direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.9D4, C22⋊C4⋊3C28, (C22×C4)⋊1C28, (C22×C28)⋊3C4, C23.2(C7×Q8), (C2×C14).8C42, C22.1(C4×C28), C24.4(C2×C14), C23.6(C2×C28), C23.32(C7×D4), (C22×C14).2Q8, C14.31(C23⋊C4), (C22×C14).152D4, (C23×C14).1C22, C14.26(C2.C42), (C7×C22⋊C4)⋊6C4, C22.4(C7×C4⋊C4), C2.3(C7×C23⋊C4), (C2×C14).21(C4⋊C4), (C2×C22⋊C4).2C14, (C14×C22⋊C4).4C2, C22.8(C7×C22⋊C4), (C22×C14).17(C2×C4), C2.7(C7×C2.C42), (C2×C14).131(C22⋊C4), SmallGroup(448,146)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.9D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=bcd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=bde-1 >
Subgroups: 298 in 142 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C22×C4, C22×C4, C24, C28, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C22⋊C4, C2×C28, C22×C14, C22×C14, C22×C14, C23.9D4, C7×C22⋊C4, C7×C22⋊C4, C22×C28, C22×C28, C23×C14, C14×C22⋊C4, C14×C22⋊C4, C7×C23.9D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, Q8, C14, C42, C22⋊C4, C4⋊C4, C28, C2×C14, C2.C42, C23⋊C4, C2×C28, C7×D4, C7×Q8, C23.9D4, C4×C28, C7×C22⋊C4, C7×C4⋊C4, C7×C2.C42, C7×C23⋊C4, C7×C23.9D4
►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 87)(2 88)(3 89)(4 90)(5 91)(6 85)(7 86)(8 26)(9 27)(10 28)(11 22)(12 23)(13 24)(14 25)(15 108)(16 109)(17 110)(18 111)(19 112)(20 106)(21 107)(29 46)(30 47)(31 48)(32 49)(33 43)(34 44)(35 45)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 99)(50 67)(51 68)(52 69)(53 70)(54 64)(55 65)(56 66)(57 74)(58 75)(59 76)(60 77)(61 71)(62 72)(63 73)(78 95)(79 96)(80 97)(81 98)(82 92)(83 93)(84 94)
(8 110)(9 111)(10 112)(11 106)(12 107)(13 108)(14 109)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 38)(30 39)(31 40)(32 41)(33 42)(34 36)(35 37)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)
(1 78)(2 79)(3 80)(4 81)(5 82)(6 83)(7 84)(8 110)(9 111)(10 112)(11 106)(12 107)(13 108)(14 109)(15 24)(16 25)(17 26)(18 27)(19 28)(20 22)(21 23)(29 38)(30 39)(31 40)(32 41)(33 42)(34 36)(35 37)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 59)(51 60)(52 61)(53 62)(54 63)(55 57)(56 58)(64 73)(65 74)(66 75)(67 76)(68 77)(69 71)(70 72)(85 93)(86 94)(87 95)(88 96)(89 97)(90 98)(91 92)
(1 22 76 40)(2 23 77 41)(3 24 71 42)(4 25 72 36)(5 26 73 37)(6 27 74 38)(7 28 75 39)(8 63 101 91)(9 57 102 85)(10 58 103 86)(11 59 104 87)(12 60 105 88)(13 61 99 89)(14 62 100 90)(15 69 33 80)(16 70 34 81)(17 64 35 82)(18 65 29 83)(19 66 30 84)(20 67 31 78)(21 68 32 79)(43 97 108 52)(44 98 109 53)(45 92 110 54)(46 93 111 55)(47 94 112 56)(48 95 106 50)(49 96 107 51)
(1 11 95 22)(2 12 96 23)(3 13 97 24)(4 14 98 25)(5 8 92 26)(6 9 93 27)(7 10 94 28)(15 80 108 89)(16 81 109 90)(17 82 110 91)(18 83 111 85)(19 84 112 86)(20 78 106 87)(21 79 107 88)(29 65 46 57)(30 66 47 58)(31 67 48 59)(32 68 49 60)(33 69 43 61)(34 70 44 62)(35 64 45 63)(36 72 100 53)(37 73 101 54)(38 74 102 55)(39 75 103 56)(40 76 104 50)(41 77 105 51)(42 71 99 52)
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), (1,87)(2,88)(3,89)(4,90)(5,91)(6,85)(7,86)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25)(15,108)(16,109)(17,110)(18,111)(19,112)(20,106)(21,107)(29,46)(30,47)(31,48)(32,49)(33,43)(34,44)(35,45)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,74)(58,75)(59,76)(60,77)(61,71)(62,72)(63,73)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94), (8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,22,76,40)(2,23,77,41)(3,24,71,42)(4,25,72,36)(5,26,73,37)(6,27,74,38)(7,28,75,39)(8,63,101,91)(9,57,102,85)(10,58,103,86)(11,59,104,87)(12,60,105,88)(13,61,99,89)(14,62,100,90)(15,69,33,80)(16,70,34,81)(17,64,35,82)(18,65,29,83)(19,66,30,84)(20,67,31,78)(21,68,32,79)(43,97,108,52)(44,98,109,53)(45,92,110,54)(46,93,111,55)(47,94,112,56)(48,95,106,50)(49,96,107,51), (1,11,95,22)(2,12,96,23)(3,13,97,24)(4,14,98,25)(5,8,92,26)(6,9,93,27)(7,10,94,28)(15,80,108,89)(16,81,109,90)(17,82,110,91)(18,83,111,85)(19,84,112,86)(20,78,106,87)(21,79,107,88)(29,65,46,57)(30,66,47,58)(31,67,48,59)(32,68,49,60)(33,69,43,61)(34,70,44,62)(35,64,45,63)(36,72,100,53)(37,73,101,54)(38,74,102,55)(39,75,103,56)(40,76,104,50)(41,77,105,51)(42,71,99,52)>;
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)(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), (1,87)(2,88)(3,89)(4,90)(5,91)(6,85)(7,86)(8,26)(9,27)(10,28)(11,22)(12,23)(13,24)(14,25)(15,108)(16,109)(17,110)(18,111)(19,112)(20,106)(21,107)(29,46)(30,47)(31,48)(32,49)(33,43)(34,44)(35,45)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(50,67)(51,68)(52,69)(53,70)(54,64)(55,65)(56,66)(57,74)(58,75)(59,76)(60,77)(61,71)(62,72)(63,73)(78,95)(79,96)(80,97)(81,98)(82,92)(83,93)(84,94), (8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105), (1,78)(2,79)(3,80)(4,81)(5,82)(6,83)(7,84)(8,110)(9,111)(10,112)(11,106)(12,107)(13,108)(14,109)(15,24)(16,25)(17,26)(18,27)(19,28)(20,22)(21,23)(29,38)(30,39)(31,40)(32,41)(33,42)(34,36)(35,37)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,59)(51,60)(52,61)(53,62)(54,63)(55,57)(56,58)(64,73)(65,74)(66,75)(67,76)(68,77)(69,71)(70,72)(85,93)(86,94)(87,95)(88,96)(89,97)(90,98)(91,92), (1,22,76,40)(2,23,77,41)(3,24,71,42)(4,25,72,36)(5,26,73,37)(6,27,74,38)(7,28,75,39)(8,63,101,91)(9,57,102,85)(10,58,103,86)(11,59,104,87)(12,60,105,88)(13,61,99,89)(14,62,100,90)(15,69,33,80)(16,70,34,81)(17,64,35,82)(18,65,29,83)(19,66,30,84)(20,67,31,78)(21,68,32,79)(43,97,108,52)(44,98,109,53)(45,92,110,54)(46,93,111,55)(47,94,112,56)(48,95,106,50)(49,96,107,51), (1,11,95,22)(2,12,96,23)(3,13,97,24)(4,14,98,25)(5,8,92,26)(6,9,93,27)(7,10,94,28)(15,80,108,89)(16,81,109,90)(17,82,110,91)(18,83,111,85)(19,84,112,86)(20,78,106,87)(21,79,107,88)(29,65,46,57)(30,66,47,58)(31,67,48,59)(32,68,49,60)(33,69,43,61)(34,70,44,62)(35,64,45,63)(36,72,100,53)(37,73,101,54)(38,74,102,55)(39,75,103,56)(40,76,104,50)(41,77,105,51)(42,71,99,52) );
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),(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)], [(1,87),(2,88),(3,89),(4,90),(5,91),(6,85),(7,86),(8,26),(9,27),(10,28),(11,22),(12,23),(13,24),(14,25),(15,108),(16,109),(17,110),(18,111),(19,112),(20,106),(21,107),(29,46),(30,47),(31,48),(32,49),(33,43),(34,44),(35,45),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,99),(50,67),(51,68),(52,69),(53,70),(54,64),(55,65),(56,66),(57,74),(58,75),(59,76),(60,77),(61,71),(62,72),(63,73),(78,95),(79,96),(80,97),(81,98),(82,92),(83,93),(84,94)], [(8,110),(9,111),(10,112),(11,106),(12,107),(13,108),(14,109),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(29,38),(30,39),(31,40),(32,41),(33,42),(34,36),(35,37),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105)], [(1,78),(2,79),(3,80),(4,81),(5,82),(6,83),(7,84),(8,110),(9,111),(10,112),(11,106),(12,107),(13,108),(14,109),(15,24),(16,25),(17,26),(18,27),(19,28),(20,22),(21,23),(29,38),(30,39),(31,40),(32,41),(33,42),(34,36),(35,37),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,59),(51,60),(52,61),(53,62),(54,63),(55,57),(56,58),(64,73),(65,74),(66,75),(67,76),(68,77),(69,71),(70,72),(85,93),(86,94),(87,95),(88,96),(89,97),(90,98),(91,92)], [(1,22,76,40),(2,23,77,41),(3,24,71,42),(4,25,72,36),(5,26,73,37),(6,27,74,38),(7,28,75,39),(8,63,101,91),(9,57,102,85),(10,58,103,86),(11,59,104,87),(12,60,105,88),(13,61,99,89),(14,62,100,90),(15,69,33,80),(16,70,34,81),(17,64,35,82),(18,65,29,83),(19,66,30,84),(20,67,31,78),(21,68,32,79),(43,97,108,52),(44,98,109,53),(45,92,110,54),(46,93,111,55),(47,94,112,56),(48,95,106,50),(49,96,107,51)], [(1,11,95,22),(2,12,96,23),(3,13,97,24),(4,14,98,25),(5,8,92,26),(6,9,93,27),(7,10,94,28),(15,80,108,89),(16,81,109,90),(17,82,110,91),(18,83,111,85),(19,84,112,86),(20,78,106,87),(21,79,107,88),(29,65,46,57),(30,66,47,58),(31,67,48,59),(32,68,49,60),(33,69,43,61),(34,70,44,62),(35,64,45,63),(36,72,100,53),(37,73,101,54),(38,74,102,55),(39,75,103,56),(40,76,104,50),(41,77,105,51),(42,71,99,52)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 4A | ··· | 4L | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14BB | 28A | ··· | 28BT |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | |||||||||
| image | C1 | C2 | C4 | C4 | C7 | C14 | C28 | C28 | D4 | Q8 | C7×D4 | C7×Q8 | C23⋊C4 | C7×C23⋊C4 |
| kernel | C7×C23.9D4 | C14×C22⋊C4 | C7×C22⋊C4 | C22×C28 | C23.9D4 | C2×C22⋊C4 | C22⋊C4 | C22×C4 | C22×C14 | C22×C14 | C23 | C23 | C14 | C2 |
| # reps | 1 | 3 | 8 | 4 | 6 | 18 | 48 | 24 | 3 | 1 | 18 | 6 | 2 | 12 |
Matrix representation of C7×C23.9D4 ►in GL6(𝔽29)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 20 | 0 | 0 | 0 |
| 0 | 0 | 0 | 20 | 0 | 0 |
| 0 | 0 | 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 0 | 0 | 20 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 28 |
| 28 | 1 | 0 | 0 | 0 | 0 |
| 27 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 | 0 | 0 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 17 | 0 | 0 | 0 | 0 | 0 |
| 5 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20,0,0,0,0,0,0,20],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,28,0],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,27,0,0,0,0,1,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,1,0,0,0,0,0,0,1,0,0],[17,5,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,28,0,0,0,28,0,0,0,0,0,0,1,0,0] >;
C7×C23.9D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._9D_4
% in TeX
G:=Group("C7xC2^3.9D4"); // GroupNames label
G:=SmallGroup(448,146);
// by ID
G=gap.SmallGroup(448,146);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,792,7059,4911]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=b*c*d,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,e*c*e^-1=f*c*f^-1=c*d=d*c,d*e=e*d,d*f=f*d,f*e*f^-1=b*d*e^-1>;
// generators/relations