direct product, metabelian, soluble, monomial, A-group
Aliases: C7×C22⋊A4, C24⋊3C21, C22⋊(C7×A4), (C2×C14)⋊1A4, (C23×C14)⋊2C3, SmallGroup(336,223)
Series: Derived ►Chief ►Lower central ►Upper central
| C24 — C7×C22⋊A4 |
Generators and relations for C7×C22⋊A4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >
Subgroups: 208 in 68 conjugacy classes, 16 normal (6 characteristic)
C1, C2, C3, C22, C22, C7, C23, A4, C14, C24, C21, C2×C14, C2×C14, C22⋊A4, C22×C14, C7×A4, C23×C14, C7×C22⋊A4
Quotients: C1, C3, C7, A4, C21, C22⋊A4, C7×A4, C7×C22⋊A4
►On 84 points
Generators in S84
(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)
(8 79)(9 80)(10 81)(11 82)(12 83)(13 84)(14 78)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(1 47)(2 48)(3 49)(4 43)(5 44)(6 45)(7 46)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)
(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 83)(23 84)(24 78)(25 79)(26 80)(27 81)(28 82)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)
(1 14 74)(2 8 75)(3 9 76)(4 10 77)(5 11 71)(6 12 72)(7 13 73)(15 61 33)(16 62 34)(17 63 35)(18 57 29)(19 58 30)(20 59 31)(21 60 32)(22 68 40)(23 69 41)(24 70 42)(25 64 36)(26 65 37)(27 66 38)(28 67 39)(43 81 53)(44 82 54)(45 83 55)(46 84 56)(47 78 50)(48 79 51)(49 80 52)
G:=sub<Sym(84)| (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), (8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,78)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74), (1,14,74)(2,8,75)(3,9,76)(4,10,77)(5,11,71)(6,12,72)(7,13,73)(15,61,33)(16,62,34)(17,63,35)(18,57,29)(19,58,30)(20,59,31)(21,60,32)(22,68,40)(23,69,41)(24,70,42)(25,64,36)(26,65,37)(27,66,38)(28,67,39)(43,81,53)(44,82,54)(45,83,55)(46,84,56)(47,78,50)(48,79,51)(49,80,52)>;
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), (8,79)(9,80)(10,81)(11,82)(12,83)(13,84)(14,78)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,47)(2,48)(3,49)(4,43)(5,44)(6,45)(7,46)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47), (8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,83)(23,84)(24,78)(25,79)(26,80)(27,81)(28,82)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74), (1,14,74)(2,8,75)(3,9,76)(4,10,77)(5,11,71)(6,12,72)(7,13,73)(15,61,33)(16,62,34)(17,63,35)(18,57,29)(19,58,30)(20,59,31)(21,60,32)(22,68,40)(23,69,41)(24,70,42)(25,64,36)(26,65,37)(27,66,38)(28,67,39)(43,81,53)(44,82,54)(45,83,55)(46,84,56)(47,78,50)(48,79,51)(49,80,52) );
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)], [(8,79),(9,80),(10,81),(11,82),(12,83),(13,84),(14,78),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)], [(1,47),(2,48),(3,49),(4,43),(5,44),(6,45),(7,46),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,83),(23,84),(24,78),(25,79),(26,80),(27,81),(28,82),(36,48),(37,49),(38,43),(39,44),(40,45),(41,46),(42,47)], [(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,83),(23,84),(24,78),(25,79),(26,80),(27,81),(28,82),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74)], [(1,14,74),(2,8,75),(3,9,76),(4,10,77),(5,11,71),(6,12,72),(7,13,73),(15,61,33),(16,62,34),(17,63,35),(18,57,29),(19,58,30),(20,59,31),(21,60,32),(22,68,40),(23,69,41),(24,70,42),(25,64,36),(26,65,37),(27,66,38),(28,67,39),(43,81,53),(44,82,54),(45,83,55),(46,84,56),(47,78,50),(48,79,51),(49,80,52)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2E | 3A | 3B | 7A | ··· | 7F | 14A | ··· | 14AD | 21A | ··· | 21L |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 7 | ··· | 7 | 14 | ··· | 14 | 21 | ··· | 21 |
| size | 1 | 3 | ··· | 3 | 16 | 16 | 1 | ··· | 1 | 3 | ··· | 3 | 16 | ··· | 16 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | + | ||||
| image | C1 | C3 | C7 | C21 | A4 | C7×A4 |
| kernel | C7×C22⋊A4 | C23×C14 | C22⋊A4 | C24 | C2×C14 | C22 |
| # reps | 1 | 2 | 6 | 12 | 5 | 30 |
Matrix representation of C7×C22⋊A4 ►in GL6(𝔽43)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 21 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 0 | 21 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 42 | 0 | 0 | 0 | 0 |
| 36 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 7 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 37 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 6 | 42 | 0 | 0 | 0 | 0 |
| 36 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 6 | 41 | 0 | 0 | 0 | 0 |
| 0 | 37 | 1 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(6,GF(43))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,21,0,0,0,0,0,0,21,0,0,0,0,0,0,21],[1,6,36,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,7,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,37,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,42],[1,6,36,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[6,0,0,0,0,0,41,37,7,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
C7×C22⋊A4 in GAP, Magma, Sage, TeX
C_7\times C_2^2\rtimes A_4
% in TeX
G:=Group("C7xC2^2:A4"); // GroupNames label
G:=SmallGroup(336,223);
// by ID
G=gap.SmallGroup(336,223);
# by ID
G:=PCGroup([6,-3,-7,-2,2,-2,2,758,1515,5044,9077]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^2=f^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,f*c*f^-1=b,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations