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×C6).121D4, (C22×C4).285D6, C6.139(C22×D4), (C2×C12).642C23, Dic3⋊C4⋊72C22, (C6×D4).311C22, C6⋊5(C22.D4), C23.74(C3⋊D4), (C23×C6).73C22, C6.D4⋊57C22, (C22×C6).227C23, C23.143(C22×S3), C22.305(S3×C23), 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
Generators and relations for C2×C23.23D6
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 >
Subgroups: 744 in 342 conjugacy classes, 127 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C23×C4, C22×D4, Dic3⋊C4, C6.D4, C22×Dic3, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C2×C22.D4, C2×Dic3⋊C4, C23.23D6, C2×C6.D4, C2×C6.D4, C23×Dic3, D4×C2×C6, C2×C23.23D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C24, C3⋊D4, C22×S3, C22.D4, C22×D4, C2×C4○D4, D4⋊2S3, C2×C3⋊D4, S3×C23, C2×C22.D4, C23.23D6, C2×D4⋊2S3, C22×C3⋊D4, C2×C23.23D6
►On 96 points
Generators in S96
(1 17)(2 18)(3 16)(4 13)(5 14)(6 15)(7 23)(8 24)(9 22)(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 77)(50 78)(51 73)(52 74)(53 75)(54 76)(55 79)(56 80)(57 81)(58 82)(59 83)(60 84)(61 68)(62 69)(63 70)(64 71)(65 72)(66 67)(85 93)(86 94)(87 95)(88 96)(89 91)(90 92)
(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 48)(23 46)(24 44)(49 79)(50 90)(51 81)(52 86)(53 83)(54 88)(55 77)(56 70)(57 73)(58 72)(59 75)(60 68)(61 84)(62 89)(63 80)(64 85)(65 82)(66 87)(67 95)(69 91)(71 93)(74 94)(76 96)(78 92)
(1 11)(2 12)(3 10)(4 7)(5 8)(6 9)(13 23)(14 24)(15 22)(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 65)(50 66)(51 61)(52 62)(53 63)(54 64)(55 58)(56 59)(57 60)(67 78)(68 73)(69 74)(70 75)(71 76)(72 77)(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 23)(20 24)(21 22)(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 62)(50 63)(51 64)(52 65)(53 66)(54 61)(55 91)(56 92)(57 93)(58 94)(59 95)(60 96)(67 75)(68 76)(69 77)(70 78)(71 73)(72 74)(79 89)(80 90)(81 85)(82 86)(83 87)(84 88)
(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 81 8 88)(2 83 9 90)(3 79 7 86)(4 89 10 82)(5 85 11 84)(6 87 12 80)(13 91 19 58)(14 93 20 60)(15 95 21 56)(16 55 23 94)(17 57 24 96)(18 59 22 92)(25 64 35 61)(26 53 36 50)(27 62 31 65)(28 51 32 54)(29 66 33 63)(30 49 34 52)(37 77 43 74)(38 71 44 68)(39 75 45 78)(40 69 46 72)(41 73 47 76)(42 67 48 70)
G:=sub<Sym(96)| (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,23)(8,24)(9,22)(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)(49,77)(50,78)(51,73)(52,74)(53,75)(54,76)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,68)(62,69)(63,70)(64,71)(65,72)(66,67)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (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)(22,48)(23,46)(24,44)(49,79)(50,90)(51,81)(52,86)(53,83)(54,88)(55,77)(56,70)(57,73)(58,72)(59,75)(60,68)(61,84)(62,89)(63,80)(64,85)(65,82)(66,87)(67,95)(69,91)(71,93)(74,94)(76,96)(78,92), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,23)(14,24)(15,22)(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)(49,65)(50,66)(51,61)(52,62)(53,63)(54,64)(55,58)(56,59)(57,60)(67,78)(68,73)(69,74)(70,75)(71,76)(72,77)(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)(19,23)(20,24)(21,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,62)(50,63)(51,64)(52,65)(53,66)(54,61)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(67,75)(68,76)(69,77)(70,78)(71,73)(72,74)(79,89)(80,90)(81,85)(82,86)(83,87)(84,88), (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), (1,81,8,88)(2,83,9,90)(3,79,7,86)(4,89,10,82)(5,85,11,84)(6,87,12,80)(13,91,19,58)(14,93,20,60)(15,95,21,56)(16,55,23,94)(17,57,24,96)(18,59,22,92)(25,64,35,61)(26,53,36,50)(27,62,31,65)(28,51,32,54)(29,66,33,63)(30,49,34,52)(37,77,43,74)(38,71,44,68)(39,75,45,78)(40,69,46,72)(41,73,47,76)(42,67,48,70)>;
G:=Group( (1,17)(2,18)(3,16)(4,13)(5,14)(6,15)(7,23)(8,24)(9,22)(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)(49,77)(50,78)(51,73)(52,74)(53,75)(54,76)(55,79)(56,80)(57,81)(58,82)(59,83)(60,84)(61,68)(62,69)(63,70)(64,71)(65,72)(66,67)(85,93)(86,94)(87,95)(88,96)(89,91)(90,92), (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)(22,48)(23,46)(24,44)(49,79)(50,90)(51,81)(52,86)(53,83)(54,88)(55,77)(56,70)(57,73)(58,72)(59,75)(60,68)(61,84)(62,89)(63,80)(64,85)(65,82)(66,87)(67,95)(69,91)(71,93)(74,94)(76,96)(78,92), (1,11)(2,12)(3,10)(4,7)(5,8)(6,9)(13,23)(14,24)(15,22)(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)(49,65)(50,66)(51,61)(52,62)(53,63)(54,64)(55,58)(56,59)(57,60)(67,78)(68,73)(69,74)(70,75)(71,76)(72,77)(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)(19,23)(20,24)(21,22)(25,28)(26,29)(27,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,46)(44,47)(45,48)(49,62)(50,63)(51,64)(52,65)(53,66)(54,61)(55,91)(56,92)(57,93)(58,94)(59,95)(60,96)(67,75)(68,76)(69,77)(70,78)(71,73)(72,74)(79,89)(80,90)(81,85)(82,86)(83,87)(84,88), (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), (1,81,8,88)(2,83,9,90)(3,79,7,86)(4,89,10,82)(5,85,11,84)(6,87,12,80)(13,91,19,58)(14,93,20,60)(15,95,21,56)(16,55,23,94)(17,57,24,96)(18,59,22,92)(25,64,35,61)(26,53,36,50)(27,62,31,65)(28,51,32,54)(29,66,33,63)(30,49,34,52)(37,77,43,74)(38,71,44,68)(39,75,45,78)(40,69,46,72)(41,73,47,76)(42,67,48,70) );
G=PermutationGroup([[(1,17),(2,18),(3,16),(4,13),(5,14),(6,15),(7,23),(8,24),(9,22),(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),(49,77),(50,78),(51,73),(52,74),(53,75),(54,76),(55,79),(56,80),(57,81),(58,82),(59,83),(60,84),(61,68),(62,69),(63,70),(64,71),(65,72),(66,67),(85,93),(86,94),(87,95),(88,96),(89,91),(90,92)], [(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),(22,48),(23,46),(24,44),(49,79),(50,90),(51,81),(52,86),(53,83),(54,88),(55,77),(56,70),(57,73),(58,72),(59,75),(60,68),(61,84),(62,89),(63,80),(64,85),(65,82),(66,87),(67,95),(69,91),(71,93),(74,94),(76,96),(78,92)], [(1,11),(2,12),(3,10),(4,7),(5,8),(6,9),(13,23),(14,24),(15,22),(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),(49,65),(50,66),(51,61),(52,62),(53,63),(54,64),(55,58),(56,59),(57,60),(67,78),(68,73),(69,74),(70,75),(71,76),(72,77),(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),(19,23),(20,24),(21,22),(25,28),(26,29),(27,30),(31,34),(32,35),(33,36),(37,40),(38,41),(39,42),(43,46),(44,47),(45,48),(49,62),(50,63),(51,64),(52,65),(53,66),(54,61),(55,91),(56,92),(57,93),(58,94),(59,95),(60,96),(67,75),(68,76),(69,77),(70,78),(71,73),(72,74),(79,89),(80,90),(81,85),(82,86),(83,87),(84,88)], [(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)], [(1,81,8,88),(2,83,9,90),(3,79,7,86),(4,89,10,82),(5,85,11,84),(6,87,12,80),(13,91,19,58),(14,93,20,60),(15,95,21,56),(16,55,23,94),(17,57,24,96),(18,59,22,92),(25,64,35,61),(26,53,36,50),(27,62,31,65),(28,51,32,54),(29,66,33,63),(30,49,34,52),(37,77,43,74),(38,71,44,68),(39,75,45,78),(40,69,46,72),(41,73,47,76),(42,67,48,70)]])
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 |
Matrix representation of C2×C23.23D6 ►in 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] >;
C2×C23.23D6 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