direct product, metabelian, soluble, monomial, A-group
Aliases: C22×C7⋊A4, C14⋊2(C2×A4), (C2×C14)⋊3A4, C24⋊2(C7⋊C3), C7⋊2(C22×A4), (C23×C14)⋊5C3, (C22×C14)⋊8C6, C23⋊2(C2×C7⋊C3), C22⋊(C22×C7⋊C3), (C2×C14)⋊9(C2×C6), SmallGroup(336,222)
Series: Derived ►Chief ►Lower central ►Upper central
| C2×C14 — C22×C7⋊A4 |
Generators and relations for C22×C7⋊A4
G = < a,b,c,d,e,f | a2=b2=c7=d2=e2=f3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=c4, fdf-1=de=ed, fef-1=d >
Subgroups: 334 in 78 conjugacy classes, 25 normal (10 characteristic)
C1, C2, C2, C3, C22, C22, C6, C7, C23, C23, A4, C2×C6, C14, C14, C24, C7⋊C3, C2×A4, C2×C14, C2×C14, C2×C7⋊C3, C22×A4, C22×C14, C22×C14, C22×C7⋊C3, C7⋊A4, C23×C14, C2×C7⋊A4, C22×C7⋊A4
Quotients: C1, C2, C3, C22, C6, A4, C2×C6, C7⋊C3, C2×A4, C2×C7⋊C3, C22×A4, C22×C7⋊C3, C7⋊A4, C2×C7⋊A4, C22×C7⋊A4
►On 84 points
Generators in S84
(1 22)(2 23)(3 24)(4 25)(5 26)(6 27)(7 28)(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)(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 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 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 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 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)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 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 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 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 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)
(1 58 32)(2 60 29)(3 62 33)(4 57 30)(5 59 34)(6 61 31)(7 63 35)(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)| (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(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)(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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (1,58,32)(2,60,29)(3,62,33)(4,57,30)(5,59,34)(6,61,31)(7,63,35)(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( (1,22)(2,23)(3,24)(4,25)(5,26)(6,27)(7,28)(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)(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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (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), (1,15)(2,16)(3,17)(4,18)(5,19)(6,20)(7,21)(8,22)(9,23)(10,24)(11,25)(12,26)(13,27)(14,28)(29,36)(30,37)(31,38)(32,39)(33,40)(34,41)(35,42)(43,50)(44,51)(45,52)(46,53)(47,54)(48,55)(49,56)(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), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(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)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84), (1,58,32)(2,60,29)(3,62,33)(4,57,30)(5,59,34)(6,61,31)(7,63,35)(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([[(1,22),(2,23),(3,24),(4,25),(5,26),(6,27),(7,28),(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),(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)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(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),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)], [(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)], [(1,15),(2,16),(3,17),(4,18),(5,19),(6,20),(7,21),(8,22),(9,23),(10,24),(11,25),(12,26),(13,27),(14,28),(29,36),(30,37),(31,38),(32,39),(33,40),(34,41),(35,42),(43,50),(44,51),(45,52),(46,53),(47,54),(48,55),(49,56),(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)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(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),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84)], [(1,58,32),(2,60,29),(3,62,33),(4,57,30),(5,59,34),(6,61,31),(7,63,35),(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 6A | ··· | 6F | 7A | 7B | 14A | ··· | 14AD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 6 | ··· | 6 | 7 | 7 | 14 | ··· | 14 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 28 | 28 | 28 | ··· | 28 | 3 | 3 | 3 | ··· | 3 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C3 | C6 | A4 | C7⋊C3 | C2×A4 | C2×C7⋊C3 | C7⋊A4 | C2×C7⋊A4 |
| kernel | C22×C7⋊A4 | C2×C7⋊A4 | C23×C14 | C22×C14 | C2×C14 | C24 | C14 | C23 | C22 | C2 |
| # reps | 1 | 3 | 2 | 6 | 1 | 2 | 3 | 6 | 6 | 18 |
Matrix representation of C22×C7⋊A4 ►in GL4(𝔽43) generated by
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 0 | 0 | 42 |
| 42 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 35 | 0 | 0 |
| 0 | 7 | 21 | 0 |
| 0 | 25 | 0 | 11 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 1 | 42 | 0 |
| 0 | 20 | 0 | 42 |
| 1 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 |
| 0 | 0 | 42 | 0 |
| 0 | 23 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 35 | 16 | 0 |
| 0 | 29 | 8 | 1 |
| 0 | 41 | 31 | 0 |
G:=sub<GL(4,GF(43))| [1,0,0,0,0,42,0,0,0,0,42,0,0,0,0,42],[42,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,35,7,25,0,0,21,0,0,0,0,11],[1,0,0,0,0,1,1,20,0,0,42,0,0,0,0,42],[1,0,0,0,0,42,0,23,0,0,42,0,0,0,0,1],[6,0,0,0,0,35,29,41,0,16,8,31,0,0,1,0] >;
C22×C7⋊A4 in GAP, Magma, Sage, TeX
C_2^2\times C_7\rtimes A_4
% in TeX
G:=Group("C2^2xC7:A4"); // GroupNames label
G:=SmallGroup(336,222);
// by ID
G=gap.SmallGroup(336,222);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,2,-7,159,286,881]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^7=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,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=c^4,f*d*f^-1=d*e=e*d,f*e*f^-1=d>;
// generators/relations