metabelian, soluble, monomial, A-group
Aliases: C7⋊(C22⋊A4), C22⋊(C7⋊A4), (C2×C14)⋊2A4, C24⋊3(C7⋊C3), (C23×C14)⋊3C3, SmallGroup(336,224)
Series: Derived ►Chief ►Lower central ►Upper central
| C23×C14 — 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, faf-1=a4, fbf-1=bc=cb, bd=db, be=eb, cd=dc, ce=ec, fcf-1=b, fdf-1=de=ed, fef-1=d >
Subgroups: 430 in 68 conjugacy classes, 15 normal (5 characteristic)
C1, C2, C3, C22, C22, C7, C23, A4, C14, C24, C7⋊C3, C2×C14, C2×C14, C22⋊A4, C22×C14, C7⋊A4, C23×C14, C7⋊(C22⋊A4)
Quotients: C1, C3, A4, C7⋊C3, 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)
(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 27)(2 28)(3 22)(4 23)(5 24)(6 25)(7 26)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)
(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 50)(30 51)(31 52)(32 53)(33 54)(34 55)(35 56)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)
(1 61 31)(2 63 35)(3 58 32)(4 60 29)(5 62 33)(6 57 30)(7 59 34)(8 65 39)(9 67 36)(10 69 40)(11 64 37)(12 66 41)(13 68 38)(14 70 42)(15 72 46)(16 74 43)(17 76 47)(18 71 44)(19 73 48)(20 75 45)(21 77 49)(22 79 53)(23 81 50)(24 83 54)(25 78 51)(26 80 55)(27 82 52)(28 84 56)
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), (29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,61,31)(2,63,35)(3,58,32)(4,60,29)(5,62,33)(6,57,30)(7,59,34)(8,65,39)(9,67,36)(10,69,40)(11,64,37)(12,66,41)(13,68,38)(14,70,42)(15,72,46)(16,74,43)(17,76,47)(18,71,44)(19,73,48)(20,75,45)(21,77,49)(22,79,53)(23,81,50)(24,83,54)(25,78,51)(26,80,55)(27,82,52)(28,84,56)>;
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), (29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,27)(2,28)(3,22)(4,23)(5,24)(6,25)(7,26)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49), (1,13)(2,14)(3,8)(4,9)(5,10)(6,11)(7,12)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(29,43)(30,44)(31,45)(32,46)(33,47)(34,48)(35,49)(36,50)(37,51)(38,52)(39,53)(40,54)(41,55)(42,56)(57,78)(58,79)(59,80)(60,81)(61,82)(62,83)(63,84)(64,71)(65,72)(66,73)(67,74)(68,75)(69,76)(70,77), (1,20)(2,21)(3,15)(4,16)(5,17)(6,18)(7,19)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,50)(30,51)(31,52)(32,53)(33,54)(34,55)(35,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(71,78)(72,79)(73,80)(74,81)(75,82)(76,83)(77,84), (1,61,31)(2,63,35)(3,58,32)(4,60,29)(5,62,33)(6,57,30)(7,59,34)(8,65,39)(9,67,36)(10,69,40)(11,64,37)(12,66,41)(13,68,38)(14,70,42)(15,72,46)(16,74,43)(17,76,47)(18,71,44)(19,73,48)(20,75,45)(21,77,49)(22,79,53)(23,81,50)(24,83,54)(25,78,51)(26,80,55)(27,82,52)(28,84,56) );
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)], [(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,27),(2,28),(3,22),(4,23),(5,24),(6,25),(7,26),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49)], [(1,13),(2,14),(3,8),(4,9),(5,10),(6,11),(7,12),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(29,43),(30,44),(31,45),(32,46),(33,47),(34,48),(35,49),(36,50),(37,51),(38,52),(39,53),(40,54),(41,55),(42,56),(57,78),(58,79),(59,80),(60,81),(61,82),(62,83),(63,84),(64,71),(65,72),(66,73),(67,74),(68,75),(69,76),(70,77)], [(1,20),(2,21),(3,15),(4,16),(5,17),(6,18),(7,19),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,50),(30,51),(31,52),(32,53),(33,54),(34,55),(35,56),(36,43),(37,44),(38,45),(39,46),(40,47),(41,48),(42,49),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(71,78),(72,79),(73,80),(74,81),(75,82),(76,83),(77,84)], [(1,61,31),(2,63,35),(3,58,32),(4,60,29),(5,62,33),(6,57,30),(7,59,34),(8,65,39),(9,67,36),(10,69,40),(11,64,37),(12,66,41),(13,68,38),(14,70,42),(15,72,46),(16,74,43),(17,76,47),(18,71,44),(19,73,48),(20,75,45),(21,77,49),(22,79,53),(23,81,50),(24,83,54),(25,78,51),(26,80,55),(27,82,52),(28,84,56)]])
40 conjugacy classes
| class | 1 | 2A | ··· | 2E | 3A | 3B | 7A | 7B | 14A | ··· | 14AD |
| order | 1 | 2 | ··· | 2 | 3 | 3 | 7 | 7 | 14 | ··· | 14 |
| size | 1 | 3 | ··· | 3 | 112 | 112 | 3 | 3 | 3 | ··· | 3 |
40 irreducible representations
| dim | 1 | 1 | 3 | 3 | 3 |
| type | + | + | |||
| image | C1 | C3 | A4 | C7⋊C3 | C7⋊A4 |
| kernel | C7⋊(C22⋊A4) | C23×C14 | C2×C14 | C24 | C22 |
| # reps | 1 | 2 | 5 | 2 | 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 | 35 | 0 | 0 |
| 0 | 0 | 0 | 0 | 21 | 0 |
| 0 | 0 | 0 | 0 | 0 | 11 |
| 1 | 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 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 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 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 37 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 7 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 36 | 41 | 0 | 0 | 0 | 0 |
| 0 | 7 | 1 | 0 | 0 | 0 |
| 0 | 37 | 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))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,0,0,0,0,0,0,21,0,0,0,0,0,0,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1],[42,0,37,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42],[42,7,0,0,0,0,0,1,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1],[36,0,0,0,0,0,41,7,37,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0] >;
C7⋊(C22⋊A4) in GAP, Magma, Sage, TeX
C_7\rtimes (C_2^2\rtimes A_4)
% in TeX
G:=Group("C7:(C2^2:A4)"); // GroupNames label
G:=SmallGroup(336,224);
// by ID
G=gap.SmallGroup(336,224);
# by ID
G:=PCGroup([6,-3,-2,2,-2,2,-7,73,164,579,1084,3461]);
// 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,f*a*f^-1=a^4,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