metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊2D7, (C4×C28)⋊1C2, Dic7⋊C4⋊1C2, D14⋊C4.1C2, (C2×C4).63D14, C7⋊1(C42⋊2C2), C14.6(C4○D4), C2.8(C4○D28), (C2×C14).17C23, (C2×C28).75C22, (C2×Dic7).4C22, (C22×D7).3C22, C22.38(C22×D7), SmallGroup(224,71)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊2D7
G = < a,b,c,d | a4=b4=c7=d2=1, ab=ba, ac=ca, dad=ab2, bc=cb, dbd=a2b-1, dcd=c-1 >
Subgroups: 246 in 60 conjugacy classes, 29 normal (8 characteristic)
C1, C2, C2, C4, C22, C22, C7, C2×C4, C2×C4, C23, D7, C14, C42, C22⋊C4, C4⋊C4, Dic7, C28, D14, C2×C14, C42⋊2C2, C2×Dic7, C2×C28, C22×D7, Dic7⋊C4, D14⋊C4, C4×C28, C42⋊2D7
Quotients: C1, C2, C22, C23, D7, C4○D4, D14, C42⋊2C2, C22×D7, C4○D28, C42⋊2D7
►On 112 points
Generators in S112
(1 76 20 62)(2 77 21 63)(3 71 15 57)(4 72 16 58)(5 73 17 59)(6 74 18 60)(7 75 19 61)(8 78 22 64)(9 79 23 65)(10 80 24 66)(11 81 25 67)(12 82 26 68)(13 83 27 69)(14 84 28 70)(29 99 43 85)(30 100 44 86)(31 101 45 87)(32 102 46 88)(33 103 47 89)(34 104 48 90)(35 105 49 91)(36 106 50 92)(37 107 51 93)(38 108 52 94)(39 109 53 95)(40 110 54 96)(41 111 55 97)(42 112 56 98)
(1 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(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 7)(2 6)(3 5)(8 10)(11 14)(12 13)(15 17)(18 21)(19 20)(22 24)(25 28)(26 27)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 66)(58 65)(59 64)(60 70)(61 69)(62 68)(63 67)(71 80)(72 79)(73 78)(74 84)(75 83)(76 82)(77 81)(85 101)(86 100)(87 99)(88 105)(89 104)(90 103)(91 102)(92 108)(93 107)(94 106)(95 112)(96 111)(97 110)(98 109)
G:=sub<Sym(112)| (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,101)(86,100)(87,99)(88,105)(89,104)(90,103)(91,102)(92,108)(93,107)(94,106)(95,112)(96,111)(97,110)(98,109)>;
G:=Group( (1,76,20,62)(2,77,21,63)(3,71,15,57)(4,72,16,58)(5,73,17,59)(6,74,18,60)(7,75,19,61)(8,78,22,64)(9,79,23,65)(10,80,24,66)(11,81,25,67)(12,82,26,68)(13,83,27,69)(14,84,28,70)(29,99,43,85)(30,100,44,86)(31,101,45,87)(32,102,46,88)(33,103,47,89)(34,104,48,90)(35,105,49,91)(36,106,50,92)(37,107,51,93)(38,108,52,94)(39,109,53,95)(40,110,54,96)(41,111,55,97)(42,112,56,98), (1,41,13,34)(2,42,14,35)(3,36,8,29)(4,37,9,30)(5,38,10,31)(6,39,11,32)(7,40,12,33)(15,50,22,43)(16,51,23,44)(17,52,24,45)(18,53,25,46)(19,54,26,47)(20,55,27,48)(21,56,28,49)(57,92,64,85)(58,93,65,86)(59,94,66,87)(60,95,67,88)(61,96,68,89)(62,97,69,90)(63,98,70,91)(71,106,78,99)(72,107,79,100)(73,108,80,101)(74,109,81,102)(75,110,82,103)(76,111,83,104)(77,112,84,105), (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), (1,7)(2,6)(3,5)(8,10)(11,14)(12,13)(15,17)(18,21)(19,20)(22,24)(25,28)(26,27)(29,52)(30,51)(31,50)(32,56)(33,55)(34,54)(35,53)(36,45)(37,44)(38,43)(39,49)(40,48)(41,47)(42,46)(57,66)(58,65)(59,64)(60,70)(61,69)(62,68)(63,67)(71,80)(72,79)(73,78)(74,84)(75,83)(76,82)(77,81)(85,101)(86,100)(87,99)(88,105)(89,104)(90,103)(91,102)(92,108)(93,107)(94,106)(95,112)(96,111)(97,110)(98,109) );
G=PermutationGroup([[(1,76,20,62),(2,77,21,63),(3,71,15,57),(4,72,16,58),(5,73,17,59),(6,74,18,60),(7,75,19,61),(8,78,22,64),(9,79,23,65),(10,80,24,66),(11,81,25,67),(12,82,26,68),(13,83,27,69),(14,84,28,70),(29,99,43,85),(30,100,44,86),(31,101,45,87),(32,102,46,88),(33,103,47,89),(34,104,48,90),(35,105,49,91),(36,106,50,92),(37,107,51,93),(38,108,52,94),(39,109,53,95),(40,110,54,96),(41,111,55,97),(42,112,56,98)], [(1,41,13,34),(2,42,14,35),(3,36,8,29),(4,37,9,30),(5,38,10,31),(6,39,11,32),(7,40,12,33),(15,50,22,43),(16,51,23,44),(17,52,24,45),(18,53,25,46),(19,54,26,47),(20,55,27,48),(21,56,28,49),(57,92,64,85),(58,93,65,86),(59,94,66,87),(60,95,67,88),(61,96,68,89),(62,97,69,90),(63,98,70,91),(71,106,78,99),(72,107,79,100),(73,108,80,101),(74,109,81,102),(75,110,82,103),(76,111,83,104),(77,112,84,105)], [(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)], [(1,7),(2,6),(3,5),(8,10),(11,14),(12,13),(15,17),(18,21),(19,20),(22,24),(25,28),(26,27),(29,52),(30,51),(31,50),(32,56),(33,55),(34,54),(35,53),(36,45),(37,44),(38,43),(39,49),(40,48),(41,47),(42,46),(57,66),(58,65),(59,64),(60,70),(61,69),(62,68),(63,67),(71,80),(72,79),(73,78),(74,84),(75,83),(76,82),(77,81),(85,101),(86,100),(87,99),(88,105),(89,104),(90,103),(91,102),(92,108),(93,107),(94,106),(95,112),(96,111),(97,110),(98,109)]])
C42⋊2D7 is a maximal subgroup of
C42.277D14 C42⋊10D14 C42.95D14 C42.96D14 C42.98D14 C42.104D14 C42⋊16D14 C42⋊17D14 C42.118D14 C42.122D14 C42.132D14 C42.133D14 C42.134D14 C42.137D14 C42⋊20D14 C42.150D14 C42.154D14 D7×C42⋊2C2 C42.189D14 C42⋊25D14 C42.165D14 C42⋊28D14 C42.180D14
C42⋊2D7 is a maximal quotient of
(C2×Dic7).Q8 (C22×C4).D14 (C22×D7).9D4 (C22×D7).Q8 (C2×C42).D7 C42⋊5Dic7 (C2×C42)⋊D7
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 4A | ··· | 4F | 4G | 4H | 4I | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 28 | 2 | ··· | 2 | 28 | 28 | 28 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | D7 | C4○D4 | D14 | C4○D28 |
| kernel | C42⋊2D7 | Dic7⋊C4 | D14⋊C4 | C4×C28 | C42 | C14 | C2×C4 | C2 |
| # reps | 1 | 3 | 3 | 1 | 3 | 6 | 9 | 36 |
Matrix representation of C42⋊2D7 ►in GL4(𝔽29) generated by
| 9 | 14 | 0 | 0 |
| 15 | 20 | 0 | 0 |
| 0 | 0 | 27 | 5 |
| 0 | 0 | 28 | 2 |
| 21 | 23 | 0 | 0 |
| 6 | 8 | 0 | 0 |
| 0 | 0 | 17 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 |
| 28 | 7 | 0 | 0 |
| 0 | 0 | 26 | 1 |
| 0 | 0 | 23 | 21 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 21 | 28 |
| 0 | 0 | 5 | 8 |
G:=sub<GL(4,GF(29))| [9,15,0,0,14,20,0,0,0,0,27,28,0,0,5,2],[21,6,0,0,23,8,0,0,0,0,17,0,0,0,0,17],[0,28,0,0,1,7,0,0,0,0,26,23,0,0,1,21],[0,1,0,0,1,0,0,0,0,0,21,5,0,0,28,8] >;
C42⋊2D7 in GAP, Magma, Sage, TeX
C_4^2\rtimes_2D_7
% in TeX
G:=Group("C4^2:2D7"); // GroupNames label
G:=SmallGroup(224,71);
// by ID
G=gap.SmallGroup(224,71);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,217,55,506,86,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,b*c=c*b,d*b*d=a^2*b^-1,d*c*d=c^-1>;
// generators/relations