direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C7×C23.D4, C22⋊C4⋊2C28, (C22×C28)⋊4C4, (C22×C4)⋊2C28, (C2×C28).18D4, C4.D4.C14, C23.2(C7×D4), C23⋊C4.1C14, C23.2(C2×C28), (C22×C14).2D4, C14.33(C23⋊C4), (D4×C14).175C22, C22.D4.1C14, (C2×C4).2(C7×D4), (C7×C22⋊C4)⋊4C4, C2.7(C7×C23⋊C4), (C2×D4).2(C2×C14), (C7×C23⋊C4).3C2, (C22×C14).9(C2×C4), (C7×C4.D4).2C2, C22.11(C7×C22⋊C4), (C2×C14).74(C22⋊C4), (C7×C22.D4).4C2, SmallGroup(448,156)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=1, e4=d, f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=bcd, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=be3 >
Subgroups: 162 in 68 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, C23, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, M4(2), C22×C4, C2×D4, C28, C2×C14, C2×C14, C23⋊C4, C4.D4, C22.D4, C56, C2×C28, C2×C28, C7×D4, C22×C14, C23.D4, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C7×M4(2), C22×C28, D4×C14, C7×C23⋊C4, C7×C4.D4, C7×C22.D4, C7×C23.D4
Quotients: C1, C2, C4, C22, C7, C2×C4, D4, C14, C22⋊C4, C28, C2×C14, C23⋊C4, C2×C28, C7×D4, C23.D4, C7×C22⋊C4, C7×C23⋊C4, C7×C23.D4
►On 112 points
Generators in S112
(1 86 27 78 19 70 11)(2 87 28 79 20 71 12)(3 88 29 80 21 72 13)(4 81 30 73 22 65 14)(5 82 31 74 23 66 15)(6 83 32 75 24 67 16)(7 84 25 76 17 68 9)(8 85 26 77 18 69 10)(33 58 108 49 100 41 92)(34 59 109 50 101 42 93)(35 60 110 51 102 43 94)(36 61 111 52 103 44 95)(37 62 112 53 104 45 96)(38 63 105 54 97 46 89)(39 64 106 55 98 47 90)(40 57 107 56 99 48 91)
(1 40)(2 33)(3 38)(4 39)(5 36)(6 37)(7 34)(8 35)(9 93)(10 94)(11 91)(12 92)(13 89)(14 90)(15 95)(16 96)(17 101)(18 102)(19 99)(20 100)(21 97)(22 98)(23 103)(24 104)(25 109)(26 110)(27 107)(28 108)(29 105)(30 106)(31 111)(32 112)(41 71)(42 68)(43 69)(44 66)(45 67)(46 72)(47 65)(48 70)(49 79)(50 76)(51 77)(52 74)(53 75)(54 80)(55 73)(56 78)(57 86)(58 87)(59 84)(60 85)(61 82)(62 83)(63 88)(64 81)
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(25 29)(27 31)(34 38)(36 40)(42 46)(44 48)(50 54)(52 56)(57 61)(59 63)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(89 93)(91 95)(97 101)(99 103)(105 109)(107 111)
(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)
(1 4 40 39)(2 34 33 7)(3 6 38 37)(5 8 36 35)(9 12 93 92)(10 95 94 15)(11 14 91 90)(13 16 89 96)(17 20 101 100)(18 103 102 23)(19 22 99 98)(21 24 97 104)(25 28 109 108)(26 111 110 31)(27 30 107 106)(29 32 105 112)(41 68 71 42)(43 66 69 44)(45 72 67 46)(47 70 65 48)(49 76 79 50)(51 74 77 52)(53 80 75 54)(55 78 73 56)(57 64 86 81)(58 84 87 59)(60 82 85 61)(62 88 83 63)
G:=sub<Sym(112)| (1,86,27,78,19,70,11)(2,87,28,79,20,71,12)(3,88,29,80,21,72,13)(4,81,30,73,22,65,14)(5,82,31,74,23,66,15)(6,83,32,75,24,67,16)(7,84,25,76,17,68,9)(8,85,26,77,18,69,10)(33,58,108,49,100,41,92)(34,59,109,50,101,42,93)(35,60,110,51,102,43,94)(36,61,111,52,103,44,95)(37,62,112,53,104,45,96)(38,63,105,54,97,46,89)(39,64,106,55,98,47,90)(40,57,107,56,99,48,91), (1,40)(2,33)(3,38)(4,39)(5,36)(6,37)(7,34)(8,35)(9,93)(10,94)(11,91)(12,92)(13,89)(14,90)(15,95)(16,96)(17,101)(18,102)(19,99)(20,100)(21,97)(22,98)(23,103)(24,104)(25,109)(26,110)(27,107)(28,108)(29,105)(30,106)(31,111)(32,112)(41,71)(42,68)(43,69)(44,66)(45,67)(46,72)(47,65)(48,70)(49,79)(50,76)(51,77)(52,74)(53,75)(54,80)(55,73)(56,78)(57,86)(58,87)(59,84)(60,85)(61,82)(62,83)(63,88)(64,81), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (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), (1,4,40,39)(2,34,33,7)(3,6,38,37)(5,8,36,35)(9,12,93,92)(10,95,94,15)(11,14,91,90)(13,16,89,96)(17,20,101,100)(18,103,102,23)(19,22,99,98)(21,24,97,104)(25,28,109,108)(26,111,110,31)(27,30,107,106)(29,32,105,112)(41,68,71,42)(43,66,69,44)(45,72,67,46)(47,70,65,48)(49,76,79,50)(51,74,77,52)(53,80,75,54)(55,78,73,56)(57,64,86,81)(58,84,87,59)(60,82,85,61)(62,88,83,63)>;
G:=Group( (1,86,27,78,19,70,11)(2,87,28,79,20,71,12)(3,88,29,80,21,72,13)(4,81,30,73,22,65,14)(5,82,31,74,23,66,15)(6,83,32,75,24,67,16)(7,84,25,76,17,68,9)(8,85,26,77,18,69,10)(33,58,108,49,100,41,92)(34,59,109,50,101,42,93)(35,60,110,51,102,43,94)(36,61,111,52,103,44,95)(37,62,112,53,104,45,96)(38,63,105,54,97,46,89)(39,64,106,55,98,47,90)(40,57,107,56,99,48,91), (1,40)(2,33)(3,38)(4,39)(5,36)(6,37)(7,34)(8,35)(9,93)(10,94)(11,91)(12,92)(13,89)(14,90)(15,95)(16,96)(17,101)(18,102)(19,99)(20,100)(21,97)(22,98)(23,103)(24,104)(25,109)(26,110)(27,107)(28,108)(29,105)(30,106)(31,111)(32,112)(41,71)(42,68)(43,69)(44,66)(45,67)(46,72)(47,65)(48,70)(49,79)(50,76)(51,77)(52,74)(53,75)(54,80)(55,73)(56,78)(57,86)(58,87)(59,84)(60,85)(61,82)(62,83)(63,88)(64,81), (1,5)(3,7)(9,13)(11,15)(17,21)(19,23)(25,29)(27,31)(34,38)(36,40)(42,46)(44,48)(50,54)(52,56)(57,61)(59,63)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(89,93)(91,95)(97,101)(99,103)(105,109)(107,111), (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), (1,4,40,39)(2,34,33,7)(3,6,38,37)(5,8,36,35)(9,12,93,92)(10,95,94,15)(11,14,91,90)(13,16,89,96)(17,20,101,100)(18,103,102,23)(19,22,99,98)(21,24,97,104)(25,28,109,108)(26,111,110,31)(27,30,107,106)(29,32,105,112)(41,68,71,42)(43,66,69,44)(45,72,67,46)(47,70,65,48)(49,76,79,50)(51,74,77,52)(53,80,75,54)(55,78,73,56)(57,64,86,81)(58,84,87,59)(60,82,85,61)(62,88,83,63) );
G=PermutationGroup([[(1,86,27,78,19,70,11),(2,87,28,79,20,71,12),(3,88,29,80,21,72,13),(4,81,30,73,22,65,14),(5,82,31,74,23,66,15),(6,83,32,75,24,67,16),(7,84,25,76,17,68,9),(8,85,26,77,18,69,10),(33,58,108,49,100,41,92),(34,59,109,50,101,42,93),(35,60,110,51,102,43,94),(36,61,111,52,103,44,95),(37,62,112,53,104,45,96),(38,63,105,54,97,46,89),(39,64,106,55,98,47,90),(40,57,107,56,99,48,91)], [(1,40),(2,33),(3,38),(4,39),(5,36),(6,37),(7,34),(8,35),(9,93),(10,94),(11,91),(12,92),(13,89),(14,90),(15,95),(16,96),(17,101),(18,102),(19,99),(20,100),(21,97),(22,98),(23,103),(24,104),(25,109),(26,110),(27,107),(28,108),(29,105),(30,106),(31,111),(32,112),(41,71),(42,68),(43,69),(44,66),(45,67),(46,72),(47,65),(48,70),(49,79),(50,76),(51,77),(52,74),(53,75),(54,80),(55,73),(56,78),(57,86),(58,87),(59,84),(60,85),(61,82),(62,83),(63,88),(64,81)], [(1,5),(3,7),(9,13),(11,15),(17,21),(19,23),(25,29),(27,31),(34,38),(36,40),(42,46),(44,48),(50,54),(52,56),(57,61),(59,63),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(89,93),(91,95),(97,101),(99,103),(105,109),(107,111)], [(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)], [(1,4,40,39),(2,34,33,7),(3,6,38,37),(5,8,36,35),(9,12,93,92),(10,95,94,15),(11,14,91,90),(13,16,89,96),(17,20,101,100),(18,103,102,23),(19,22,99,98),(21,24,97,104),(25,28,109,108),(26,111,110,31),(27,30,107,106),(29,32,105,112),(41,68,71,42),(43,66,69,44),(45,72,67,46),(47,70,65,48),(49,76,79,50),(51,74,77,52),(53,80,75,54),(55,78,73,56),(57,64,86,81),(58,84,87,59),(60,82,85,61),(62,88,83,63)]])
91 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | 4B | 4C | 4D | 4E | 4F | 7A | ··· | 7F | 8A | 8B | 14A | ··· | 14F | 14G | ··· | 14L | 14M | ··· | 14X | 28A | ··· | 28R | 28S | ··· | 28AJ | 56A | ··· | 56L |
| order | 1 | 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 | 2 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | ··· | 1 | 8 | 8 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 | 8 | ··· | 8 |
91 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 | C2 | C4 | C4 | C7 | C14 | C14 | C14 | C28 | C28 | D4 | D4 | C7×D4 | C7×D4 | C23⋊C4 | C23.D4 | C7×C23⋊C4 | C7×C23.D4 |
| kernel | C7×C23.D4 | C7×C23⋊C4 | C7×C4.D4 | C7×C22.D4 | C7×C22⋊C4 | C22×C28 | C23.D4 | C23⋊C4 | C4.D4 | C22.D4 | C22⋊C4 | C22×C4 | C2×C28 | C22×C14 | C2×C4 | C23 | C14 | C7 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 12 | 1 | 1 | 6 | 6 | 1 | 2 | 6 | 12 |
Matrix representation of C7×C23.D4 ►in GL4(𝔽113) generated by
| 49 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 49 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 112 | 0 | 0 | 0 |
| 0 | 112 | 0 | 0 |
| 0 | 0 | 112 | 0 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 0 | 112 |
| 0 | 0 | 1 | 0 |
| 0 | 98 | 0 | 0 |
| 98 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
G:=sub<GL(4,GF(113))| [49,0,0,0,0,49,0,0,0,0,49,0,0,0,0,49],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0],[112,0,0,0,0,112,0,0,0,0,1,0,0,0,0,1],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,0,0,98,0,0,98,0,0,1,0,0,112,0,0,0],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;
C7×C23.D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3.D_4
% in TeX
G:=Group("C7xC2^3.D4"); // GroupNames label
G:=SmallGroup(448,156);
// by ID
G=gap.SmallGroup(448,156);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,1576,3923,2951,375,14117]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=1,e^4=d,f^2=b,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,e*b*e^-1=b*c*d,b*f=f*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*e^3>;
// generators/relations