G = C6.442- 1+4 order 192 = 26·3
44th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.442- 1+4, D6⋊3Q8⋊41C2, C12.261(C2×D4), (C2×C12).216D4, (C2×Q8).212D6, (C22×Q8)⋊15S3, (C2×C6).308C24, D6⋊C4.77C22, C6.156(C22×D4), (C22×C4).294D6, Dic3⋊Q8⋊30C2, C12.23D4⋊29C2, (C2×C12).648C23, (C6×Q8).235C22, (C2×D12).279C22, C23.26D6⋊34C2, C23.28D6⋊29C2, Dic3⋊C4.90C22, C4⋊Dic3.390C22, C22.319(S3×C23), C23.249(C22×S3), (C22×C6).426C23, (C22×S3).134C23, (C22×C12).440C22, C2.44(Q8.15D6), C3⋊6(C23.38C23), (C2×Dic6).308C22, (C4×Dic3).171C22, (C2×Dic3).159C23, C6.D4.132C22, (Q8×C2×C6)⋊7C2, C4.99(C2×C3⋊D4), (C2×C6).590(C2×D4), (C2×C4○D12).25C2, (C2×C4).94(C3⋊D4), (S3×C2×C4).165C22, C2.29(C22×C3⋊D4), C22.37(C2×C3⋊D4), (C2×C4).634(C22×S3), (C2×C3⋊D4).138C22, SmallGroup(192,1375)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.442- 1+4
G = < a,b,c,d,e | a6=b4=1, c2=a3, d2=e2=a3b2, bab-1=cac-1=a-1, ad=da, ae=ea, cbc-1=a3b-1, dbd-1=ebe-1=a3b, dcd-1=ece-1=a3c, ede-1=a3b2d >
Subgroups: 616 in 270 conjugacy classes, 111 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C2×C12, C3×Q8, C22×S3, C22×C6, C42⋊C2, C22⋊Q8, C22.D4, C4.4D4, C4⋊Q8, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C22×C12, C6×Q8, C6×Q8, C23.38C23, C23.26D6, C23.28D6, Dic3⋊Q8, D6⋊3Q8, C12.23D4, C2×C4○D12, Q8×C2×C6, C6.442- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C3⋊D4, C22×S3, C22×D4, 2- 1+4, C2×C3⋊D4, S3×C23, C23.38C23, Q8.15D6, C22×C3⋊D4, C6.442- 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 21 14 29)(2 20 15 28)(3 19 16 27)(4 24 17 26)(5 23 18 25)(6 22 13 30)(7 80 91 88)(8 79 92 87)(9 84 93 86)(10 83 94 85)(11 82 95 90)(12 81 96 89)(31 49 39 47)(32 54 40 46)(33 53 41 45)(34 52 42 44)(35 51 37 43)(36 50 38 48)(55 73 63 71)(56 78 64 70)(57 77 65 69)(58 76 66 68)(59 75 61 67)(60 74 62 72)
(1 45 4 48)(2 44 5 47)(3 43 6 46)(7 56 10 59)(8 55 11 58)(9 60 12 57)(13 54 16 51)(14 53 17 50)(15 52 18 49)(19 40 22 37)(20 39 23 42)(21 38 24 41)(25 34 28 31)(26 33 29 36)(27 32 30 35)(61 91 64 94)(62 96 65 93)(63 95 66 92)(67 83 70 80)(68 82 71 79)(69 81 72 84)(73 87 76 90)(74 86 77 89)(75 85 78 88)
(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 87 30 82)(20 88 25 83)(21 89 26 84)(22 90 27 79)(23 85 28 80)(24 86 29 81)(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 63 54 58)(44 64 49 59)(45 65 50 60)(46 66 51 55)(47 61 52 56)(48 62 53 57)
(1 26 17 21)(2 27 18 22)(3 28 13 23)(4 29 14 24)(5 30 15 19)(6 25 16 20)(7 79 94 90)(8 80 95 85)(9 81 96 86)(10 82 91 87)(11 83 92 88)(12 84 93 89)(31 51 42 46)(32 52 37 47)(33 53 38 48)(34 54 39 43)(35 49 40 44)(36 50 41 45)(55 70 66 75)(56 71 61 76)(57 72 62 77)(58 67 63 78)(59 68 64 73)(60 69 65 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), (1,21,14,29)(2,20,15,28)(3,19,16,27)(4,24,17,26)(5,23,18,25)(6,22,13,30)(7,80,91,88)(8,79,92,87)(9,84,93,86)(10,83,94,85)(11,82,95,90)(12,81,96,89)(31,49,39,47)(32,54,40,46)(33,53,41,45)(34,52,42,44)(35,51,37,43)(36,50,38,48)(55,73,63,71)(56,78,64,70)(57,77,65,69)(58,76,66,68)(59,75,61,67)(60,74,62,72), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,56,10,59)(8,55,11,58)(9,60,12,57)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,40,22,37)(20,39,23,42)(21,38,24,41)(25,34,28,31)(26,33,29,36)(27,32,30,35)(61,91,64,94)(62,96,65,93)(63,95,66,92)(67,83,70,80)(68,82,71,79)(69,81,72,84)(73,87,76,90)(74,86,77,89)(75,85,78,88), (1,93,17,12)(2,94,18,7)(3,95,13,8)(4,96,14,9)(5,91,15,10)(6,92,16,11)(19,87,30,82)(20,88,25,83)(21,89,26,84)(22,90,27,79)(23,85,28,80)(24,86,29,81)(31,70,42,75)(32,71,37,76)(33,72,38,77)(34,67,39,78)(35,68,40,73)(36,69,41,74)(43,63,54,58)(44,64,49,59)(45,65,50,60)(46,66,51,55)(47,61,52,56)(48,62,53,57), (1,26,17,21)(2,27,18,22)(3,28,13,23)(4,29,14,24)(5,30,15,19)(6,25,16,20)(7,79,94,90)(8,80,95,85)(9,81,96,86)(10,82,91,87)(11,83,92,88)(12,84,93,89)(31,51,42,46)(32,52,37,47)(33,53,38,48)(34,54,39,43)(35,49,40,44)(36,50,41,45)(55,70,66,75)(56,71,61,76)(57,72,62,77)(58,67,63,78)(59,68,64,73)(60,69,65,74)>;
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), (1,21,14,29)(2,20,15,28)(3,19,16,27)(4,24,17,26)(5,23,18,25)(6,22,13,30)(7,80,91,88)(8,79,92,87)(9,84,93,86)(10,83,94,85)(11,82,95,90)(12,81,96,89)(31,49,39,47)(32,54,40,46)(33,53,41,45)(34,52,42,44)(35,51,37,43)(36,50,38,48)(55,73,63,71)(56,78,64,70)(57,77,65,69)(58,76,66,68)(59,75,61,67)(60,74,62,72), (1,45,4,48)(2,44,5,47)(3,43,6,46)(7,56,10,59)(8,55,11,58)(9,60,12,57)(13,54,16,51)(14,53,17,50)(15,52,18,49)(19,40,22,37)(20,39,23,42)(21,38,24,41)(25,34,28,31)(26,33,29,36)(27,32,30,35)(61,91,64,94)(62,96,65,93)(63,95,66,92)(67,83,70,80)(68,82,71,79)(69,81,72,84)(73,87,76,90)(74,86,77,89)(75,85,78,88), (1,93,17,12)(2,94,18,7)(3,95,13,8)(4,96,14,9)(5,91,15,10)(6,92,16,11)(19,87,30,82)(20,88,25,83)(21,89,26,84)(22,90,27,79)(23,85,28,80)(24,86,29,81)(31,70,42,75)(32,71,37,76)(33,72,38,77)(34,67,39,78)(35,68,40,73)(36,69,41,74)(43,63,54,58)(44,64,49,59)(45,65,50,60)(46,66,51,55)(47,61,52,56)(48,62,53,57), (1,26,17,21)(2,27,18,22)(3,28,13,23)(4,29,14,24)(5,30,15,19)(6,25,16,20)(7,79,94,90)(8,80,95,85)(9,81,96,86)(10,82,91,87)(11,83,92,88)(12,84,93,89)(31,51,42,46)(32,52,37,47)(33,53,38,48)(34,54,39,43)(35,49,40,44)(36,50,41,45)(55,70,66,75)(56,71,61,76)(57,72,62,77)(58,67,63,78)(59,68,64,73)(60,69,65,74) );
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)], [(1,21,14,29),(2,20,15,28),(3,19,16,27),(4,24,17,26),(5,23,18,25),(6,22,13,30),(7,80,91,88),(8,79,92,87),(9,84,93,86),(10,83,94,85),(11,82,95,90),(12,81,96,89),(31,49,39,47),(32,54,40,46),(33,53,41,45),(34,52,42,44),(35,51,37,43),(36,50,38,48),(55,73,63,71),(56,78,64,70),(57,77,65,69),(58,76,66,68),(59,75,61,67),(60,74,62,72)], [(1,45,4,48),(2,44,5,47),(3,43,6,46),(7,56,10,59),(8,55,11,58),(9,60,12,57),(13,54,16,51),(14,53,17,50),(15,52,18,49),(19,40,22,37),(20,39,23,42),(21,38,24,41),(25,34,28,31),(26,33,29,36),(27,32,30,35),(61,91,64,94),(62,96,65,93),(63,95,66,92),(67,83,70,80),(68,82,71,79),(69,81,72,84),(73,87,76,90),(74,86,77,89),(75,85,78,88)], [(1,93,17,12),(2,94,18,7),(3,95,13,8),(4,96,14,9),(5,91,15,10),(6,92,16,11),(19,87,30,82),(20,88,25,83),(21,89,26,84),(22,90,27,79),(23,85,28,80),(24,86,29,81),(31,70,42,75),(32,71,37,76),(33,72,38,77),(34,67,39,78),(35,68,40,73),(36,69,41,74),(43,63,54,58),(44,64,49,59),(45,65,50,60),(46,66,51,55),(47,61,52,56),(48,62,53,57)], [(1,26,17,21),(2,27,18,22),(3,28,13,23),(4,29,14,24),(5,30,15,19),(6,25,16,20),(7,79,94,90),(8,80,95,85),(9,81,96,86),(10,82,91,87),(11,83,92,88),(12,84,93,89),(31,51,42,46),(32,52,37,47),(33,53,38,48),(34,54,39,43),(35,49,40,44),(36,50,41,45),(55,70,66,75),(56,71,61,76),(57,72,62,77),(58,67,63,78),(59,68,64,73),(60,69,65,74)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C3⋊D4 | 2- 1+4 | Q8.15D6 |
| kernel | C6.442- 1+4 | C23.26D6 | C23.28D6 | Dic3⋊Q8 | D6⋊3Q8 | C12.23D4 | C2×C4○D12 | Q8×C2×C6 | C22×Q8 | C2×C12 | C22×C4 | C2×Q8 | C2×C4 | C6 | C2 |
| # reps | 1 | 1 | 4 | 2 | 4 | 2 | 1 | 1 | 1 | 4 | 3 | 4 | 8 | 2 | 4 |
Matrix representation of C6.442- 1+4 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 2 | 11 | 12 | 1 |
| 0 | 0 | 0 | 11 | 0 | 1 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 4 | 11 | 0 | 0 |
| 0 | 0 | 4 | 4 | 11 | 11 |
| 0 | 0 | 8 | 9 | 9 | 2 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 0 | 5 | 0 |
| 0 | 0 | 0 | 8 | 0 | 5 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 1 | 0 |
| 0 | 0 | 0 | 12 | 0 | 1 |
| 0 | 0 | 11 | 0 | 1 | 0 |
| 0 | 0 | 0 | 11 | 0 | 1 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,1,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,2,0,0,0,12,12,11,11,0,0,0,0,12,0,0,0,0,0,1,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,2,4,4,8,0,0,2,11,4,9,0,0,0,0,11,9,0,0,0,0,11,2],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,0,0,5,0,5,0,0,0,0,5,0,5],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,12,0,11,0,0,0,0,12,0,11,0,0,1,0,1,0,0,0,0,1,0,1] >;
C6.442- 1+4 in GAP, Magma, Sage, TeX
C_6._{44}2_-^{1+4} % in TeX
G:=Group("C6.44ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1375);
// by ID
G=gap.SmallGroup(192,1375);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,675,570,136,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=a^3,d^2=e^2=a^3*b^2,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=a^3*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=a^3*b^2*d>;
// generators/relations