direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C24⋊3C4, C24⋊4C28, C25.2C14, (C23×C14)⋊5C4, (C24×C14).1C2, C23.33(C7×D4), C14.87C22≀C2, C23.24(C2×C28), C24.26(C2×C14), (C22×C28)⋊3C22, C22.29(D4×C14), (C22×C14).153D4, C23.52(C22×C14), C22.28(C22×C28), (C23×C14).83C22, (C22×C14).443C23, (C2×C22⋊C4)⋊1C14, (C14×C22⋊C4)⋊5C2, (C22×C4)⋊1(C2×C14), C2.1(C7×C22≀C2), C2.4(C14×C22⋊C4), C22⋊2(C7×C22⋊C4), (C2×C14)⋊7(C22⋊C4), (C2×C14).596(C2×D4), C14.91(C2×C22⋊C4), (C2×C14).215(C22×C4), (C22×C14).110(C2×C4), SmallGroup(448,787)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C24⋊3C4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e2=f4=1, 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: 898 in 506 conjugacy classes, 130 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C22⋊C4, C22×C4, C24, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C25, C2×C28, C22×C14, C22×C14, C22×C14, C24⋊3C4, C7×C22⋊C4, C22×C28, C23×C14, C23×C14, C14×C22⋊C4, C24×C14, C7×C24⋊3C4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C23, C14, C22⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C22⋊C4, C22≀C2, C2×C28, C7×D4, C22×C14, C24⋊3C4, C7×C22⋊C4, C22×C28, D4×C14, C14×C22⋊C4, C7×C22≀C2, C7×C24⋊3C4
►On 112 points
Generators in S112
(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)
(1 95)(2 96)(3 97)(4 98)(5 92)(6 93)(7 94)(8 42)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 99)(22 32)(23 33)(24 34)(25 35)(26 29)(27 30)(28 31)(43 108)(44 109)(45 110)(46 111)(47 112)(48 106)(49 107)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 66)(58 67)(59 68)(60 69)(61 70)(62 64)(63 65)(78 87)(79 88)(80 89)(81 90)(82 91)(83 85)(84 86)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 109)(9 110)(10 111)(11 112)(12 106)(13 107)(14 108)(15 25)(16 26)(17 27)(18 28)(19 22)(20 23)(21 24)(29 101)(30 102)(31 103)(32 104)(33 105)(34 99)(35 100)(36 45)(37 46)(38 47)(39 48)(40 49)(41 43)(42 44)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(64 98)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 90)(72 91)(73 85)(74 86)(75 87)(76 88)(77 89)
(1 59)(2 60)(3 61)(4 62)(5 63)(6 57)(7 58)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 43)(15 35)(16 29)(17 30)(18 31)(19 32)(20 33)(21 34)(22 104)(23 105)(24 99)(25 100)(26 101)(27 102)(28 103)(36 110)(37 111)(38 112)(39 106)(40 107)(41 108)(42 109)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(64 98)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 90)(72 91)(73 85)(74 86)(75 87)(76 88)(77 89)
(1 68)(2 69)(3 70)(4 64)(5 65)(6 66)(7 67)(8 42)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 100)(16 101)(17 102)(18 103)(19 104)(20 105)(21 99)(22 32)(23 33)(24 34)(25 35)(26 29)(27 30)(28 31)(43 108)(44 109)(45 110)(46 111)(47 112)(48 106)(49 107)(50 87)(51 88)(52 89)(53 90)(54 91)(55 85)(56 86)(57 93)(58 94)(59 95)(60 96)(61 97)(62 98)(63 92)(71 81)(72 82)(73 83)(74 84)(75 78)(76 79)(77 80)
(1 39 78 32)(2 40 79 33)(3 41 80 34)(4 42 81 35)(5 36 82 29)(6 37 83 30)(7 38 84 31)(8 71 25 64)(9 72 26 65)(10 73 27 66)(11 74 28 67)(12 75 22 68)(13 76 23 69)(14 77 24 70)(15 62 109 53)(16 63 110 54)(17 57 111 55)(18 58 112 56)(19 59 106 50)(20 60 107 51)(21 61 108 52)(43 89 99 97)(44 90 100 98)(45 91 101 92)(46 85 102 93)(47 86 103 94)(48 87 104 95)(49 88 105 96)
G:=sub<Sym(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), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,99)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(43,108)(44,109)(45,110)(46,111)(47,112)(48,106)(49,107)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,66)(58,67)(59,68)(60,69)(61,70)(62,64)(63,65)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,109)(9,110)(10,111)(11,112)(12,106)(13,107)(14,108)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(29,101)(30,102)(31,103)(32,104)(33,105)(34,99)(35,100)(36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,98)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,90)(72,91)(73,85)(74,86)(75,87)(76,88)(77,89), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,104)(23,105)(24,99)(25,100)(26,101)(27,102)(28,103)(36,110)(37,111)(38,112)(39,106)(40,107)(41,108)(42,109)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,98)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,90)(72,91)(73,85)(74,86)(75,87)(76,88)(77,89), (1,68)(2,69)(3,70)(4,64)(5,65)(6,66)(7,67)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,99)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(43,108)(44,109)(45,110)(46,111)(47,112)(48,106)(49,107)(50,87)(51,88)(52,89)(53,90)(54,91)(55,85)(56,86)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(71,81)(72,82)(73,83)(74,84)(75,78)(76,79)(77,80), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96)>;
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)(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), (1,95)(2,96)(3,97)(4,98)(5,92)(6,93)(7,94)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,99)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(43,108)(44,109)(45,110)(46,111)(47,112)(48,106)(49,107)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,66)(58,67)(59,68)(60,69)(61,70)(62,64)(63,65)(78,87)(79,88)(80,89)(81,90)(82,91)(83,85)(84,86), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,109)(9,110)(10,111)(11,112)(12,106)(13,107)(14,108)(15,25)(16,26)(17,27)(18,28)(19,22)(20,23)(21,24)(29,101)(30,102)(31,103)(32,104)(33,105)(34,99)(35,100)(36,45)(37,46)(38,47)(39,48)(40,49)(41,43)(42,44)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,98)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,90)(72,91)(73,85)(74,86)(75,87)(76,88)(77,89), (1,59)(2,60)(3,61)(4,62)(5,63)(6,57)(7,58)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,43)(15,35)(16,29)(17,30)(18,31)(19,32)(20,33)(21,34)(22,104)(23,105)(24,99)(25,100)(26,101)(27,102)(28,103)(36,110)(37,111)(38,112)(39,106)(40,107)(41,108)(42,109)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(64,98)(65,92)(66,93)(67,94)(68,95)(69,96)(70,97)(71,90)(72,91)(73,85)(74,86)(75,87)(76,88)(77,89), (1,68)(2,69)(3,70)(4,64)(5,65)(6,66)(7,67)(8,42)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,100)(16,101)(17,102)(18,103)(19,104)(20,105)(21,99)(22,32)(23,33)(24,34)(25,35)(26,29)(27,30)(28,31)(43,108)(44,109)(45,110)(46,111)(47,112)(48,106)(49,107)(50,87)(51,88)(52,89)(53,90)(54,91)(55,85)(56,86)(57,93)(58,94)(59,95)(60,96)(61,97)(62,98)(63,92)(71,81)(72,82)(73,83)(74,84)(75,78)(76,79)(77,80), (1,39,78,32)(2,40,79,33)(3,41,80,34)(4,42,81,35)(5,36,82,29)(6,37,83,30)(7,38,84,31)(8,71,25,64)(9,72,26,65)(10,73,27,66)(11,74,28,67)(12,75,22,68)(13,76,23,69)(14,77,24,70)(15,62,109,53)(16,63,110,54)(17,57,111,55)(18,58,112,56)(19,59,106,50)(20,60,107,51)(21,61,108,52)(43,89,99,97)(44,90,100,98)(45,91,101,92)(46,85,102,93)(47,86,103,94)(48,87,104,95)(49,88,105,96) );
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),(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)], [(1,95),(2,96),(3,97),(4,98),(5,92),(6,93),(7,94),(8,42),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,99),(22,32),(23,33),(24,34),(25,35),(26,29),(27,30),(28,31),(43,108),(44,109),(45,110),(46,111),(47,112),(48,106),(49,107),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,66),(58,67),(59,68),(60,69),(61,70),(62,64),(63,65),(78,87),(79,88),(80,89),(81,90),(82,91),(83,85),(84,86)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,109),(9,110),(10,111),(11,112),(12,106),(13,107),(14,108),(15,25),(16,26),(17,27),(18,28),(19,22),(20,23),(21,24),(29,101),(30,102),(31,103),(32,104),(33,105),(34,99),(35,100),(36,45),(37,46),(38,47),(39,48),(40,49),(41,43),(42,44),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(64,98),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,90),(72,91),(73,85),(74,86),(75,87),(76,88),(77,89)], [(1,59),(2,60),(3,61),(4,62),(5,63),(6,57),(7,58),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,43),(15,35),(16,29),(17,30),(18,31),(19,32),(20,33),(21,34),(22,104),(23,105),(24,99),(25,100),(26,101),(27,102),(28,103),(36,110),(37,111),(38,112),(39,106),(40,107),(41,108),(42,109),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(64,98),(65,92),(66,93),(67,94),(68,95),(69,96),(70,97),(71,90),(72,91),(73,85),(74,86),(75,87),(76,88),(77,89)], [(1,68),(2,69),(3,70),(4,64),(5,65),(6,66),(7,67),(8,42),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,100),(16,101),(17,102),(18,103),(19,104),(20,105),(21,99),(22,32),(23,33),(24,34),(25,35),(26,29),(27,30),(28,31),(43,108),(44,109),(45,110),(46,111),(47,112),(48,106),(49,107),(50,87),(51,88),(52,89),(53,90),(54,91),(55,85),(56,86),(57,93),(58,94),(59,95),(60,96),(61,97),(62,98),(63,92),(71,81),(72,82),(73,83),(74,84),(75,78),(76,79),(77,80)], [(1,39,78,32),(2,40,79,33),(3,41,80,34),(4,42,81,35),(5,36,82,29),(6,37,83,30),(7,38,84,31),(8,71,25,64),(9,72,26,65),(10,73,27,66),(11,74,28,67),(12,75,22,68),(13,76,23,69),(14,77,24,70),(15,62,109,53),(16,63,110,54),(17,57,111,55),(18,58,112,56),(19,59,106,50),(20,60,107,51),(21,61,108,52),(43,89,99,97),(44,90,100,98),(45,91,101,92),(46,85,102,93),(47,86,103,94),(48,87,104,95),(49,88,105,96)]])
196 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2S | 4A | ··· | 4H | 7A | ··· | 7F | 14A | ··· | 14AP | 14AQ | ··· | 14DJ | 28A | ··· | 28AV |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
196 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C4 | C7 | C14 | C14 | C28 | D4 | C7×D4 |
| kernel | C7×C24⋊3C4 | C14×C22⋊C4 | C24×C14 | C23×C14 | C24⋊3C4 | C2×C22⋊C4 | C25 | C24 | C22×C14 | C23 |
| # reps | 1 | 6 | 1 | 8 | 6 | 36 | 6 | 48 | 12 | 72 |
Matrix representation of C7×C24⋊3C4 ►in GL5(𝔽29)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 7 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 23 | 0 |
| 0 | 0 | 0 | 0 | 23 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 28 | 1 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 28 | 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 | 28 | 0 |
| 0 | 0 | 0 | 0 | 28 |
| 17 | 0 | 0 | 0 | 0 |
| 0 | 1 | 27 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 |
G:=sub<GL(5,GF(29))| [1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,23,0,0,0,0,0,23],[28,0,0,0,0,0,28,28,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,28],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[17,0,0,0,0,0,1,0,0,0,0,27,28,0,0,0,0,0,0,28,0,0,0,1,0] >;
C7×C24⋊3C4 in GAP, Magma, Sage, TeX
C_7\times C_2^4\rtimes_3C_4
% in TeX
G:=Group("C7xC2^4:3C4"); // GroupNames label
G:=SmallGroup(448,787);
// by ID
G=gap.SmallGroup(448,787);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^2=f^4=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,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