direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23⋊C8, C23⋊2C56, C24.2C28, C22⋊C8⋊1C14, (C22×C14)⋊1C8, (C2×C28).441D4, (C22×C4).1C28, C22.2(C2×C56), (C23×C14).1C4, (C22×C28).4C4, C23.21(C2×C28), C14.27(C23⋊C4), C14.20(C22⋊C8), (C2×C14).14M4(2), (C22×C28).1C22, C22.2(C7×M4(2)), C14.11(C4.D4), (C7×C22⋊C8)⋊3C2, (C2×C4).91(C7×D4), C2.3(C7×C22⋊C8), C2.1(C7×C23⋊C4), (C2×C14).20(C2×C8), C2.1(C7×C4.D4), (C14×C22⋊C4).3C2, (C2×C22⋊C4).1C14, (C22×C4).1(C2×C14), C22.23(C7×C22⋊C4), (C22×C14).106(C2×C4), (C2×C14).118(C22⋊C4), SmallGroup(448,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23⋊C8
G = < a,b,c,d,e | a7=b2=c2=d2=e8=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 218 in 98 conjugacy classes, 38 normal (26 characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C23, C23, C14, C14, C22⋊C4, C2×C8, C22×C4, C24, C28, C2×C14, C2×C14, C22⋊C8, C2×C22⋊C4, C56, C2×C28, C2×C28, C22×C14, C22×C14, C22×C14, C23⋊C8, C7×C22⋊C4, C2×C56, C22×C28, C23×C14, C7×C22⋊C8, C14×C22⋊C4, C7×C23⋊C8
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, D4, C14, C22⋊C4, C2×C8, M4(2), C28, C2×C14, C22⋊C8, C23⋊C4, C4.D4, C56, C2×C28, C7×D4, C23⋊C8, C7×C22⋊C4, C2×C56, C7×M4(2), C7×C22⋊C8, C7×C23⋊C4, C7×C4.D4, C7×C23⋊C8
►On 112 points
Generators in S112
(1 70 10 23 87 31 79)(2 71 11 24 88 32 80)(3 72 12 17 81 25 73)(4 65 13 18 82 26 74)(5 66 14 19 83 27 75)(6 67 15 20 84 28 76)(7 68 16 21 85 29 77)(8 69 9 22 86 30 78)(33 63 105 49 97 41 89)(34 64 106 50 98 42 90)(35 57 107 51 99 43 91)(36 58 108 52 100 44 92)(37 59 109 53 101 45 93)(38 60 110 54 102 46 94)(39 61 111 55 103 47 95)(40 62 112 56 104 48 96)
(1 5)(2 40)(3 37)(6 36)(7 33)(10 14)(11 112)(12 109)(15 108)(16 105)(17 53)(19 23)(20 52)(21 49)(24 56)(25 45)(27 31)(28 44)(29 41)(32 48)(35 39)(43 47)(51 55)(57 61)(58 67)(59 72)(62 71)(63 68)(66 70)(73 93)(75 79)(76 92)(77 89)(80 96)(81 101)(83 87)(84 100)(85 97)(88 104)(91 95)(99 103)(107 111)
(1 5)(2 36)(3 7)(4 38)(6 40)(8 34)(9 106)(10 14)(11 108)(12 16)(13 110)(15 112)(17 21)(18 54)(19 23)(20 56)(22 50)(24 52)(25 29)(26 46)(27 31)(28 48)(30 42)(32 44)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(58 71)(59 63)(60 65)(62 67)(64 69)(66 70)(68 72)(73 77)(74 94)(75 79)(76 96)(78 90)(80 92)(81 85)(82 102)(83 87)(84 104)(86 98)(88 100)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(1 39)(2 40)(3 33)(4 34)(5 35)(6 36)(7 37)(8 38)(9 110)(10 111)(11 112)(12 105)(13 106)(14 107)(15 108)(16 109)(17 49)(18 50)(19 51)(20 52)(21 53)(22 54)(23 55)(24 56)(25 41)(26 42)(27 43)(28 44)(29 45)(30 46)(31 47)(32 48)(57 66)(58 67)(59 68)(60 69)(61 70)(62 71)(63 72)(64 65)(73 89)(74 90)(75 91)(76 92)(77 93)(78 94)(79 95)(80 96)(81 97)(82 98)(83 99)(84 100)(85 101)(86 102)(87 103)(88 104)
(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)| (1,70,10,23,87,31,79)(2,71,11,24,88,32,80)(3,72,12,17,81,25,73)(4,65,13,18,82,26,74)(5,66,14,19,83,27,75)(6,67,15,20,84,28,76)(7,68,16,21,85,29,77)(8,69,9,22,86,30,78)(33,63,105,49,97,41,89)(34,64,106,50,98,42,90)(35,57,107,51,99,43,91)(36,58,108,52,100,44,92)(37,59,109,53,101,45,93)(38,60,110,54,102,46,94)(39,61,111,55,103,47,95)(40,62,112,56,104,48,96), (1,5)(2,40)(3,37)(6,36)(7,33)(10,14)(11,112)(12,109)(15,108)(16,105)(17,53)(19,23)(20,52)(21,49)(24,56)(25,45)(27,31)(28,44)(29,41)(32,48)(35,39)(43,47)(51,55)(57,61)(58,67)(59,72)(62,71)(63,68)(66,70)(73,93)(75,79)(76,92)(77,89)(80,96)(81,101)(83,87)(84,100)(85,97)(88,104)(91,95)(99,103)(107,111), (1,5)(2,36)(3,7)(4,38)(6,40)(8,34)(9,106)(10,14)(11,108)(12,16)(13,110)(15,112)(17,21)(18,54)(19,23)(20,56)(22,50)(24,52)(25,29)(26,46)(27,31)(28,48)(30,42)(32,44)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(58,71)(59,63)(60,65)(62,67)(64,69)(66,70)(68,72)(73,77)(74,94)(75,79)(76,96)(78,90)(80,92)(81,85)(82,102)(83,87)(84,104)(86,98)(88,100)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104), (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( (1,70,10,23,87,31,79)(2,71,11,24,88,32,80)(3,72,12,17,81,25,73)(4,65,13,18,82,26,74)(5,66,14,19,83,27,75)(6,67,15,20,84,28,76)(7,68,16,21,85,29,77)(8,69,9,22,86,30,78)(33,63,105,49,97,41,89)(34,64,106,50,98,42,90)(35,57,107,51,99,43,91)(36,58,108,52,100,44,92)(37,59,109,53,101,45,93)(38,60,110,54,102,46,94)(39,61,111,55,103,47,95)(40,62,112,56,104,48,96), (1,5)(2,40)(3,37)(6,36)(7,33)(10,14)(11,112)(12,109)(15,108)(16,105)(17,53)(19,23)(20,52)(21,49)(24,56)(25,45)(27,31)(28,44)(29,41)(32,48)(35,39)(43,47)(51,55)(57,61)(58,67)(59,72)(62,71)(63,68)(66,70)(73,93)(75,79)(76,92)(77,89)(80,96)(81,101)(83,87)(84,100)(85,97)(88,104)(91,95)(99,103)(107,111), (1,5)(2,36)(3,7)(4,38)(6,40)(8,34)(9,106)(10,14)(11,108)(12,16)(13,110)(15,112)(17,21)(18,54)(19,23)(20,56)(22,50)(24,52)(25,29)(26,46)(27,31)(28,48)(30,42)(32,44)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(57,61)(58,71)(59,63)(60,65)(62,67)(64,69)(66,70)(68,72)(73,77)(74,94)(75,79)(76,96)(78,90)(80,92)(81,85)(82,102)(83,87)(84,104)(86,98)(88,100)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (1,39)(2,40)(3,33)(4,34)(5,35)(6,36)(7,37)(8,38)(9,110)(10,111)(11,112)(12,105)(13,106)(14,107)(15,108)(16,109)(17,49)(18,50)(19,51)(20,52)(21,53)(22,54)(23,55)(24,56)(25,41)(26,42)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(57,66)(58,67)(59,68)(60,69)(61,70)(62,71)(63,72)(64,65)(73,89)(74,90)(75,91)(76,92)(77,93)(78,94)(79,95)(80,96)(81,97)(82,98)(83,99)(84,100)(85,101)(86,102)(87,103)(88,104), (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([[(1,70,10,23,87,31,79),(2,71,11,24,88,32,80),(3,72,12,17,81,25,73),(4,65,13,18,82,26,74),(5,66,14,19,83,27,75),(6,67,15,20,84,28,76),(7,68,16,21,85,29,77),(8,69,9,22,86,30,78),(33,63,105,49,97,41,89),(34,64,106,50,98,42,90),(35,57,107,51,99,43,91),(36,58,108,52,100,44,92),(37,59,109,53,101,45,93),(38,60,110,54,102,46,94),(39,61,111,55,103,47,95),(40,62,112,56,104,48,96)], [(1,5),(2,40),(3,37),(6,36),(7,33),(10,14),(11,112),(12,109),(15,108),(16,105),(17,53),(19,23),(20,52),(21,49),(24,56),(25,45),(27,31),(28,44),(29,41),(32,48),(35,39),(43,47),(51,55),(57,61),(58,67),(59,72),(62,71),(63,68),(66,70),(73,93),(75,79),(76,92),(77,89),(80,96),(81,101),(83,87),(84,100),(85,97),(88,104),(91,95),(99,103),(107,111)], [(1,5),(2,36),(3,7),(4,38),(6,40),(8,34),(9,106),(10,14),(11,108),(12,16),(13,110),(15,112),(17,21),(18,54),(19,23),(20,56),(22,50),(24,52),(25,29),(26,46),(27,31),(28,48),(30,42),(32,44),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(57,61),(58,71),(59,63),(60,65),(62,67),(64,69),(66,70),(68,72),(73,77),(74,94),(75,79),(76,96),(78,90),(80,92),(81,85),(82,102),(83,87),(84,104),(86,98),(88,100),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)], [(1,39),(2,40),(3,33),(4,34),(5,35),(6,36),(7,37),(8,38),(9,110),(10,111),(11,112),(12,105),(13,106),(14,107),(15,108),(16,109),(17,49),(18,50),(19,51),(20,52),(21,53),(22,54),(23,55),(24,56),(25,41),(26,42),(27,43),(28,44),(29,45),(30,46),(31,47),(32,48),(57,66),(58,67),(59,68),(60,69),(61,70),(62,71),(63,72),(64,65),(73,89),(74,90),(75,91),(76,92),(77,93),(78,94),(79,95),(80,96),(81,97),(82,98),(83,99),(84,100),(85,101),(86,102),(87,103),(88,104)], [(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)]])
154 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | ··· | 8H | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AP | 28A | ··· | 28X | 28Y | ··· | 28AJ | 56A | ··· | 56AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | ··· | 1 | 4 | ··· | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
154 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C7 | C8 | C14 | C14 | C28 | C28 | C56 | D4 | M4(2) | C7×D4 | C7×M4(2) | C23⋊C4 | C4.D4 | C7×C23⋊C4 | C7×C4.D4 |
| kernel | C7×C23⋊C8 | C7×C22⋊C8 | C14×C22⋊C4 | C22×C28 | C23×C14 | C23⋊C8 | C22×C14 | C22⋊C8 | C2×C22⋊C4 | C22×C4 | C24 | C23 | C2×C28 | C2×C14 | C2×C4 | C22 | C14 | C14 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 6 | 8 | 12 | 6 | 12 | 12 | 48 | 2 | 2 | 12 | 12 | 1 | 1 | 6 | 6 |
Matrix representation of C7×C23⋊C8 ►in GL6(𝔽113)
| 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 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 83 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 98 | 112 |
| 112 | 0 | 0 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 112 | 0 | 0 | 0 |
| 0 | 0 | 0 | 112 | 0 | 0 |
| 0 | 0 | 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 0 | 0 | 112 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 98 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 111 |
| 0 | 0 | 98 | 1 | 0 | 0 |
| 0 | 0 | 0 | 49 | 0 | 0 |
G:=sub<GL(6,GF(113))| [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],[112,0,0,0,0,0,0,1,0,0,0,0,0,0,112,83,0,0,0,0,0,1,0,0,0,0,0,0,1,98,0,0,0,0,0,112],[112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,112],[0,98,0,0,0,0,1,0,0,0,0,0,0,0,0,0,98,0,0,0,0,0,1,49,0,0,1,0,0,0,0,0,0,111,0,0] >;
C7×C23⋊C8 in GAP, Magma, Sage, TeX
C_7\times C_2^3\rtimes C_8
% in TeX
G:=Group("C7xC2^3:C8"); // GroupNames label
G:=SmallGroup(448,127);
// by ID
G=gap.SmallGroup(448,127);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,2951,172]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^2=e^8=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations