metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42.139D6, C6.882- (1+4), (C4×Dic6)⋊44C2, (Q8×Dic3)⋊18C2, (C2×D4).169D6, (C2×Q8).159D6, C22⋊C4.33D6, C4.4D4.8S3, (C2×C6).215C24, (C2×C12).77C23, C2.49(Q8○D12), (D4×Dic3).14C2, Dic3⋊Q8⋊22C2, C12.124(C4○D4), C4.15(D4⋊2S3), (C4×C12).184C22, C23.12D6.9C2, (C6×D4).151C22, C23.8D6⋊37C2, (C22×C6).45C23, C23.47(C22×S3), (C6×Q8).124C22, Dic3.28(C4○D4), Dic3.D4⋊38C2, C23.16D6⋊18C2, Dic3⋊C4.48C22, C4⋊Dic3.233C22, C22.236(S3×C23), C3⋊6(C22.50C24), (C2×Dic6).296C22, (C4×Dic3).131C22, (C2×Dic3).252C23, C6.D4.52C22, (C22×Dic3).140C22, C2.74(S3×C4○D4), C6.93(C2×C4○D4), C2.55(C2×D4⋊2S3), (C3×C4.4D4).6C2, (C2×C4).299(C22×S3), (C3×C22⋊C4).62C22, SmallGroup(192,1230)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 448 in 212 conjugacy classes, 97 normal (43 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×13], C22, C22 [×6], C6 [×3], C6 [×2], C2×C4 [×3], C2×C4 [×2], C2×C4 [×12], D4 [×2], Q8 [×6], C23 [×2], Dic3 [×2], Dic3 [×7], C12 [×2], C12 [×4], C2×C6, C2×C6 [×6], C42, C42 [×6], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×12], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], Dic6 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×3], C2×C12 [×2], C3×D4 [×2], C3×Q8 [×2], C22×C6 [×2], C42⋊C2 [×2], C4×D4, C4×Q8 [×3], C22⋊Q8 [×2], C4.4D4, C4.4D4, C42⋊2C2 [×4], C4⋊Q8, C4×Dic3 [×2], C4×Dic3 [×4], Dic3⋊C4 [×2], Dic3⋊C4 [×6], C4⋊Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×6], C4×C12, C3×C22⋊C4 [×4], C2×Dic6 [×2], C22×Dic3 [×2], C6×D4, C6×Q8, C22.50C24, C4×Dic6 [×2], C23.16D6 [×2], Dic3.D4 [×2], C23.8D6 [×4], D4×Dic3, C23.12D6, Dic3⋊Q8, Q8×Dic3, C3×C4.4D4, C42.139D6
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D6 [×7], C4○D4 [×4], C24, C22×S3 [×7], C2×C4○D4 [×2], 2- (1+4), D4⋊2S3 [×2], S3×C23, C22.50C24, C2×D4⋊2S3, S3×C4○D4, Q8○D12, C42.139D6
Generators and relations
G = < a,b,c,d | a4=b4=c6=1, d2=a2b2, ab=ba, cac-1=dad-1=a-1, cbc-1=a2b-1, bd=db, dcd-1=c-1 >
►On 96 points
(1 70 13 67)(2 68 14 71)(3 72 15 69)(4 38 7 41)(5 42 8 39)(6 40 9 37)(10 46 19 43)(11 44 20 47)(12 48 21 45)(16 26 24 29)(17 30 22 27)(18 28 23 25)(31 55 50 75)(32 76 51 56)(33 57 52 77)(34 78 53 58)(35 59 54 73)(36 74 49 60)(61 81 94 89)(62 90 95 82)(63 83 96 85)(64 86 91 84)(65 79 92 87)(66 88 93 80)
(1 89 18 84)(2 87 16 82)(3 85 17 80)(4 58 10 75)(5 56 11 73)(6 60 12 77)(7 78 19 55)(8 76 20 59)(9 74 21 57)(13 81 23 86)(14 79 24 90)(15 83 22 88)(25 91 67 94)(26 62 68 65)(27 93 69 96)(28 64 70 61)(29 95 71 92)(30 66 72 63)(31 38 34 46)(32 44 35 42)(33 40 36 48)(37 49 45 52)(39 51 47 54)(41 53 43 50)
(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 20 23 5)(2 19 24 4)(3 21 22 6)(7 14 10 16)(8 13 11 18)(9 15 12 17)(25 39 70 44)(26 38 71 43)(27 37 72 48)(28 42 67 47)(29 41 68 46)(30 40 69 45)(31 95 53 65)(32 94 54 64)(33 93 49 63)(34 92 50 62)(35 91 51 61)(36 96 52 66)(55 90 58 87)(56 89 59 86)(57 88 60 85)(73 84 76 81)(74 83 77 80)(75 82 78 79)
G:=sub<Sym(96)| (1,70,13,67)(2,68,14,71)(3,72,15,69)(4,38,7,41)(5,42,8,39)(6,40,9,37)(10,46,19,43)(11,44,20,47)(12,48,21,45)(16,26,24,29)(17,30,22,27)(18,28,23,25)(31,55,50,75)(32,76,51,56)(33,57,52,77)(34,78,53,58)(35,59,54,73)(36,74,49,60)(61,81,94,89)(62,90,95,82)(63,83,96,85)(64,86,91,84)(65,79,92,87)(66,88,93,80), (1,89,18,84)(2,87,16,82)(3,85,17,80)(4,58,10,75)(5,56,11,73)(6,60,12,77)(7,78,19,55)(8,76,20,59)(9,74,21,57)(13,81,23,86)(14,79,24,90)(15,83,22,88)(25,91,67,94)(26,62,68,65)(27,93,69,96)(28,64,70,61)(29,95,71,92)(30,66,72,63)(31,38,34,46)(32,44,35,42)(33,40,36,48)(37,49,45,52)(39,51,47,54)(41,53,43,50), (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), (1,20,23,5)(2,19,24,4)(3,21,22,6)(7,14,10,16)(8,13,11,18)(9,15,12,17)(25,39,70,44)(26,38,71,43)(27,37,72,48)(28,42,67,47)(29,41,68,46)(30,40,69,45)(31,95,53,65)(32,94,54,64)(33,93,49,63)(34,92,50,62)(35,91,51,61)(36,96,52,66)(55,90,58,87)(56,89,59,86)(57,88,60,85)(73,84,76,81)(74,83,77,80)(75,82,78,79)>;
G:=Group( (1,70,13,67)(2,68,14,71)(3,72,15,69)(4,38,7,41)(5,42,8,39)(6,40,9,37)(10,46,19,43)(11,44,20,47)(12,48,21,45)(16,26,24,29)(17,30,22,27)(18,28,23,25)(31,55,50,75)(32,76,51,56)(33,57,52,77)(34,78,53,58)(35,59,54,73)(36,74,49,60)(61,81,94,89)(62,90,95,82)(63,83,96,85)(64,86,91,84)(65,79,92,87)(66,88,93,80), (1,89,18,84)(2,87,16,82)(3,85,17,80)(4,58,10,75)(5,56,11,73)(6,60,12,77)(7,78,19,55)(8,76,20,59)(9,74,21,57)(13,81,23,86)(14,79,24,90)(15,83,22,88)(25,91,67,94)(26,62,68,65)(27,93,69,96)(28,64,70,61)(29,95,71,92)(30,66,72,63)(31,38,34,46)(32,44,35,42)(33,40,36,48)(37,49,45,52)(39,51,47,54)(41,53,43,50), (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), (1,20,23,5)(2,19,24,4)(3,21,22,6)(7,14,10,16)(8,13,11,18)(9,15,12,17)(25,39,70,44)(26,38,71,43)(27,37,72,48)(28,42,67,47)(29,41,68,46)(30,40,69,45)(31,95,53,65)(32,94,54,64)(33,93,49,63)(34,92,50,62)(35,91,51,61)(36,96,52,66)(55,90,58,87)(56,89,59,86)(57,88,60,85)(73,84,76,81)(74,83,77,80)(75,82,78,79) );
G=PermutationGroup([(1,70,13,67),(2,68,14,71),(3,72,15,69),(4,38,7,41),(5,42,8,39),(6,40,9,37),(10,46,19,43),(11,44,20,47),(12,48,21,45),(16,26,24,29),(17,30,22,27),(18,28,23,25),(31,55,50,75),(32,76,51,56),(33,57,52,77),(34,78,53,58),(35,59,54,73),(36,74,49,60),(61,81,94,89),(62,90,95,82),(63,83,96,85),(64,86,91,84),(65,79,92,87),(66,88,93,80)], [(1,89,18,84),(2,87,16,82),(3,85,17,80),(4,58,10,75),(5,56,11,73),(6,60,12,77),(7,78,19,55),(8,76,20,59),(9,74,21,57),(13,81,23,86),(14,79,24,90),(15,83,22,88),(25,91,67,94),(26,62,68,65),(27,93,69,96),(28,64,70,61),(29,95,71,92),(30,66,72,63),(31,38,34,46),(32,44,35,42),(33,40,36,48),(37,49,45,52),(39,51,47,54),(41,53,43,50)], [(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)], [(1,20,23,5),(2,19,24,4),(3,21,22,6),(7,14,10,16),(8,13,11,18),(9,15,12,17),(25,39,70,44),(26,38,71,43),(27,37,72,48),(28,42,67,47),(29,41,68,46),(30,40,69,45),(31,95,53,65),(32,94,54,64),(33,93,49,63),(34,92,50,62),(35,91,51,61),(36,96,52,66),(55,90,58,87),(56,89,59,86),(57,88,60,85),(73,84,76,81),(74,83,77,80),(75,82,78,79)])
Matrix representation ►G ⊆ GL6(𝔽13)
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 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 | 5 | 11 |
| 0 | 0 | 0 | 0 | 0 | 8 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 11 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 7 | 0 | 0 |
| 0 | 0 | 7 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 0 | 8 |
G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,11,8],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,9,11,0,0,0,0,0,3,0,0,0,0,0,0,1,5,0,0,0,0,0,12],[1,0,0,0,0,0,0,12,0,0,0,0,0,0,11,7,0,0,0,0,7,2,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | ··· | 4O | 4P | 4Q | 4R | 4S | 6A | 6B | 6C | 6D | 6E | 12A | ··· | 12F | 12G | 12H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 8 | 8 | 4 | ··· | 4 | 8 | 8 |
39 irreducible representations
| dim | 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 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D4 | 2- (1+4) | D4⋊2S3 | S3×C4○D4 | Q8○D12 |
| kernel | C42.139D6 | C4×Dic6 | C23.16D6 | Dic3.D4 | C23.8D6 | D4×Dic3 | C23.12D6 | Dic3⋊Q8 | Q8×Dic3 | C3×C4.4D4 | C4.4D4 | C42 | C22⋊C4 | C2×D4 | C2×Q8 | Dic3 | C12 | C6 | C4 | C2 | C2 |
| # reps | 1 | 2 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 4 | 1 | 2 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{139}D_6 % in TeX
G:=Group("C4^2.139D6"); // GroupNames label
G:=SmallGroup(192,1230);
// by ID
G=gap.SmallGroup(192,1230);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,758,387,100,794,297,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^6=1,d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations