G = C6.722- 1+4 order 192 = 26·3
27th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C4○D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.722- 1+4, C6.392+ 1+4, (C4×S3)⋊2D4, D6.3(C2×D4), C4⋊D4⋊11S3, C4⋊C4.180D6, (C2×D4).92D6, C4.184(S3×D4), C22⋊C4.8D6, Dic3⋊D4⋊19C2, D6⋊3D4⋊19C2, C4.D12⋊21C2, C12.228(C2×D4), C6.67(C22×D4), (C2×C6).152C24, (C2×C12).39C23, D6⋊C4.15C22, C2.30(Q8○D12), Dic3.46(C2×D4), (C22×C4).239D6, C23.14D6⋊13C2, C2.41(D4⋊6D6), C12.48D4⋊34C2, (C6×D4).122C22, C23.26(C22×S3), Dic3.D4⋊19C2, (C2×D12).220C22, Dic3⋊C4.18C22, C4⋊Dic3.207C22, (C22×C6).187C23, C22.173(S3×C23), (C2×Dic3).73C23, (C22×S3).187C23, (C22×C12).241C22, C3⋊3(C22.31C24), (C2×Dic6).154C22, C6.D4.25C22, (C22×Dic3).109C22, (S3×C4⋊C4)⋊21C2, C2.40(C2×S3×D4), (C2×C4○D12)⋊21C2, (C3×C4⋊D4)⋊14C2, (S3×C2×C4).83C22, (C2×D4⋊2S3)⋊13C2, (C3×C4⋊C4).144C22, (C2×C4).586(C22×S3), (C2×C3⋊D4).28C22, (C3×C22⋊C4).13C22, SmallGroup(192,1167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.722- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=dad-1=a-1, ac=ca, ae=ea, cbc=b-1, dbd-1=ebe-1=a3b, dcd-1=ece-1=a3c, ede-1=b2d >
Subgroups: 784 in 294 conjugacy classes, 103 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C2×C4⋊C4, C4⋊D4, C4⋊D4, C22⋊Q8, C2×C4○D4, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, D4⋊2S3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, C22.31C24, Dic3.D4, Dic3⋊D4, S3×C4⋊C4, C4.D12, C12.48D4, D6⋊3D4, D6⋊3D4, C23.14D6, C3×C4⋊D4, C2×C4○D12, C2×D4⋊2S3, C6.722- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, 2- 1+4, S3×D4, S3×C23, C22.31C24, C2×S3×D4, D4⋊6D6, Q8○D12, C6.722- 1+4
►On 96 points
(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 91 18 9)(2 96 13 8)(3 95 14 7)(4 94 15 12)(5 93 16 11)(6 92 17 10)(19 79 29 89)(20 84 30 88)(21 83 25 87)(22 82 26 86)(23 81 27 85)(24 80 28 90)(31 77 41 67)(32 76 42 72)(33 75 37 71)(34 74 38 70)(35 73 39 69)(36 78 40 68)(43 58 53 62)(44 57 54 61)(45 56 49 66)(46 55 50 65)(47 60 51 64)(48 59 52 63)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 90)(8 85)(9 86)(10 87)(11 88)(12 89)(13 23)(14 24)(15 19)(16 20)(17 21)(18 22)(31 50)(32 51)(33 52)(34 53)(35 54)(36 49)(37 48)(38 43)(39 44)(40 45)(41 46)(42 47)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(79 94)(80 95)(81 96)(82 91)(83 92)(84 93)
(1 70 15 77)(2 69 16 76)(3 68 17 75)(4 67 18 74)(5 72 13 73)(6 71 14 78)(7 37 92 36)(8 42 93 35)(9 41 94 34)(10 40 95 33)(11 39 96 32)(12 38 91 31)(19 58 26 65)(20 57 27 64)(21 56 28 63)(22 55 29 62)(23 60 30 61)(24 59 25 66)(43 79 50 86)(44 84 51 85)(45 83 52 90)(46 82 53 89)(47 81 54 88)(48 80 49 87)
(1 91 15 12)(2 92 16 7)(3 93 17 8)(4 94 18 9)(5 95 13 10)(6 96 14 11)(19 86 26 79)(20 87 27 80)(21 88 28 81)(22 89 29 82)(23 90 30 83)(24 85 25 84)(31 67 38 74)(32 68 39 75)(33 69 40 76)(34 70 41 77)(35 71 42 78)(36 72 37 73)(43 55 50 62)(44 56 51 63)(45 57 52 64)(46 58 53 65)(47 59 54 66)(48 60 49 61)
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), (1,91,18,9)(2,96,13,8)(3,95,14,7)(4,94,15,12)(5,93,16,11)(6,92,17,10)(19,79,29,89)(20,84,30,88)(21,83,25,87)(22,82,26,86)(23,81,27,85)(24,80,28,90)(31,77,41,67)(32,76,42,72)(33,75,37,71)(34,74,38,70)(35,73,39,69)(36,78,40,68)(43,58,53,62)(44,57,54,61)(45,56,49,66)(46,55,50,65)(47,60,51,64)(48,59,52,63), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,70,15,77)(2,69,16,76)(3,68,17,75)(4,67,18,74)(5,72,13,73)(6,71,14,78)(7,37,92,36)(8,42,93,35)(9,41,94,34)(10,40,95,33)(11,39,96,32)(12,38,91,31)(19,58,26,65)(20,57,27,64)(21,56,28,63)(22,55,29,62)(23,60,30,61)(24,59,25,66)(43,79,50,86)(44,84,51,85)(45,83,52,90)(46,82,53,89)(47,81,54,88)(48,80,49,87), (1,91,15,12)(2,92,16,7)(3,93,17,8)(4,94,18,9)(5,95,13,10)(6,96,14,11)(19,86,26,79)(20,87,27,80)(21,88,28,81)(22,89,29,82)(23,90,30,83)(24,85,25,84)(31,67,38,74)(32,68,39,75)(33,69,40,76)(34,70,41,77)(35,71,42,78)(36,72,37,73)(43,55,50,62)(44,56,51,63)(45,57,52,64)(46,58,53,65)(47,59,54,66)(48,60,49,61)>;
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), (1,91,18,9)(2,96,13,8)(3,95,14,7)(4,94,15,12)(5,93,16,11)(6,92,17,10)(19,79,29,89)(20,84,30,88)(21,83,25,87)(22,82,26,86)(23,81,27,85)(24,80,28,90)(31,77,41,67)(32,76,42,72)(33,75,37,71)(34,74,38,70)(35,73,39,69)(36,78,40,68)(43,58,53,62)(44,57,54,61)(45,56,49,66)(46,55,50,65)(47,60,51,64)(48,59,52,63), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,90)(8,85)(9,86)(10,87)(11,88)(12,89)(13,23)(14,24)(15,19)(16,20)(17,21)(18,22)(31,50)(32,51)(33,52)(34,53)(35,54)(36,49)(37,48)(38,43)(39,44)(40,45)(41,46)(42,47)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(79,94)(80,95)(81,96)(82,91)(83,92)(84,93), (1,70,15,77)(2,69,16,76)(3,68,17,75)(4,67,18,74)(5,72,13,73)(6,71,14,78)(7,37,92,36)(8,42,93,35)(9,41,94,34)(10,40,95,33)(11,39,96,32)(12,38,91,31)(19,58,26,65)(20,57,27,64)(21,56,28,63)(22,55,29,62)(23,60,30,61)(24,59,25,66)(43,79,50,86)(44,84,51,85)(45,83,52,90)(46,82,53,89)(47,81,54,88)(48,80,49,87), (1,91,15,12)(2,92,16,7)(3,93,17,8)(4,94,18,9)(5,95,13,10)(6,96,14,11)(19,86,26,79)(20,87,27,80)(21,88,28,81)(22,89,29,82)(23,90,30,83)(24,85,25,84)(31,67,38,74)(32,68,39,75)(33,69,40,76)(34,70,41,77)(35,71,42,78)(36,72,37,73)(43,55,50,62)(44,56,51,63)(45,57,52,64)(46,58,53,65)(47,59,54,66)(48,60,49,61) );
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)], [(1,91,18,9),(2,96,13,8),(3,95,14,7),(4,94,15,12),(5,93,16,11),(6,92,17,10),(19,79,29,89),(20,84,30,88),(21,83,25,87),(22,82,26,86),(23,81,27,85),(24,80,28,90),(31,77,41,67),(32,76,42,72),(33,75,37,71),(34,74,38,70),(35,73,39,69),(36,78,40,68),(43,58,53,62),(44,57,54,61),(45,56,49,66),(46,55,50,65),(47,60,51,64),(48,59,52,63)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,25),(7,90),(8,85),(9,86),(10,87),(11,88),(12,89),(13,23),(14,24),(15,19),(16,20),(17,21),(18,22),(31,50),(32,51),(33,52),(34,53),(35,54),(36,49),(37,48),(38,43),(39,44),(40,45),(41,46),(42,47),(55,77),(56,78),(57,73),(58,74),(59,75),(60,76),(61,69),(62,70),(63,71),(64,72),(65,67),(66,68),(79,94),(80,95),(81,96),(82,91),(83,92),(84,93)], [(1,70,15,77),(2,69,16,76),(3,68,17,75),(4,67,18,74),(5,72,13,73),(6,71,14,78),(7,37,92,36),(8,42,93,35),(9,41,94,34),(10,40,95,33),(11,39,96,32),(12,38,91,31),(19,58,26,65),(20,57,27,64),(21,56,28,63),(22,55,29,62),(23,60,30,61),(24,59,25,66),(43,79,50,86),(44,84,51,85),(45,83,52,90),(46,82,53,89),(47,81,54,88),(48,80,49,87)], [(1,91,15,12),(2,92,16,7),(3,93,17,8),(4,94,18,9),(5,95,13,10),(6,96,14,11),(19,86,26,79),(20,87,27,80),(21,88,28,81),(22,89,29,82),(23,90,30,83),(24,85,25,84),(31,67,38,74),(32,68,39,75),(33,69,40,76),(34,70,41,77),(35,71,42,78),(36,72,37,73),(43,55,50,62),(44,56,51,63),(45,57,52,64),(46,58,53,65),(47,59,54,66),(48,60,49,61)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4L | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | ··· | 12 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 4 | 4 | 4 | 4 | 8 | 8 |
36 irreducible representations
| dim | 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 | S3 | D4 | D6 | D6 | D6 | D6 | 2+ 1+4 | 2- 1+4 | S3×D4 | D4⋊6D6 | Q8○D12 |
| kernel | C6.722- 1+4 | Dic3.D4 | Dic3⋊D4 | S3×C4⋊C4 | C4.D12 | C12.48D4 | D6⋊3D4 | C23.14D6 | C3×C4⋊D4 | C2×C4○D12 | C2×D4⋊2S3 | C4⋊D4 | C4×S3 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C6 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 3 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 1 | 1 | 3 | 1 | 1 | 2 | 2 | 2 |
Matrix representation of C6.722- 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 3 | 10 |
| 0 | 0 | 10 | 7 | 0 | 0 |
| 0 | 0 | 10 | 3 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 8 | 4 | 0 |
| 0 | 0 | 4 | 9 | 9 | 9 |
| 0 | 0 | 4 | 0 | 4 | 8 |
| 0 | 0 | 9 | 9 | 4 | 9 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 6 |
| 0 | 0 | 0 | 0 | 7 | 10 |
| 0 | 0 | 3 | 6 | 0 | 0 |
| 0 | 0 | 7 | 10 | 0 | 0 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,10,10,0,0,0,0,7,3,0,0,3,3,0,0,0,0,6,10,0,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,4,4,4,9,0,0,8,9,0,9,0,0,4,9,4,4,0,0,0,9,8,9],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,6,10,0,0,3,7,0,0,0,0,6,10,0,0] >;
C6.722- 1+4 in GAP, Magma, Sage, TeX
C_6._{72}2_-^{1+4} % in TeX
G:=Group("C6.72ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1167);
// by ID
G=gap.SmallGroup(192,1167);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,1123,570,185,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=c^2=1,d^2=e^2=a^3*b^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=a^3*b,d*c*d^-1=e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;
// generators/relations