?
G = C6.522+ (1+4) order 192 = 26·3
52nd non-split extension by C6 of 2+ (1+4) acting via 2+ (1+4)/C2×D4=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C6.522+ (1+4), C3⋊D4⋊3Q8, C4⋊C4.97D6, C3⋊6(D4⋊3Q8), C22⋊Q8⋊13S3, D6.10(C2×Q8), (C2×Q8).99D6, D6⋊Q8⋊22C2, D6⋊3Q8⋊18C2, C22.2(S3×Q8), C22⋊C4.61D6, C6.38(C22×Q8), Dic3.Q8⋊19C2, (C2×C12).59C23, (C2×C6).180C24, D6⋊C4.25C22, Dic3.12(C2×Q8), (C22×C4).258D6, Dic6⋊C4⋊27C2, Dic3⋊Q8⋊16C2, C2.54(D4⋊6D6), Dic3⋊4D4.3C2, (C6×Q8).111C22, Dic3.24(C4○D4), Dic3.D4⋊26C2, Dic3⋊C4.30C22, C4⋊Dic3.217C22, C23.203(C22×S3), C22.201(S3×C23), (C22×C6).208C23, (C2×Dic3).91C23, (C22×S3).202C23, (C22×C12).380C22, (C2×Dic6).161C22, (C4×Dic3).109C22, C6.D4.120C22, (C22×Dic3).121C22, (S3×C4⋊C4)⋊28C2, C2.21(C2×S3×Q8), (C2×C6).9(C2×Q8), C2.51(S3×C4○D4), C6.163(C2×C4○D4), (C4×C3⋊D4).18C2, (C3×C22⋊Q8)⋊16C2, (C2×Dic3⋊C4)⋊41C2, (S3×C2×C4).100C22, (C2×C4).50(C22×S3), (C3×C4⋊C4).162C22, (C2×C3⋊D4).127C22, (C3×C22⋊C4).35C22, SmallGroup(192,1195)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 528 in 228 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×15], C22, C22 [×2], C22 [×6], S3 [×2], C6 [×3], C6 [×2], C2×C4 [×6], C2×C4 [×15], D4 [×4], Q8 [×4], C23, C23, Dic3 [×4], Dic3 [×5], C12 [×6], D6 [×2], D6 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×4], C4⋊C4 [×3], C4⋊C4 [×13], C22×C4, C22×C4 [×5], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×3], C4×S3 [×4], C2×Dic3 [×7], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×6], C2×C12, C3×Q8, C22×S3, C22×C6, C2×C4⋊C4 [×2], C4×D4 [×3], C4×Q8, C22⋊Q8, C22⋊Q8 [×5], C42.C2 [×2], C4⋊Q8, C4×Dic3 [×3], Dic3⋊C4 [×11], C4⋊Dic3 [×2], D6⋊C4 [×3], C6.D4, C3×C22⋊C4 [×2], C3×C4⋊C4 [×3], C2×Dic6 [×2], S3×C2×C4 [×3], C22×Dic3 [×2], C2×C3⋊D4, C22×C12, C6×Q8, D4⋊3Q8, Dic3.D4 [×2], Dic3⋊4D4 [×2], Dic6⋊C4, Dic3.Q8 [×2], S3×C4⋊C4, D6⋊Q8 [×2], C2×Dic3⋊C4, C4×C3⋊D4, Dic3⋊Q8, D6⋊3Q8, C3×C22⋊Q8, C6.522+ (1+4)
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D6 [×7], C2×Q8 [×6], C4○D4 [×2], C24, C22×S3 [×7], C22×Q8, C2×C4○D4, 2+ (1+4), S3×Q8 [×2], S3×C23, D4⋊3Q8, D4⋊6D6, C2×S3×Q8, S3×C4○D4, C6.522+ (1+4)
Generators and relations
G = < a,b,c,d,e | a6=b4=e2=1, c2=a3, d2=b2, ab=ba, ac=ca, dad-1=eae=a-1, cbc-1=b-1, dbd-1=a3b, be=eb, cd=dc, ce=ec, ede=a3b2d >
►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 53 14 43)(2 54 15 44)(3 49 16 45)(4 50 17 46)(5 51 18 47)(6 52 13 48)(7 59 95 63)(8 60 96 64)(9 55 91 65)(10 56 92 66)(11 57 93 61)(12 58 94 62)(19 41 29 31)(20 42 30 32)(21 37 25 33)(22 38 26 34)(23 39 27 35)(24 40 28 36)(67 82 77 86)(68 83 78 87)(69 84 73 88)(70 79 74 89)(71 80 75 90)(72 81 76 85)
(1 31 4 34)(2 32 5 35)(3 33 6 36)(7 75 10 78)(8 76 11 73)(9 77 12 74)(13 40 16 37)(14 41 17 38)(15 42 18 39)(19 46 22 43)(20 47 23 44)(21 48 24 45)(25 52 28 49)(26 53 29 50)(27 54 30 51)(55 82 58 79)(56 83 59 80)(57 84 60 81)(61 88 64 85)(62 89 65 86)(63 90 66 87)(67 94 70 91)(68 95 71 92)(69 96 72 93)
(1 79 14 89)(2 84 15 88)(3 83 16 87)(4 82 17 86)(5 81 18 85)(6 80 13 90)(7 24 95 28)(8 23 96 27)(9 22 91 26)(10 21 92 25)(11 20 93 30)(12 19 94 29)(31 55 41 65)(32 60 42 64)(33 59 37 63)(34 58 38 62)(35 57 39 61)(36 56 40 66)(43 67 53 77)(44 72 54 76)(45 71 49 75)(46 70 50 74)(47 69 51 73)(48 68 52 78)
(2 6)(3 5)(7 96)(8 95)(9 94)(10 93)(11 92)(12 91)(13 15)(16 18)(20 24)(21 23)(25 27)(28 30)(32 36)(33 35)(37 39)(40 42)(44 48)(45 47)(49 51)(52 54)(55 62)(56 61)(57 66)(58 65)(59 64)(60 63)(67 74)(68 73)(69 78)(70 77)(71 76)(72 75)(79 86)(80 85)(81 90)(82 89)(83 88)(84 87)
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), (1,53,14,43)(2,54,15,44)(3,49,16,45)(4,50,17,46)(5,51,18,47)(6,52,13,48)(7,59,95,63)(8,60,96,64)(9,55,91,65)(10,56,92,66)(11,57,93,61)(12,58,94,62)(19,41,29,31)(20,42,30,32)(21,37,25,33)(22,38,26,34)(23,39,27,35)(24,40,28,36)(67,82,77,86)(68,83,78,87)(69,84,73,88)(70,79,74,89)(71,80,75,90)(72,81,76,85), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,75,10,78)(8,76,11,73)(9,77,12,74)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,52,28,49)(26,53,29,50)(27,54,30,51)(55,82,58,79)(56,83,59,80)(57,84,60,81)(61,88,64,85)(62,89,65,86)(63,90,66,87)(67,94,70,91)(68,95,71,92)(69,96,72,93), (1,79,14,89)(2,84,15,88)(3,83,16,87)(4,82,17,86)(5,81,18,85)(6,80,13,90)(7,24,95,28)(8,23,96,27)(9,22,91,26)(10,21,92,25)(11,20,93,30)(12,19,94,29)(31,55,41,65)(32,60,42,64)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,56,40,66)(43,67,53,77)(44,72,54,76)(45,71,49,75)(46,70,50,74)(47,69,51,73)(48,68,52,78), (2,6)(3,5)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,15)(16,18)(20,24)(21,23)(25,27)(28,30)(32,36)(33,35)(37,39)(40,42)(44,48)(45,47)(49,51)(52,54)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,74)(68,73)(69,78)(70,77)(71,76)(72,75)(79,86)(80,85)(81,90)(82,89)(83,88)(84,87)>;
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), (1,53,14,43)(2,54,15,44)(3,49,16,45)(4,50,17,46)(5,51,18,47)(6,52,13,48)(7,59,95,63)(8,60,96,64)(9,55,91,65)(10,56,92,66)(11,57,93,61)(12,58,94,62)(19,41,29,31)(20,42,30,32)(21,37,25,33)(22,38,26,34)(23,39,27,35)(24,40,28,36)(67,82,77,86)(68,83,78,87)(69,84,73,88)(70,79,74,89)(71,80,75,90)(72,81,76,85), (1,31,4,34)(2,32,5,35)(3,33,6,36)(7,75,10,78)(8,76,11,73)(9,77,12,74)(13,40,16,37)(14,41,17,38)(15,42,18,39)(19,46,22,43)(20,47,23,44)(21,48,24,45)(25,52,28,49)(26,53,29,50)(27,54,30,51)(55,82,58,79)(56,83,59,80)(57,84,60,81)(61,88,64,85)(62,89,65,86)(63,90,66,87)(67,94,70,91)(68,95,71,92)(69,96,72,93), (1,79,14,89)(2,84,15,88)(3,83,16,87)(4,82,17,86)(5,81,18,85)(6,80,13,90)(7,24,95,28)(8,23,96,27)(9,22,91,26)(10,21,92,25)(11,20,93,30)(12,19,94,29)(31,55,41,65)(32,60,42,64)(33,59,37,63)(34,58,38,62)(35,57,39,61)(36,56,40,66)(43,67,53,77)(44,72,54,76)(45,71,49,75)(46,70,50,74)(47,69,51,73)(48,68,52,78), (2,6)(3,5)(7,96)(8,95)(9,94)(10,93)(11,92)(12,91)(13,15)(16,18)(20,24)(21,23)(25,27)(28,30)(32,36)(33,35)(37,39)(40,42)(44,48)(45,47)(49,51)(52,54)(55,62)(56,61)(57,66)(58,65)(59,64)(60,63)(67,74)(68,73)(69,78)(70,77)(71,76)(72,75)(79,86)(80,85)(81,90)(82,89)(83,88)(84,87) );
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)], [(1,53,14,43),(2,54,15,44),(3,49,16,45),(4,50,17,46),(5,51,18,47),(6,52,13,48),(7,59,95,63),(8,60,96,64),(9,55,91,65),(10,56,92,66),(11,57,93,61),(12,58,94,62),(19,41,29,31),(20,42,30,32),(21,37,25,33),(22,38,26,34),(23,39,27,35),(24,40,28,36),(67,82,77,86),(68,83,78,87),(69,84,73,88),(70,79,74,89),(71,80,75,90),(72,81,76,85)], [(1,31,4,34),(2,32,5,35),(3,33,6,36),(7,75,10,78),(8,76,11,73),(9,77,12,74),(13,40,16,37),(14,41,17,38),(15,42,18,39),(19,46,22,43),(20,47,23,44),(21,48,24,45),(25,52,28,49),(26,53,29,50),(27,54,30,51),(55,82,58,79),(56,83,59,80),(57,84,60,81),(61,88,64,85),(62,89,65,86),(63,90,66,87),(67,94,70,91),(68,95,71,92),(69,96,72,93)], [(1,79,14,89),(2,84,15,88),(3,83,16,87),(4,82,17,86),(5,81,18,85),(6,80,13,90),(7,24,95,28),(8,23,96,27),(9,22,91,26),(10,21,92,25),(11,20,93,30),(12,19,94,29),(31,55,41,65),(32,60,42,64),(33,59,37,63),(34,58,38,62),(35,57,39,61),(36,56,40,66),(43,67,53,77),(44,72,54,76),(45,71,49,75),(46,70,50,74),(47,69,51,73),(48,68,52,78)], [(2,6),(3,5),(7,96),(8,95),(9,94),(10,93),(11,92),(12,91),(13,15),(16,18),(20,24),(21,23),(25,27),(28,30),(32,36),(33,35),(37,39),(40,42),(44,48),(45,47),(49,51),(52,54),(55,62),(56,61),(57,66),(58,65),(59,64),(60,63),(67,74),(68,73),(69,78),(70,77),(71,76),(72,75),(79,86),(80,85),(81,90),(82,89),(83,88),(84,87)])
Matrix representation ►G ⊆ 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 6 | 1 |
| 8 | 0 | 0 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 9 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 7 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,6,0,0,0,0,0,1],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,9,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,1,7,0,0,0,0,0,12] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | ··· | 4G | 4H | ··· | 4M | 4N | 4O | 4P | 4Q | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 2 | 2 | 6 | 6 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D6 | D6 | D6 | D6 | C4○D4 | 2+ (1+4) | S3×Q8 | D4⋊6D6 | S3×C4○D4 |
| kernel | C6.522+ (1+4) | Dic3.D4 | Dic3⋊4D4 | Dic6⋊C4 | Dic3.Q8 | S3×C4⋊C4 | D6⋊Q8 | C2×Dic3⋊C4 | C4×C3⋊D4 | Dic3⋊Q8 | D6⋊3Q8 | C3×C22⋊Q8 | C22⋊Q8 | C3⋊D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×Q8 | Dic3 | C6 | C22 | C2 | C2 |
| # reps | 1 | 2 | 2 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 2 | 3 | 1 | 1 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_6._{52}2_+^{(1+4)} % in TeX
G:=Group("C6.52ES+(2,2)"); // GroupNames label
G:=SmallGroup(192,1195);
// by ID
G=gap.SmallGroup(192,1195);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,100,570,409,80,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^4=e^2=1,c^2=a^3,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e=a^-1,c*b*c^-1=b^-1,d*b*d^-1=a^3*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a^3*b^2*d>;
// generators/relations