G = C6.452- 1+4 order 192 = 26·3
45th non-split extension by C6 of 2- 1+4 acting via 2- 1+4/C2×Q8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.452- 1+4, (C3×Q8)⋊17D4, Q8⋊9(C3⋊D4), C3⋊6(Q8⋊5D4), C12⋊7D4⋊38C2, D6⋊3Q8⋊42C2, (Q8×Dic3)⋊27C2, C12.262(C2×D4), (C2×Q8).213D6, (C22×Q8)⋊16S3, (C2×C6).309C24, D6⋊C4.78C22, (C22×C4).295D6, C6.157(C22×D4), C12.23D4⋊30C2, (C2×C12).649C23, C22⋊4(Q8⋊3S3), (C6×Q8).236C22, (C2×D12).183C22, C4⋊Dic3.258C22, (C22×C6).427C23, C22.320(S3×C23), C23.250(C22×S3), Dic3⋊C4.171C22, (C22×S3).135C23, (C22×C12).287C22, C2.45(Q8.15D6), (C4×Dic3).172C22, (C2×Dic3).290C23, C6.D4.146C22, (Q8×C2×C6)⋊8C2, (C4×C3⋊D4)⋊27C2, C4.70(C2×C3⋊D4), (C2×C6)⋊18(C4○D4), C6.129(C2×C4○D4), (C2×Q8⋊3S3)⋊18C2, (S3×C2×C4).166C22, C2.36(C2×Q8⋊3S3), C2.30(C22×C3⋊D4), (C2×C4).244(C22×S3), (C2×C3⋊D4).139C22, SmallGroup(192,1376)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C6.452- 1+4
G = < a,b,c,d,e | a6=b4=c2=1, d2=e2=a3b2, bab-1=a-1, ac=ca, ad=da, ae=ea, cbc=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede-1=a3b2d >
Subgroups: 696 in 290 conjugacy classes, 115 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×C6, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C4.4D4, C22×Q8, C2×C4○D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, S3×C2×C4, C2×D12, Q8⋊3S3, C2×C3⋊D4, C22×C12, C6×Q8, C6×Q8, C6×Q8, Q8⋊5D4, C4×C3⋊D4, C12⋊7D4, Q8×Dic3, D6⋊3Q8, C12.23D4, C2×Q8⋊3S3, Q8×C2×C6, C6.452- 1+4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22×D4, C2×C4○D4, 2- 1+4, Q8⋊3S3, C2×C3⋊D4, S3×C23, Q8⋊5D4, C2×Q8⋊3S3, Q8.15D6, C22×C3⋊D4, C6.452- 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 12 15 91)(2 11 16 96)(3 10 17 95)(4 9 18 94)(5 8 13 93)(6 7 14 92)(19 86 26 79)(20 85 27 84)(21 90 28 83)(22 89 29 82)(23 88 30 81)(24 87 25 80)(31 77 38 70)(32 76 39 69)(33 75 40 68)(34 74 41 67)(35 73 42 72)(36 78 37 71)(43 58 50 65)(44 57 51 64)(45 56 52 63)(46 55 53 62)(47 60 54 61)(48 59 49 66)
(1 22)(2 23)(3 24)(4 19)(5 20)(6 21)(7 83)(8 84)(9 79)(10 80)(11 81)(12 82)(13 27)(14 28)(15 29)(16 30)(17 25)(18 26)(31 53)(32 54)(33 49)(34 50)(35 51)(36 52)(37 45)(38 46)(39 47)(40 48)(41 43)(42 44)(55 77)(56 78)(57 73)(58 74)(59 75)(60 76)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)
(1 50 18 46)(2 51 13 47)(3 52 14 48)(4 53 15 43)(5 54 16 44)(6 49 17 45)(7 63 95 59)(8 64 96 60)(9 65 91 55)(10 66 92 56)(11 61 93 57)(12 62 94 58)(19 31 29 41)(20 32 30 42)(21 33 25 37)(22 34 26 38)(23 35 27 39)(24 36 28 40)(67 89 77 79)(68 90 78 80)(69 85 73 81)(70 86 74 82)(71 87 75 83)(72 88 76 84)
(1 58 18 62)(2 59 13 63)(3 60 14 64)(4 55 15 65)(5 56 16 66)(6 57 17 61)(7 51 95 47)(8 52 96 48)(9 53 91 43)(10 54 92 44)(11 49 93 45)(12 50 94 46)(19 77 29 67)(20 78 30 68)(21 73 25 69)(22 74 26 70)(23 75 27 71)(24 76 28 72)(31 89 41 79)(32 90 42 80)(33 85 37 81)(34 86 38 82)(35 87 39 83)(36 88 40 84)
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), (1,12,15,91)(2,11,16,96)(3,10,17,95)(4,9,18,94)(5,8,13,93)(6,7,14,92)(19,86,26,79)(20,85,27,84)(21,90,28,83)(22,89,29,82)(23,88,30,81)(24,87,25,80)(31,77,38,70)(32,76,39,69)(33,75,40,68)(34,74,41,67)(35,73,42,72)(36,78,37,71)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,27)(14,28)(15,29)(16,30)(17,25)(18,26)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,63,95,59)(8,64,96,60)(9,65,91,55)(10,66,92,56)(11,61,93,57)(12,62,94,58)(19,31,29,41)(20,32,30,42)(21,33,25,37)(22,34,26,38)(23,35,27,39)(24,36,28,40)(67,89,77,79)(68,90,78,80)(69,85,73,81)(70,86,74,82)(71,87,75,83)(72,88,76,84), (1,58,18,62)(2,59,13,63)(3,60,14,64)(4,55,15,65)(5,56,16,66)(6,57,17,61)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84)>;
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), (1,12,15,91)(2,11,16,96)(3,10,17,95)(4,9,18,94)(5,8,13,93)(6,7,14,92)(19,86,26,79)(20,85,27,84)(21,90,28,83)(22,89,29,82)(23,88,30,81)(24,87,25,80)(31,77,38,70)(32,76,39,69)(33,75,40,68)(34,74,41,67)(35,73,42,72)(36,78,37,71)(43,58,50,65)(44,57,51,64)(45,56,52,63)(46,55,53,62)(47,60,54,61)(48,59,49,66), (1,22)(2,23)(3,24)(4,19)(5,20)(6,21)(7,83)(8,84)(9,79)(10,80)(11,81)(12,82)(13,27)(14,28)(15,29)(16,30)(17,25)(18,26)(31,53)(32,54)(33,49)(34,50)(35,51)(36,52)(37,45)(38,46)(39,47)(40,48)(41,43)(42,44)(55,77)(56,78)(57,73)(58,74)(59,75)(60,76)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (1,50,18,46)(2,51,13,47)(3,52,14,48)(4,53,15,43)(5,54,16,44)(6,49,17,45)(7,63,95,59)(8,64,96,60)(9,65,91,55)(10,66,92,56)(11,61,93,57)(12,62,94,58)(19,31,29,41)(20,32,30,42)(21,33,25,37)(22,34,26,38)(23,35,27,39)(24,36,28,40)(67,89,77,79)(68,90,78,80)(69,85,73,81)(70,86,74,82)(71,87,75,83)(72,88,76,84), (1,58,18,62)(2,59,13,63)(3,60,14,64)(4,55,15,65)(5,56,16,66)(6,57,17,61)(7,51,95,47)(8,52,96,48)(9,53,91,43)(10,54,92,44)(11,49,93,45)(12,50,94,46)(19,77,29,67)(20,78,30,68)(21,73,25,69)(22,74,26,70)(23,75,27,71)(24,76,28,72)(31,89,41,79)(32,90,42,80)(33,85,37,81)(34,86,38,82)(35,87,39,83)(36,88,40,84) );
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)], [(1,12,15,91),(2,11,16,96),(3,10,17,95),(4,9,18,94),(5,8,13,93),(6,7,14,92),(19,86,26,79),(20,85,27,84),(21,90,28,83),(22,89,29,82),(23,88,30,81),(24,87,25,80),(31,77,38,70),(32,76,39,69),(33,75,40,68),(34,74,41,67),(35,73,42,72),(36,78,37,71),(43,58,50,65),(44,57,51,64),(45,56,52,63),(46,55,53,62),(47,60,54,61),(48,59,49,66)], [(1,22),(2,23),(3,24),(4,19),(5,20),(6,21),(7,83),(8,84),(9,79),(10,80),(11,81),(12,82),(13,27),(14,28),(15,29),(16,30),(17,25),(18,26),(31,53),(32,54),(33,49),(34,50),(35,51),(36,52),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44),(55,77),(56,78),(57,73),(58,74),(59,75),(60,76),(61,69),(62,70),(63,71),(64,72),(65,67),(66,68),(85,93),(86,94),(87,95),(88,96),(89,91),(90,92)], [(1,50,18,46),(2,51,13,47),(3,52,14,48),(4,53,15,43),(5,54,16,44),(6,49,17,45),(7,63,95,59),(8,64,96,60),(9,65,91,55),(10,66,92,56),(11,61,93,57),(12,62,94,58),(19,31,29,41),(20,32,30,42),(21,33,25,37),(22,34,26,38),(23,35,27,39),(24,36,28,40),(67,89,77,79),(68,90,78,80),(69,85,73,81),(70,86,74,82),(71,87,75,83),(72,88,76,84)], [(1,58,18,62),(2,59,13,63),(3,60,14,64),(4,55,15,65),(5,56,16,66),(6,57,17,61),(7,51,95,47),(8,52,96,48),(9,53,91,43),(10,54,92,44),(11,49,93,45),(12,50,94,46),(19,77,29,67),(20,78,30,68),(21,73,25,69),(22,74,26,70),(23,75,27,71),(24,76,28,72),(31,89,41,79),(32,90,42,80),(33,85,37,81),(34,86,38,82),(35,87,39,83),(36,88,40,84)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 4O | 4P | 6A | ··· | 6G | 12A | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 12 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | 2- 1+4 | Q8⋊3S3 | Q8.15D6 |
| kernel | C6.452- 1+4 | C4×C3⋊D4 | C12⋊7D4 | Q8×Dic3 | D6⋊3Q8 | C12.23D4 | C2×Q8⋊3S3 | Q8×C2×C6 | C22×Q8 | C3×Q8 | C22×C4 | C2×Q8 | C2×C6 | Q8 | C6 | C22 | C2 |
| # reps | 1 | 3 | 3 | 1 | 3 | 3 | 1 | 1 | 1 | 4 | 3 | 4 | 4 | 8 | 1 | 2 | 2 |
Matrix representation of C6.452- 1+4 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 4 | 11 | 0 | 0 |
| 2 | 9 | 0 | 0 |
| 0 | 0 | 5 | 5 |
| 0 | 0 | 3 | 8 |
| 2 | 9 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 11 | 4 | 0 | 0 |
| 9 | 2 | 0 | 0 |
| 0 | 0 | 8 | 8 |
| 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 11 | 12 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,1,0,0,0,0,12,0,0,0,0,12],[4,2,0,0,11,9,0,0,0,0,5,3,0,0,5,8],[2,4,0,0,9,11,0,0,0,0,12,0,0,0,0,12],[11,9,0,0,4,2,0,0,0,0,8,0,0,0,8,5],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,1,12] >;
C6.452- 1+4 in GAP, Magma, Sage, TeX
C_6._{45}2_-^{1+4} % in TeX
G:=Group("C6.45ES-(2,2)"); // GroupNames label
G:=SmallGroup(192,1376);
// by ID
G=gap.SmallGroup(192,1376);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,387,184,675,6278]);
// 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=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^3*b^2*d>;
// generators/relations