43)(36 53 41 48)(55 76 66 71)(56 75 61 70)(57 74 62 69)(58 73 63 68)(59 78 64 67)(60 77 65 72)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 59)(8 58)(9 57)(10 56)(11 55)(12 60)(13 51)(14 50)(15 49)(16 54)(17 53)(18 52)(19 40)(20 39)(21 38)(22 37)(23 42)(24 41)(25 34)(26 33)(27 32)(28 31)(29 36)(30 35)(61 91)(62 96)(63 95)(64 94)(65 93)(66 92)(67 80)(68 79)(69 84)(70 83)(71 82)(72 81)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)
(1 93 17 12)(2 94 18 7)(3 95 13 8)(4 96 14 9)(5 91 15 10)(6 92 16 11)(19 90 30 79)(20 85 25 80)(21 86 26 81)(22 87 27 82)(23 88 28 83)(24 89 29 84)(31 70 42 75)(32 71 37 76)(33 72 38 77)(34 67 39 78)(35 68 40 73)(36 69 41 74)(43 66 54 55)(44 61 49 56)(45 62 50 57)(46 63 51 58)(47 64 52 59)(48 65 53 60)
(1 29 14 21)(2 30 15 22)(3 25 16 23)(4 26 17 24)(5 27 18 19)(6 28 13 20)(7 79 91 87)(8 80 92 88)(9 81 93 89)(10 82 94 90)(11 83 95 85)(12 84 96 86)(31 54 39 46)(32 49 40 47)(33 50 41 48)(34 51 42 43)(35 52 37 44)(36 53 38 45)(55 67 63 75)(56 68 64 76)(57 69 65 77)(58 70 66 78)(59 71 61 73)(60 72 62 74)
G:=sub<Sym(96)| (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), (1,24,17,29)(2,23,18,28)(3,22,13,27)(4,21,14,26)(5,20,15,25)(6,19,16,30)(7,80,94,85)(8,79,95,90)(9,84,96,89)(10,83,91,88)(11,82,92,87)(12,81,93,86)(31,52,42,47)(32,51,37,46)(33,50,38,45)(34,49,39,44)(35,54,40,43)(36,53,41,48)(55,76,66,71)(56,75,61,70)(57,74,62,69)(58,73,63,68)(59,78,64,67)(60,77,65,72), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,59)(8,58)(9,57)(10,56)(11,55)(12,60)(13,51)(14,50)(15,49)(16,54)(17,53)(18,52)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41)(25,34)(26,33)(27,32)(28,31)(29,36)(30,35)(61,91)(62,96)(63,95)(64,94)(65,93)(66,92)(67,80)(68,79)(69,84)(70,83)(71,82)(72,81)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85), (1,93,17,12)(2,94,18,7)(3,95,13,8)(4,96,14,9)(5,91,15,10)(6,92,16,11)(19,90,30,79)(20,85,25,80)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(31,70,42,75)(32,71,37,76)(33,72,38,77)(34,67,39,78)(35,68,40,73)(36,69,41,74)(43,66,54,55)(44,61,49,56)(45,62,50,57)(46,63,51,58)(47,64,52,59)(48,65,53,60), (1,29,14,21)(2,30,15,22)(3,25,16,23)(4,26,17,24)(5,27,18,19)(6,28,13,20)(7,79,91,87)(8,80,92,88)(9,81,93,89)(10,82,94,90)(11,83,95,85)(12,84,96,86)(31,54,39,46)(32,49,40,47)(33,50,41,48)(34,51,42,43)(35,52,37,44)(36,53,38,45)(55,67,63,75)(56,68,64,76)(57,69,65,77)(58,70,66,78)(59,71,61,73)(60,72,62,74)>;
G:=Group( (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), (1,24,17,29)(2,23,18,28)(3,22,13,27)(4,21,14,26)(5,20,15,25)(6,19,16,30)(7,80,94,85)(8,79,95,90)(9,84,96,89)(10,83,91,88)(11,82,92,87)(12,81,93,86)(31,52,42,47)(32,51,37,46)(33,50,38,45)(34,49,39,44)(35,54,40,43)(36,53,41,48)(55,76,66,71)(56,75,61,70)(57,74,62,69)(58,73,63,68)(59,78,64,67)(60,77,65,72), (1,48)(2,47)(3,46)(4,45)(5,44)(6,43)(7,59)(8,58)(9,57)(10,56)(11,55)(12,60)(13,51)(14,50)(15,49)(16,54)(17,53)(18,52)(19,40)(20,39)(21,38)(22,37)(23,42)(24,41)(25,34)(26,33)(27,32)(28,31)(29,36)(30,35)(61,91)(62,96)(63,95)(64,94)(65,93)(66,92)(67,80)(68,79)(69,84)(70,83)(71,82)(72,81)(73,90)(74,89)(75,88)(76,87)(77,86)(78,85), (1,93,17,12)(2,94,18,7)(3,95,13,8)(4,96,14,9)(5,91,15,10)(6,92,16,11)(19,90,30,79)(20,85,25,80)(21,86,26,81)(22,87,27,82)(23,88,28,83)(24,89,29,84)(31,70,42,75)(32,71,37,76)(33,72,38,77)(34,67,39,78)(35,68,40,73)(36,69,41,74)(43,66,54,55)(44,61,49,56)(45,62,50,57)(46,63,51,58)(47,64,52,59)(48,65,53,60), (1,29,14,21)(2,30,15,22)(3,25,16,23)(4,26,17,24)(5,27,18,19)(6,28,13,20)(7,79,91,87)(8,80,92,88)(9,81,93,89)(10,82,94,90)(11,83,95,85)(12,84,96,86)(31,54,39,46)(32,49,40,47)(33,50,41,48)(34,51,42,43)(35,52,37,44)(36,53,38,45)(55,67,63,75)(56,68,64,76)(57,69,65,77)(58,70,66,78)(59,71,61,73)(60,72,62,74) );
G=PermutationGroup([[(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)], [(1,24,17,29),(2,23,18,28),(3,22,13,27),(4,21,14,26),(5,20,15,25),(6,19,16,30),(7,80,94,85),(8,79,95,90),(9,84,96,89),(10,83,91,88),(11,82,92,87),(12,81,93,86),(31,52,42,47),(32,51,37,46),(33,50,38,45),(34,49,39,44),(35,54,40,43),(36,53,41,48),(55,76,66,71),(56,75,61,70),(57,74,62,69),(58,73,63,68),(59,78,64,67),(60,77,65,72)], [(1,48),(2,47),(3,46),(4,45),(5,44),(6,43),(7,59),(8,58),(9,57),(10,56),(11,55),(12,60),(13,51),(14,50),(15,49),(16,54),(17,53),(18,52),(19,40),(20,39),(21,38),(22,37),(23,42),(24,41),(25,34),(26,33),(27,32),(28,31),(29,36),(30,35),(61,91),(62,96),(63,95),(64,94),(65,93),(66,92),(67,80),(68,79),(69,84),(70,83),(71,82),(72,81),(73,90),(74,89),(75,88),(76,87),(77,86),(78,85)], [(1,93,17,12),(2,94,18,7),(3,95,13,8),(4,96,14,9),(5,91,15,10),(6,92,16,11),(19,90,30,79),(20,85,25,80),(21,86,26,81),(22,87,27,82),(23,88,28,83),(24,89,29,84),(31,70,42,75),(32,71,37,76),(33,72,38,77),(34,67,39,78),(35,68,40,73),(36,69,41,74),(43,66,54,55),(44,61,49,56),(45,62,50,57),(46,63,51,58),(47,64,52,59),(48,65,53,60)], [(1,29,14,21),(2,30,15,22),(3,25,16,23),(4,26,17,24),(5,27,18,19),(6,28,13,20),(7,79,91,87),(8,80,92,88),(9,81,93,89),(10,82,94,90),(11,83,95,85),(12,84,96,86),(31,54,39,46),(32,49,40,47),(33,50,41,48),(34,51,42,43),(35,52,37,44),(36,53,38,45),(55,67,63,75),(56,68,64,76),(57,69,65,77),(58,70,66,78),(59,71,61,73),(60,72,62,74)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | ··· | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 12 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | D6 | C4○D4 | 2+ 1+4 | 2- 1+4 | D4⋊2S3 | Q8.15D6 | D4○D12 |
| kernel | C6.242- 1+4 | C23.9D6 | C23.11D6 | C12⋊Q8 | Dic3⋊5D4 | C4.D12 | C4⋊C4⋊S3 | C23.26D6 | C12⋊7D4 | Q8×Dic3 | C12.23D4 | C3×C22⋊Q8 | C22⋊Q8 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | C12 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 3 | 1 | 1 | 4 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.242- 1+4 ►in GL8(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 12 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 0 | 12 | 11 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12 |
| 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 10 | 7 | 8 |
| 0 | 0 | 0 | 0 | 3 | 4 | 8 | 6 |
| 0 | 0 | 0 | 0 | 10 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 3 | 3 | 9 |
| 0 | 12 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 12 | 12 | 0 |
G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1,0,0,0,0,0,2,0,0,12],[8,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,3,10,0,0,0,0,0,10,4,0,3,0,0,0,0,7,8,9,3,0,0,0,0,8,6,10,9],[0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0] >;
C6.242- 1+4 in GAP, Magma, Sage, TeX
C_6._{24}2_-^{1+4} % in TeX
G:=Group("C6.24ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1202);
// by ID
G=gap.SmallGroup(192,1202);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,675,570,192,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=b^2,e^2=a^3*b^2,b*a*b^-1=c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;
// generators/relations