direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C23.23D6, C24.69D6, (C2×D4).228D6, (C2×C6).291C24, (C23×Dic3)⋊8C2, (C22×D4).11S3, (C22×C4).285D6, C6.139(C22×D4), (C22×C6).121D4, (C2×C12).642C23, Dic3⋊C4⋊72C22, (C6×D4).311C22, C6⋊5(C22.D4), C23.74(C3⋊D4), (C23×C6).73C22, C6.D4⋊57C22, C22.305(S3×C23), C23.143(C22×S3), (C22×C6).227C23, C22.77(D4⋊2S3), (C22×C12).437C22, (C2×Dic3).281C23, (C22×Dic3)⋊48C22, (D4×C2×C6).21C2, (C2×C6).73(C2×D4), C6.103(C2×C4○D4), C3⋊6(C2×C22.D4), (C2×Dic3⋊C4)⋊47C2, C2.67(C2×D4⋊2S3), C2.12(C22×C3⋊D4), (C2×C6).175(C4○D4), (C2×C6.D4)⋊24C2, (C2×C4).236(C22×S3), C22.108(C2×C3⋊D4), SmallGroup(192,1355)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 744 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×6], C3, C4 [×10], C22, C22 [×10], C22 [×22], C6, C6 [×6], C6 [×6], C2×C4 [×2], C2×C4 [×26], D4 [×8], C23, C23 [×8], C23 [×10], Dic3 [×8], C12 [×2], C2×C6, C2×C6 [×10], C2×C6 [×22], C22⋊C4 [×12], C4⋊C4 [×8], C22×C4, C22×C4 [×12], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×Dic3 [×8], C2×Dic3 [×16], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×C6, C22×C6 [×8], C22×C6 [×10], C2×C22⋊C4 [×3], C2×C4⋊C4 [×2], C22.D4 [×8], C23×C4, C22×D4, Dic3⋊C4 [×8], C6.D4 [×12], C22×Dic3 [×8], C22×Dic3 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C2×C22.D4, C2×Dic3⋊C4 [×2], C23.23D6 [×8], C2×C6.D4, C2×C6.D4 [×2], C23×Dic3, D4×C2×C6, C2×C23.23D6
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C4○D4 [×4], C24, C3⋊D4 [×4], C22×S3 [×7], C22.D4 [×4], C22×D4, C2×C4○D4 [×2], D4⋊2S3 [×4], C2×C3⋊D4 [×6], S3×C23, C2×C22.D4, C23.23D6 [×4], C2×D4⋊2S3 [×2], C22×C3⋊D4, C2×C23.23D6
Generators and relations
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
►On 96 points
Generators in S96
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 22)(8 23)(9 24)(10 19)(11 20)(12 21)(25 38)(26 39)(27 40)(28 41)(29 42)(30 37)(31 46)(32 47)(33 48)(34 43)(35 44)(36 45)(49 72)(50 67)(51 68)(52 69)(53 70)(54 71)(55 62)(56 63)(57 64)(58 65)(59 66)(60 61)(73 96)(74 91)(75 92)(76 93)(77 94)(78 95)(79 85)(80 86)(81 87)(82 88)(83 89)(84 90)
(1 25)(2 29)(3 27)(4 30)(5 28)(6 26)(7 31)(8 35)(9 33)(10 34)(11 32)(12 36)(13 37)(14 41)(15 39)(16 40)(17 38)(18 42)(19 43)(20 47)(21 45)(22 46)(23 44)(24 48)(49 76)(50 79)(51 78)(52 81)(53 74)(54 83)(55 77)(56 80)(57 73)(58 82)(59 75)(60 84)(61 90)(62 94)(63 86)(64 96)(65 88)(66 92)(67 85)(68 95)(69 87)(70 91)(71 89)(72 93)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 22)(14 23)(15 24)(16 19)(17 20)(18 21)(25 32)(26 33)(27 34)(28 35)(29 36)(30 31)(37 46)(38 47)(39 48)(40 43)(41 44)(42 45)(49 57)(50 58)(51 59)(52 60)(53 55)(54 56)(61 69)(62 70)(63 71)(64 72)(65 67)(66 68)(73 76)(74 77)(75 78)(79 82)(80 83)(81 84)(85 88)(86 89)(87 90)(91 94)(92 95)(93 96)
(1 5)(2 6)(3 4)(7 10)(8 11)(9 12)(13 16)(14 17)(15 18)(19 22)(20 23)(21 24)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 48)(49 60)(50 55)(51 56)(52 57)(53 58)(54 59)(61 72)(62 67)(63 68)(64 69)(65 70)(66 71)(73 81)(74 82)(75 83)(76 84)(77 79)(78 80)(85 94)(86 95)(87 96)(88 91)(89 92)(90 93)
(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 76 8 81)(2 78 9 83)(3 74 7 79)(4 82 10 77)(5 84 11 73)(6 80 12 75)(13 88 19 94)(14 90 20 96)(15 86 21 92)(16 91 22 85)(17 93 23 87)(18 95 24 89)(25 60 35 57)(26 51 36 54)(27 58 31 55)(28 49 32 52)(29 56 33 59)(30 53 34 50)(37 70 43 67)(38 61 44 64)(39 68 45 71)(40 65 46 62)(41 72 47 69)(42 63 48 66)
G:=sub<Sym(96)| (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,72)(50,67)(51,68)(52,69)(53,70)(54,71)(55,62)(56,63)(57,64)(58,65)(59,66)(60,61)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,25)(2,29)(3,27)(4,30)(5,28)(6,26)(7,31)(8,35)(9,33)(10,34)(11,32)(12,36)(13,37)(14,41)(15,39)(16,40)(17,38)(18,42)(19,43)(20,47)(21,45)(22,46)(23,44)(24,48)(49,76)(50,79)(51,78)(52,81)(53,74)(54,83)(55,77)(56,80)(57,73)(58,82)(59,75)(60,84)(61,90)(62,94)(63,86)(64,96)(65,88)(66,92)(67,85)(68,95)(69,87)(70,91)(71,89)(72,93), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,57)(50,58)(51,59)(52,60)(53,55)(54,56)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,60)(50,55)(51,56)(52,57)(53,58)(54,59)(61,72)(62,67)(63,68)(64,69)(65,70)(66,71)(73,81)(74,82)(75,83)(76,84)(77,79)(78,80)(85,94)(86,95)(87,96)(88,91)(89,92)(90,93), (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), (1,76,8,81)(2,78,9,83)(3,74,7,79)(4,82,10,77)(5,84,11,73)(6,80,12,75)(13,88,19,94)(14,90,20,96)(15,86,21,92)(16,91,22,85)(17,93,23,87)(18,95,24,89)(25,60,35,57)(26,51,36,54)(27,58,31,55)(28,49,32,52)(29,56,33,59)(30,53,34,50)(37,70,43,67)(38,61,44,64)(39,68,45,71)(40,65,46,62)(41,72,47,69)(42,63,48,66)>;
G:=Group( (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,22)(8,23)(9,24)(10,19)(11,20)(12,21)(25,38)(26,39)(27,40)(28,41)(29,42)(30,37)(31,46)(32,47)(33,48)(34,43)(35,44)(36,45)(49,72)(50,67)(51,68)(52,69)(53,70)(54,71)(55,62)(56,63)(57,64)(58,65)(59,66)(60,61)(73,96)(74,91)(75,92)(76,93)(77,94)(78,95)(79,85)(80,86)(81,87)(82,88)(83,89)(84,90), (1,25)(2,29)(3,27)(4,30)(5,28)(6,26)(7,31)(8,35)(9,33)(10,34)(11,32)(12,36)(13,37)(14,41)(15,39)(16,40)(17,38)(18,42)(19,43)(20,47)(21,45)(22,46)(23,44)(24,48)(49,76)(50,79)(51,78)(52,81)(53,74)(54,83)(55,77)(56,80)(57,73)(58,82)(59,75)(60,84)(61,90)(62,94)(63,86)(64,96)(65,88)(66,92)(67,85)(68,95)(69,87)(70,91)(71,89)(72,93), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,22)(14,23)(15,24)(16,19)(17,20)(18,21)(25,32)(26,33)(27,34)(28,35)(29,36)(30,31)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45)(49,57)(50,58)(51,59)(52,60)(53,55)(54,56)(61,69)(62,70)(63,71)(64,72)(65,67)(66,68)(73,76)(74,77)(75,78)(79,82)(80,83)(81,84)(85,88)(86,89)(87,90)(91,94)(92,95)(93,96), (1,5)(2,6)(3,4)(7,10)(8,11)(9,12)(13,16)(14,17)(15,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,60)(50,55)(51,56)(52,57)(53,58)(54,59)(61,72)(62,67)(63,68)(64,69)(65,70)(66,71)(73,81)(74,82)(75,83)(76,84)(77,79)(78,80)(85,94)(86,95)(87,96)(88,91)(89,92)(90,93), (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), (1,76,8,81)(2,78,9,83)(3,74,7,79)(4,82,10,77)(5,84,11,73)(6,80,12,75)(13,88,19,94)(14,90,20,96)(15,86,21,92)(16,91,22,85)(17,93,23,87)(18,95,24,89)(25,60,35,57)(26,51,36,54)(27,58,31,55)(28,49,32,52)(29,56,33,59)(30,53,34,50)(37,70,43,67)(38,61,44,64)(39,68,45,71)(40,65,46,62)(41,72,47,69)(42,63,48,66) );
G=PermutationGroup([(1,17),(2,18),(3,16),(4,13),(5,14),(6,15),(7,22),(8,23),(9,24),(10,19),(11,20),(12,21),(25,38),(26,39),(27,40),(28,41),(29,42),(30,37),(31,46),(32,47),(33,48),(34,43),(35,44),(36,45),(49,72),(50,67),(51,68),(52,69),(53,70),(54,71),(55,62),(56,63),(57,64),(58,65),(59,66),(60,61),(73,96),(74,91),(75,92),(76,93),(77,94),(78,95),(79,85),(80,86),(81,87),(82,88),(83,89),(84,90)], [(1,25),(2,29),(3,27),(4,30),(5,28),(6,26),(7,31),(8,35),(9,33),(10,34),(11,32),(12,36),(13,37),(14,41),(15,39),(16,40),(17,38),(18,42),(19,43),(20,47),(21,45),(22,46),(23,44),(24,48),(49,76),(50,79),(51,78),(52,81),(53,74),(54,83),(55,77),(56,80),(57,73),(58,82),(59,75),(60,84),(61,90),(62,94),(63,86),(64,96),(65,88),(66,92),(67,85),(68,95),(69,87),(70,91),(71,89),(72,93)], [(1,11),(2,12),(3,10),(4,7),(5,8),(6,9),(13,22),(14,23),(15,24),(16,19),(17,20),(18,21),(25,32),(26,33),(27,34),(28,35),(29,36),(30,31),(37,46),(38,47),(39,48),(40,43),(41,44),(42,45),(49,57),(50,58),(51,59),(52,60),(53,55),(54,56),(61,69),(62,70),(63,71),(64,72),(65,67),(66,68),(73,76),(74,77),(75,78),(79,82),(80,83),(81,84),(85,88),(86,89),(87,90),(91,94),(92,95),(93,96)], [(1,5),(2,6),(3,4),(7,10),(8,11),(9,12),(13,16),(14,17),(15,18),(19,22),(20,23),(21,24),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(49,60),(50,55),(51,56),(52,57),(53,58),(54,59),(61,72),(62,67),(63,68),(64,69),(65,70),(66,71),(73,81),(74,82),(75,83),(76,84),(77,79),(78,80),(85,94),(86,95),(87,96),(88,91),(89,92),(90,93)], [(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)], [(1,76,8,81),(2,78,9,83),(3,74,7,79),(4,82,10,77),(5,84,11,73),(6,80,12,75),(13,88,19,94),(14,90,20,96),(15,86,21,92),(16,91,22,85),(17,93,23,87),(18,95,24,89),(25,60,35,57),(26,51,36,54),(27,58,31,55),(28,49,32,52),(29,56,33,59),(30,53,34,50),(37,70,43,67),(38,61,44,64),(39,68,45,71),(40,65,46,62),(41,72,47,69),(42,63,48,66)])
Matrix representation ►G ⊆ GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,0,3,0,0,0,4,0,0,0,0,0,0,8,0,0,0,0,0,5] >;
48 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 3 | 4A | 4B | 4C | ··· | 4J | 4K | 4L | 4M | 4N | 6A | ··· | 6G | 6H | ··· | 6O | 12A | 12B | 12C | 12D |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D4⋊2S3 |
| kernel | C2×C23.23D6 | C2×Dic3⋊C4 | C23.23D6 | C2×C6.D4 | C23×Dic3 | D4×C2×C6 | C22×D4 | C22×C6 | C22×C4 | C2×D4 | C24 | C2×C6 | C23 | C22 |
| # reps | 1 | 2 | 8 | 3 | 1 | 1 | 1 | 4 | 1 | 4 | 2 | 8 | 8 | 4 |
In GAP, Magma, Sage, TeX
C_2\times C_2^3._{23}D_6 % in TeX
G:=Group("C2xC2^3.23D6"); // GroupNames label
G:=SmallGroup(192,1355);
// by ID
G=gap.SmallGroup(192,1355);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,675,297,6278]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=1,f^2=d*c=c*d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀