direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C7×C23.7D4, 2+ 1+4.2C14, C23⋊C4⋊4C14, (C2×C28).26D4, C23.7(C7×D4), (C22×C14).7D4, (C22×C28)⋊4C22, C22.19(D4×C14), C14.105C22≀C2, C22.D4⋊1C14, C23.4(C22×C14), (D4×C14).184C22, (C22×C14).83C23, (C7×2+ 1+4).4C2, (C2×C4).7(C7×D4), C22⋊C4⋊2(C2×C14), (C7×C23⋊C4)⋊10C2, (C22×C4)⋊2(C2×C14), (C2×D4).9(C2×C14), C2.19(C7×C22≀C2), (C2×C14).414(C2×D4), (C7×C22⋊C4)⋊37C22, (C7×C22.D4)⋊20C2, SmallGroup(448,866)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C23.7D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=1, f2=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf-1=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef-1=de-1 >
Subgroups: 322 in 160 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C4○D4, C28, C2×C14, C2×C14, C23⋊C4, C22.D4, 2+ 1+4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×C14, C22×C14, C22×C14, C23.7D4, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, D4×C14, C7×C4○D4, C7×C23⋊C4, C7×C22.D4, C7×2+ 1+4, C7×C23.7D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C2×C14, C22≀C2, C7×D4, C22×C14, C23.7D4, D4×C14, C7×C22≀C2, C7×C23.7D4
►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 14)(2 8)(3 9)(4 10)(5 11)(6 12)(7 13)(15 33)(16 34)(17 35)(18 29)(19 30)(20 31)(21 32)(22 40)(23 41)(24 42)(25 36)(26 37)(27 38)(28 39)(43 109)(44 110)(45 111)(46 112)(47 106)(48 107)(49 108)(50 78)(51 79)(52 80)(53 81)(54 82)(55 83)(56 84)(57 85)(58 86)(59 87)(60 88)(61 89)(62 90)(63 91)(64 92)(65 93)(66 94)(67 95)(68 96)(69 97)(70 98)(71 99)(72 100)(73 101)(74 102)(75 103)(76 104)(77 105)
(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 98)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)
(1 35)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 111)(23 112)(24 106)(25 107)(26 108)(27 109)(28 110)(36 48)(37 49)(38 43)(39 44)(40 45)(41 46)(42 47)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 98)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)
(1 74 42 50)(2 75 36 51)(3 76 37 52)(4 77 38 53)(5 71 39 54)(6 72 40 55)(7 73 41 56)(8 85 107 79)(9 86 108 80)(10 87 109 81)(11 88 110 82)(12 89 111 83)(13 90 112 84)(14 91 106 78)(15 100 22 96)(16 101 23 97)(17 102 24 98)(18 103 25 92)(19 104 26 93)(20 105 27 94)(21 99 28 95)(29 57 48 64)(30 58 49 65)(31 59 43 66)(32 60 44 67)(33 61 45 68)(34 62 46 69)(35 63 47 70)
(1 14 35 17)(2 8 29 18)(3 9 30 19)(4 10 31 20)(5 11 32 21)(6 12 33 15)(7 13 34 16)(22 40 111 45)(23 41 112 46)(24 42 106 47)(25 36 107 48)(26 37 108 49)(27 38 109 43)(28 39 110 44)(50 102 70 91)(51 103 64 85)(52 104 65 86)(53 105 66 87)(54 99 67 88)(55 100 68 89)(56 101 69 90)(57 79 75 92)(58 80 76 93)(59 81 77 94)(60 82 71 95)(61 83 72 96)(62 84 73 97)(63 78 74 98)
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), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,109)(44,110)(45,111)(46,112)(47,106)(48,107)(49,108)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105), (50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102), (1,74,42,50)(2,75,36,51)(3,76,37,52)(4,77,38,53)(5,71,39,54)(6,72,40,55)(7,73,41,56)(8,85,107,79)(9,86,108,80)(10,87,109,81)(11,88,110,82)(12,89,111,83)(13,90,112,84)(14,91,106,78)(15,100,22,96)(16,101,23,97)(17,102,24,98)(18,103,25,92)(19,104,26,93)(20,105,27,94)(21,99,28,95)(29,57,48,64)(30,58,49,65)(31,59,43,66)(32,60,44,67)(33,61,45,68)(34,62,46,69)(35,63,47,70), (1,14,35,17)(2,8,29,18)(3,9,30,19)(4,10,31,20)(5,11,32,21)(6,12,33,15)(7,13,34,16)(22,40,111,45)(23,41,112,46)(24,42,106,47)(25,36,107,48)(26,37,108,49)(27,38,109,43)(28,39,110,44)(50,102,70,91)(51,103,64,85)(52,104,65,86)(53,105,66,87)(54,99,67,88)(55,100,68,89)(56,101,69,90)(57,79,75,92)(58,80,76,93)(59,81,77,94)(60,82,71,95)(61,83,72,96)(62,84,73,97)(63,78,74,98)>;
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), (1,14)(2,8)(3,9)(4,10)(5,11)(6,12)(7,13)(15,33)(16,34)(17,35)(18,29)(19,30)(20,31)(21,32)(22,40)(23,41)(24,42)(25,36)(26,37)(27,38)(28,39)(43,109)(44,110)(45,111)(46,112)(47,106)(48,107)(49,108)(50,78)(51,79)(52,80)(53,81)(54,82)(55,83)(56,84)(57,85)(58,86)(59,87)(60,88)(61,89)(62,90)(63,91)(64,92)(65,93)(66,94)(67,95)(68,96)(69,97)(70,98)(71,99)(72,100)(73,101)(74,102)(75,103)(76,104)(77,105), (50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102), (1,35)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,111)(23,112)(24,106)(25,107)(26,108)(27,109)(28,110)(36,48)(37,49)(38,43)(39,44)(40,45)(41,46)(42,47)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102), (1,74,42,50)(2,75,36,51)(3,76,37,52)(4,77,38,53)(5,71,39,54)(6,72,40,55)(7,73,41,56)(8,85,107,79)(9,86,108,80)(10,87,109,81)(11,88,110,82)(12,89,111,83)(13,90,112,84)(14,91,106,78)(15,100,22,96)(16,101,23,97)(17,102,24,98)(18,103,25,92)(19,104,26,93)(20,105,27,94)(21,99,28,95)(29,57,48,64)(30,58,49,65)(31,59,43,66)(32,60,44,67)(33,61,45,68)(34,62,46,69)(35,63,47,70), (1,14,35,17)(2,8,29,18)(3,9,30,19)(4,10,31,20)(5,11,32,21)(6,12,33,15)(7,13,34,16)(22,40,111,45)(23,41,112,46)(24,42,106,47)(25,36,107,48)(26,37,108,49)(27,38,109,43)(28,39,110,44)(50,102,70,91)(51,103,64,85)(52,104,65,86)(53,105,66,87)(54,99,67,88)(55,100,68,89)(56,101,69,90)(57,79,75,92)(58,80,76,93)(59,81,77,94)(60,82,71,95)(61,83,72,96)(62,84,73,97)(63,78,74,98) );
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)], [(1,14),(2,8),(3,9),(4,10),(5,11),(6,12),(7,13),(15,33),(16,34),(17,35),(18,29),(19,30),(20,31),(21,32),(22,40),(23,41),(24,42),(25,36),(26,37),(27,38),(28,39),(43,109),(44,110),(45,111),(46,112),(47,106),(48,107),(49,108),(50,78),(51,79),(52,80),(53,81),(54,82),(55,83),(56,84),(57,85),(58,86),(59,87),(60,88),(61,89),(62,90),(63,91),(64,92),(65,93),(66,94),(67,95),(68,96),(69,97),(70,98),(71,99),(72,100),(73,101),(74,102),(75,103),(76,104),(77,105)], [(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,98),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102)], [(1,35),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,111),(23,112),(24,106),(25,107),(26,108),(27,109),(28,110),(36,48),(37,49),(38,43),(39,44),(40,45),(41,46),(42,47),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,98),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102)], [(1,74,42,50),(2,75,36,51),(3,76,37,52),(4,77,38,53),(5,71,39,54),(6,72,40,55),(7,73,41,56),(8,85,107,79),(9,86,108,80),(10,87,109,81),(11,88,110,82),(12,89,111,83),(13,90,112,84),(14,91,106,78),(15,100,22,96),(16,101,23,97),(17,102,24,98),(18,103,25,92),(19,104,26,93),(20,105,27,94),(21,99,28,95),(29,57,48,64),(30,58,49,65),(31,59,43,66),(32,60,44,67),(33,61,45,68),(34,62,46,69),(35,63,47,70)], [(1,14,35,17),(2,8,29,18),(3,9,30,19),(4,10,31,20),(5,11,32,21),(6,12,33,15),(7,13,34,16),(22,40,111,45),(23,41,112,46),(24,42,106,47),(25,36,107,48),(26,37,108,49),(27,38,109,43),(28,39,110,44),(50,102,70,91),(51,103,64,85),(52,104,65,86),(53,105,66,87),(54,99,67,88),(55,100,68,89),(56,101,69,90),(57,79,75,92),(58,80,76,93),(59,81,77,94),(60,82,71,95),(61,83,72,96),(62,84,73,97),(63,78,74,98)]])
112 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4E | 4F | 4G | 4H | 7A | ··· | 7F | 14A | ··· | 14F | 14G | ··· | 14X | 14Y | ··· | 14AP | 28A | ··· | 28AD | 28AE | ··· | 28AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | ··· | 4 | 8 | 8 | 8 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
112 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | D4 | D4 | C7×D4 | C7×D4 | C23.7D4 | C7×C23.7D4 |
| kernel | C7×C23.7D4 | C7×C23⋊C4 | C7×C22.D4 | C7×2+ 1+4 | C23.7D4 | C23⋊C4 | C22.D4 | 2+ 1+4 | C2×C28 | C22×C14 | C2×C4 | C23 | C7 | C1 |
| # reps | 1 | 3 | 3 | 1 | 6 | 18 | 18 | 6 | 3 | 3 | 18 | 18 | 2 | 12 |
Matrix representation of C7×C23.7D4 ►in GL4(𝔽29) generated by
| 20 | 0 | 0 | 0 |
| 0 | 20 | 0 | 0 |
| 0 | 0 | 20 | 0 |
| 0 | 0 | 0 | 20 |
| 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 17 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 17 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
G:=sub<GL(4,GF(29))| [20,0,0,0,0,20,0,0,0,0,20,0,0,0,0,20],[0,17,0,0,12,0,0,0,0,0,0,17,0,0,12,0],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[0,0,1,0,0,0,0,28,0,28,0,0,1,0,0,0],[0,17,0,0,17,0,0,0,0,0,17,0,0,0,0,17] >;
C7×C23.7D4 in GAP, Magma, Sage, TeX
C_7\times C_2^3._7D_4
% in TeX
G:=Group("C7xC2^3.7D4"); // GroupNames label
G:=SmallGroup(448,866);
// by ID
G=gap.SmallGroup(448,866);
# by ID
G:=PCGroup([7,-2,-2,-2,-7,-2,-2,-2,1568,813,2438,2468,7068]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^7=b^2=c^2=d^2=e^4=1,f^2=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,b*d=d*b,e*b*e^-1=f*b*f^-1=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=d*e^-1>;
// generators/relations