direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C4.Dic3, C12.74C24, C24.7Dic3, C3⋊C8⋊13C23, C6⋊3(C2×M4(2)), (C2×C6)⋊9M4(2), C4.73(S3×C23), (C23×C6).11C4, (C23×C4).15S3, C6.41(C23×C4), C3⋊3(C22×M4(2)), (C22×C12).28C4, (C23×C12).16C2, (C22×C4).488D6, C2.3(C23×Dic3), (C2×C12).799C23, C12.180(C22×C4), C4.38(C22×Dic3), (C22×C4).21Dic3, C23.40(C2×Dic3), (C22×C12).546C22, C22.28(C22×Dic3), (C2×C3⋊C8)⋊48C22, (C22×C3⋊C8)⋊23C2, (C2×C12).300(C2×C4), (C2×C4).86(C2×Dic3), (C2×C4).827(C22×S3), (C22×C6).140(C2×C4), (C2×C6).205(C22×C4), SmallGroup(192,1340)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 440 in 298 conjugacy classes, 207 normal (17 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4, C4 [×7], C22 [×11], C22 [×12], C6, C6 [×6], C6 [×4], C8 [×8], C2×C4 [×28], C23, C23 [×6], C23 [×4], C12, C12 [×7], C2×C6 [×11], C2×C6 [×12], C2×C8 [×12], M4(2) [×16], C22×C4 [×2], C22×C4 [×12], C24, C3⋊C8 [×8], C2×C12 [×28], C22×C6, C22×C6 [×6], C22×C6 [×4], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C2×C3⋊C8 [×12], C4.Dic3 [×16], C22×C12 [×2], C22×C12 [×12], C23×C6, C22×M4(2), C22×C3⋊C8 [×2], C2×C4.Dic3 [×12], C23×C12, C22×C4.Dic3
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], Dic3 [×8], D6 [×7], M4(2) [×4], C22×C4 [×14], C24, C2×Dic3 [×28], C22×S3 [×7], C2×M4(2) [×6], C23×C4, C4.Dic3 [×4], C22×Dic3 [×14], S3×C23, C22×M4(2), C2×C4.Dic3 [×6], C23×Dic3, C22×C4.Dic3
Generators and relations
G = < a,b,c,d,e | a2=b2=c4=1, d6=c2, e2=c2d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >
►On 96 points
Generators in S96
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 37)(14 38)(15 39)(16 40)(17 41)(18 42)(19 43)(20 44)(21 45)(22 46)(23 47)(24 48)(49 81)(50 82)(51 83)(52 84)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 85)(70 86)(71 87)(72 88)
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 37)(32 38)(33 39)(34 40)(35 41)(36 42)(49 63)(50 64)(51 65)(52 66)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 61)(60 62)(73 95)(74 96)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(82 92)(83 93)(84 94)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 40 31 46)(26 41 32 47)(27 42 33 48)(28 43 34 37)(29 44 35 38)(30 45 36 39)(49 66 55 72)(50 67 56 61)(51 68 57 62)(52 69 58 63)(53 70 59 64)(54 71 60 65)(73 86 79 92)(74 87 80 93)(75 88 81 94)(76 89 82 95)(77 90 83 96)(78 91 84 85)
(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 10 50 7 59 4 56)(2 58 11 55 8 52 5 49)(3 51 12 60 9 57 6 54)(13 70 22 67 19 64 16 61)(14 63 23 72 20 69 17 66)(15 68 24 65 21 62 18 71)(25 82 34 79 31 76 28 73)(26 75 35 84 32 81 29 78)(27 80 36 77 33 74 30 83)(37 86 46 95 43 92 40 89)(38 91 47 88 44 85 41 94)(39 96 48 93 45 90 42 87)
G:=sub<Sym(96)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62)(73,95)(74,96)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,40,31,46)(26,41,32,47)(27,42,33,48)(28,43,34,37)(29,44,35,38)(30,45,36,39)(49,66,55,72)(50,67,56,61)(51,68,57,62)(52,69,58,63)(53,70,59,64)(54,71,60,65)(73,86,79,92)(74,87,80,93)(75,88,81,94)(76,89,82,95)(77,90,83,96)(78,91,84,85), (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), (1,53,10,50,7,59,4,56)(2,58,11,55,8,52,5,49)(3,51,12,60,9,57,6,54)(13,70,22,67,19,64,16,61)(14,63,23,72,20,69,17,66)(15,68,24,65,21,62,18,71)(25,82,34,79,31,76,28,73)(26,75,35,84,32,81,29,78)(27,80,36,77,33,74,30,83)(37,86,46,95,43,92,40,89)(38,91,47,88,44,85,41,94)(39,96,48,93,45,90,42,87)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,37)(14,38)(15,39)(16,40)(17,41)(18,42)(19,43)(20,44)(21,45)(22,46)(23,47)(24,48)(49,81)(50,82)(51,83)(52,84)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,85)(70,86)(71,87)(72,88), (1,22)(2,23)(3,24)(4,13)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,37)(32,38)(33,39)(34,40)(35,41)(36,42)(49,63)(50,64)(51,65)(52,66)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,61)(60,62)(73,95)(74,96)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(81,91)(82,92)(83,93)(84,94), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,40,31,46)(26,41,32,47)(27,42,33,48)(28,43,34,37)(29,44,35,38)(30,45,36,39)(49,66,55,72)(50,67,56,61)(51,68,57,62)(52,69,58,63)(53,70,59,64)(54,71,60,65)(73,86,79,92)(74,87,80,93)(75,88,81,94)(76,89,82,95)(77,90,83,96)(78,91,84,85), (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), (1,53,10,50,7,59,4,56)(2,58,11,55,8,52,5,49)(3,51,12,60,9,57,6,54)(13,70,22,67,19,64,16,61)(14,63,23,72,20,69,17,66)(15,68,24,65,21,62,18,71)(25,82,34,79,31,76,28,73)(26,75,35,84,32,81,29,78)(27,80,36,77,33,74,30,83)(37,86,46,95,43,92,40,89)(38,91,47,88,44,85,41,94)(39,96,48,93,45,90,42,87) );
G=PermutationGroup([(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,37),(14,38),(15,39),(16,40),(17,41),(18,42),(19,43),(20,44),(21,45),(22,46),(23,47),(24,48),(49,81),(50,82),(51,83),(52,84),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,85),(70,86),(71,87),(72,88)], [(1,22),(2,23),(3,24),(4,13),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,37),(32,38),(33,39),(34,40),(35,41),(36,42),(49,63),(50,64),(51,65),(52,66),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,61),(60,62),(73,95),(74,96),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(81,91),(82,92),(83,93),(84,94)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,40,31,46),(26,41,32,47),(27,42,33,48),(28,43,34,37),(29,44,35,38),(30,45,36,39),(49,66,55,72),(50,67,56,61),(51,68,57,62),(52,69,58,63),(53,70,59,64),(54,71,60,65),(73,86,79,92),(74,87,80,93),(75,88,81,94),(76,89,82,95),(77,90,83,96),(78,91,84,85)], [(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)], [(1,53,10,50,7,59,4,56),(2,58,11,55,8,52,5,49),(3,51,12,60,9,57,6,54),(13,70,22,67,19,64,16,61),(14,63,23,72,20,69,17,66),(15,68,24,65,21,62,18,71),(25,82,34,79,31,76,28,73),(26,75,35,84,32,81,29,78),(27,80,36,77,33,74,30,83),(37,86,46,95,43,92,40,89),(38,91,47,88,44,85,41,94),(39,96,48,93,45,90,42,87)])
Matrix representation ►G ⊆ GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 30 | 46 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 37 | 24 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 66 | 71 |
| 0 | 0 | 38 | 7 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,27,30,0,0,0,46],[1,0,0,0,0,1,0,0,0,0,3,37,0,0,0,24],[1,0,0,0,0,1,0,0,0,0,66,38,0,0,71,7] >;
72 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | ··· | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6O | 8A | ··· | 8P | 12A | ··· | 12P |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | M4(2) | C4.Dic3 |
| kernel | C22×C4.Dic3 | C22×C3⋊C8 | C2×C4.Dic3 | C23×C12 | C22×C12 | C23×C6 | C23×C4 | C22×C4 | C22×C4 | C24 | C2×C6 | C22 |
| # reps | 1 | 2 | 12 | 1 | 14 | 2 | 1 | 7 | 7 | 1 | 8 | 16 |
In GAP, Magma, Sage, TeX
C_2^2\times C_4.Dic_3
% in TeX
G:=Group("C2^2xC4.Dic3"); // GroupNames label
G:=SmallGroup(192,1340);
// by ID
G=gap.SmallGroup(192,1340);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,1123,102,6278]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^4=1,d^6=c^2,e^2=c^2*d^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀