direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C24.4C4, C24.5C28, C4.70(D4×C14), C22⋊C8⋊11C14, (C2×C56)⋊37C22, C28.475(C2×D4), (C2×C28).513D4, (C2×C14)⋊7M4(2), (C23×C14).6C4, (C2×M4(2))⋊6C14, (C23×C4).10C14, (C23×C28).23C2, (C22×C28).20C4, (C22×C4).12C28, C23.29(C2×C28), C2.6(C14×M4(2)), C22⋊2(C7×M4(2)), (C14×M4(2))⋊24C2, (C2×C28).982C23, C14.48(C2×M4(2)), C28.107(C22⋊C4), C22.41(C22×C28), (C22×C28).496C22, (C2×C8)⋊7(C2×C14), (C7×C22⋊C8)⋊28C2, (C2×C4).71(C2×C28), (C2×C4).118(C7×D4), C4.21(C7×C22⋊C4), (C2×C28).333(C2×C4), C2.10(C14×C22⋊C4), C14.98(C2×C22⋊C4), (C22×C4).92(C2×C14), C22.17(C7×C22⋊C4), (C2×C14).80(C22⋊C4), (C2×C4).150(C22×C14), (C2×C14).232(C22×C4), (C22×C14).115(C2×C4), SmallGroup(448,815)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C24.4C4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e2=1, f4=e, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, fcf-1=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 290 in 190 conjugacy classes, 90 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C14, C14, C14, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C24, C28, C28, C2×C14, C2×C14, C2×C14, C22⋊C8, C2×M4(2), C23×C4, C56, C2×C28, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C24.4C4, C2×C56, C7×M4(2), C22×C28, C22×C28, C22×C28, C23×C14, C7×C22⋊C8, C14×M4(2), C23×C28, C7×C24.4C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, M4(2), C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C2×M4(2), C2×C28, C7×D4, C22×C14, C24.4C4, C7×C22⋊C4, C7×M4(2), C22×C28, D4×C14, C14×C22⋊C4, C14×M4(2), C7×C24.4C4
►On 112 points
Generators in S112
(1 65 16 83 27 75 19)(2 66 9 84 28 76 20)(3 67 10 85 29 77 21)(4 68 11 86 30 78 22)(5 69 12 87 31 79 23)(6 70 13 88 32 80 24)(7 71 14 81 25 73 17)(8 72 15 82 26 74 18)(33 61 105 49 97 41 89)(34 62 106 50 98 42 90)(35 63 107 51 99 43 91)(36 64 108 52 100 44 92)(37 57 109 53 101 45 93)(38 58 110 54 102 46 94)(39 59 111 55 103 47 95)(40 60 112 56 104 48 96)
(2 40)(4 34)(6 36)(8 38)(9 112)(11 106)(13 108)(15 110)(18 94)(20 96)(22 90)(24 92)(26 102)(28 104)(30 98)(32 100)(42 78)(44 80)(46 74)(48 76)(50 86)(52 88)(54 82)(56 84)(58 72)(60 66)(62 68)(64 70)
(1 39)(2 36)(3 33)(4 38)(5 35)(6 40)(7 37)(8 34)(9 108)(10 105)(11 110)(12 107)(13 112)(14 109)(15 106)(16 111)(17 93)(18 90)(19 95)(20 92)(21 89)(22 94)(23 91)(24 96)(25 101)(26 98)(27 103)(28 100)(29 97)(30 102)(31 99)(32 104)(41 77)(42 74)(43 79)(44 76)(45 73)(46 78)(47 75)(48 80)(49 85)(50 82)(51 87)(52 84)(53 81)(54 86)(55 83)(56 88)(57 71)(58 68)(59 65)(60 70)(61 67)(62 72)(63 69)(64 66)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 112)(10 105)(11 106)(12 107)(13 108)(14 109)(15 110)(16 111)(17 93)(18 94)(19 95)(20 96)(21 89)(22 90)(23 91)(24 92)(25 101)(26 102)(27 103)(28 104)(29 97)(30 98)(31 99)(32 100)(41 77)(42 78)(43 79)(44 80)(45 73)(46 74)(47 75)(48 76)(49 85)(50 86)(51 87)(52 88)(53 81)(54 82)(55 83)(56 84)(57 71)(58 72)(59 65)(60 66)(61 67)(62 68)(63 69)(64 70)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)(73 77)(74 78)(75 79)(76 80)(81 85)(82 86)(83 87)(84 88)(89 93)(90 94)(91 95)(92 96)(97 101)(98 102)(99 103)(100 104)(105 109)(106 110)(107 111)(108 112)
(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)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)
G:=sub<Sym(112)| (1,65,16,83,27,75,19)(2,66,9,84,28,76,20)(3,67,10,85,29,77,21)(4,68,11,86,30,78,22)(5,69,12,87,31,79,23)(6,70,13,88,32,80,24)(7,71,14,81,25,73,17)(8,72,15,82,26,74,18)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (2,40)(4,34)(6,36)(8,38)(9,112)(11,106)(13,108)(15,110)(18,94)(20,96)(22,90)(24,92)(26,102)(28,104)(30,98)(32,100)(42,78)(44,80)(46,74)(48,76)(50,86)(52,88)(54,82)(56,84)(58,72)(60,66)(62,68)(64,70), (1,39)(2,36)(3,33)(4,38)(5,35)(6,40)(7,37)(8,34)(9,108)(10,105)(11,110)(12,107)(13,112)(14,109)(15,106)(16,111)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,101)(26,98)(27,103)(28,100)(29,97)(30,102)(31,99)(32,104)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,85)(50,82)(51,87)(52,84)(53,81)(54,86)(55,83)(56,88)(57,71)(58,68)(59,65)(60,70)(61,67)(62,72)(63,69)(64,66), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,93)(18,94)(19,95)(20,96)(21,89)(22,90)(23,91)(24,92)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,85)(50,86)(51,87)(52,88)(53,81)(54,82)(55,83)(56,84)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)>;
G:=Group( (1,65,16,83,27,75,19)(2,66,9,84,28,76,20)(3,67,10,85,29,77,21)(4,68,11,86,30,78,22)(5,69,12,87,31,79,23)(6,70,13,88,32,80,24)(7,71,14,81,25,73,17)(8,72,15,82,26,74,18)(33,61,105,49,97,41,89)(34,62,106,50,98,42,90)(35,63,107,51,99,43,91)(36,64,108,52,100,44,92)(37,57,109,53,101,45,93)(38,58,110,54,102,46,94)(39,59,111,55,103,47,95)(40,60,112,56,104,48,96), (2,40)(4,34)(6,36)(8,38)(9,112)(11,106)(13,108)(15,110)(18,94)(20,96)(22,90)(24,92)(26,102)(28,104)(30,98)(32,100)(42,78)(44,80)(46,74)(48,76)(50,86)(52,88)(54,82)(56,84)(58,72)(60,66)(62,68)(64,70), (1,39)(2,36)(3,33)(4,38)(5,35)(6,40)(7,37)(8,34)(9,108)(10,105)(11,110)(12,107)(13,112)(14,109)(15,106)(16,111)(17,93)(18,90)(19,95)(20,92)(21,89)(22,94)(23,91)(24,96)(25,101)(26,98)(27,103)(28,100)(29,97)(30,102)(31,99)(32,104)(41,77)(42,74)(43,79)(44,76)(45,73)(46,78)(47,75)(48,80)(49,85)(50,82)(51,87)(52,84)(53,81)(54,86)(55,83)(56,88)(57,71)(58,68)(59,65)(60,70)(61,67)(62,72)(63,69)(64,66), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,112)(10,105)(11,106)(12,107)(13,108)(14,109)(15,110)(16,111)(17,93)(18,94)(19,95)(20,96)(21,89)(22,90)(23,91)(24,92)(25,101)(26,102)(27,103)(28,104)(29,97)(30,98)(31,99)(32,100)(41,77)(42,78)(43,79)(44,80)(45,73)(46,74)(47,75)(48,76)(49,85)(50,86)(51,87)(52,88)(53,81)(54,82)(55,83)(56,84)(57,71)(58,72)(59,65)(60,66)(61,67)(62,68)(63,69)(64,70), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64)(65,69)(66,70)(67,71)(68,72)(73,77)(74,78)(75,79)(76,80)(81,85)(82,86)(83,87)(84,88)(89,93)(90,94)(91,95)(92,96)(97,101)(98,102)(99,103)(100,104)(105,109)(106,110)(107,111)(108,112), (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)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112) );
G=PermutationGroup([[(1,65,16,83,27,75,19),(2,66,9,84,28,76,20),(3,67,10,85,29,77,21),(4,68,11,86,30,78,22),(5,69,12,87,31,79,23),(6,70,13,88,32,80,24),(7,71,14,81,25,73,17),(8,72,15,82,26,74,18),(33,61,105,49,97,41,89),(34,62,106,50,98,42,90),(35,63,107,51,99,43,91),(36,64,108,52,100,44,92),(37,57,109,53,101,45,93),(38,58,110,54,102,46,94),(39,59,111,55,103,47,95),(40,60,112,56,104,48,96)], [(2,40),(4,34),(6,36),(8,38),(9,112),(11,106),(13,108),(15,110),(18,94),(20,96),(22,90),(24,92),(26,102),(28,104),(30,98),(32,100),(42,78),(44,80),(46,74),(48,76),(50,86),(52,88),(54,82),(56,84),(58,72),(60,66),(62,68),(64,70)], [(1,39),(2,36),(3,33),(4,38),(5,35),(6,40),(7,37),(8,34),(9,108),(10,105),(11,110),(12,107),(13,112),(14,109),(15,106),(16,111),(17,93),(18,90),(19,95),(20,92),(21,89),(22,94),(23,91),(24,96),(25,101),(26,98),(27,103),(28,100),(29,97),(30,102),(31,99),(32,104),(41,77),(42,74),(43,79),(44,76),(45,73),(46,78),(47,75),(48,80),(49,85),(50,82),(51,87),(52,84),(53,81),(54,86),(55,83),(56,88),(57,71),(58,68),(59,65),(60,70),(61,67),(62,72),(63,69),(64,66)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,112),(10,105),(11,106),(12,107),(13,108),(14,109),(15,110),(16,111),(17,93),(18,94),(19,95),(20,96),(21,89),(22,90),(23,91),(24,92),(25,101),(26,102),(27,103),(28,104),(29,97),(30,98),(31,99),(32,100),(41,77),(42,78),(43,79),(44,80),(45,73),(46,74),(47,75),(48,76),(49,85),(50,86),(51,87),(52,88),(53,81),(54,82),(55,83),(56,84),(57,71),(58,72),(59,65),(60,66),(61,67),(62,68),(63,69),(64,70)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64),(65,69),(66,70),(67,71),(68,72),(73,77),(74,78),(75,79),(76,80),(81,85),(82,86),(83,87),(84,88),(89,93),(90,94),(91,95),(92,96),(97,101),(98,102),(99,103),(100,104),(105,109),(106,110),(107,111),(108,112)], [(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),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112)]])
196 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28X | 28Y | ··· | 28BH | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | D4 | M4(2) | C7×D4 | C7×M4(2) |
| kernel | C7×C24.4C4 | C7×C22⋊C8 | C14×M4(2) | C23×C28 | C22×C28 | C23×C14 | C24.4C4 | C22⋊C8 | C2×M4(2) | C23×C4 | C22×C4 | C24 | C2×C28 | C2×C14 | C2×C4 | C22 |
| # reps | 1 | 4 | 2 | 1 | 6 | 2 | 6 | 24 | 12 | 6 | 36 | 12 | 4 | 8 | 24 | 48 |
Matrix representation of C7×C24.4C4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 30 | 0 |
| 0 | 0 | 0 | 30 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 112 | 0 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,30,0,0,0,0,30],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,112],[1,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,112,0,0,0,0,112],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[0,98,0,0,1,0,0,0,0,0,0,112,0,0,1,0] >;
C7×C24.4C4 in GAP, Magma, Sage, TeX
C_7\times C_2^4._4C_4
% in TeX
G:=Group("C7xC2^4.4C4"); // GroupNames label
G:=SmallGroup(448,815);
// by ID
G=gap.SmallGroup(448,815);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,784,813,4790,124]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^2=1,f^4=e,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f^-1=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations