metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C42⋊1D7, (C4×C28)⋊9C2, (C4×D7)⋊3C4, C4.22(C4×D7), C28.25(C2×C4), D14⋊C4.7C2, (C4×Dic7)⋊8C2, D14.3(C2×C4), (C2×C4).96D14, Dic7⋊C4⋊17C2, C7⋊1(C42⋊C2), C2.2(C4○D28), C14.3(C4○D4), C14.3(C22×C4), Dic7.5(C2×C4), (C2×C28).73C22, (C2×C14).13C23, C22.10(C22×D7), (C2×Dic7).24C22, (C22×D7).14C22, C2.5(C2×C4×D7), (C2×C4×D7).8C2, SmallGroup(224,67)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C42⋊D7
G = < a,b,c,d | a4=b4=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 278 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, C2×C4, C23, D7, C14, C14, C42, C42, C22⋊C4, C4⋊C4, C22×C4, Dic7, Dic7, C28, C28, D14, D14, C2×C14, C42⋊C2, C4×D7, C2×Dic7, C2×Dic7, C2×C28, C2×C28, C22×D7, C4×Dic7, Dic7⋊C4, D14⋊C4, C4×C28, C2×C4×D7, C42⋊D7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, C4○D4, D14, C42⋊C2, C4×D7, C22×D7, C2×C4×D7, C4○D28, C42⋊D7
►On 112 points
(1 69 13 62)(2 70 14 63)(3 64 8 57)(4 65 9 58)(5 66 10 59)(6 67 11 60)(7 68 12 61)(15 78 22 71)(16 79 23 72)(17 80 24 73)(18 81 25 74)(19 82 26 75)(20 83 27 76)(21 84 28 77)(29 92 36 85)(30 93 37 86)(31 94 38 87)(32 95 39 88)(33 96 40 89)(34 97 41 90)(35 98 42 91)(43 106 50 99)(44 107 51 100)(45 108 52 101)(46 109 53 102)(47 110 54 103)(48 111 55 104)(49 112 56 105)
(1 48 20 34)(2 49 21 35)(3 43 15 29)(4 44 16 30)(5 45 17 31)(6 46 18 32)(7 47 19 33)(8 50 22 36)(9 51 23 37)(10 52 24 38)(11 53 25 39)(12 54 26 40)(13 55 27 41)(14 56 28 42)(57 99 71 85)(58 100 72 86)(59 101 73 87)(60 102 74 88)(61 103 75 89)(62 104 76 90)(63 105 77 91)(64 106 78 92)(65 107 79 93)(66 108 80 94)(67 109 81 95)(68 110 82 96)(69 111 83 97)(70 112 84 98)
(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 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)(57 59)(60 63)(61 62)(64 66)(67 70)(68 69)(71 73)(74 77)(75 76)(78 80)(81 84)(82 83)(85 94)(86 93)(87 92)(88 98)(89 97)(90 96)(91 95)(99 108)(100 107)(101 106)(102 112)(103 111)(104 110)(105 109)
G:=sub<Sym(112)| (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109)>;
G:=Group( (1,69,13,62)(2,70,14,63)(3,64,8,57)(4,65,9,58)(5,66,10,59)(6,67,11,60)(7,68,12,61)(15,78,22,71)(16,79,23,72)(17,80,24,73)(18,81,25,74)(19,82,26,75)(20,83,27,76)(21,84,28,77)(29,92,36,85)(30,93,37,86)(31,94,38,87)(32,95,39,88)(33,96,40,89)(34,97,41,90)(35,98,42,91)(43,106,50,99)(44,107,51,100)(45,108,52,101)(46,109,53,102)(47,110,54,103)(48,111,55,104)(49,112,56,105), (1,48,20,34)(2,49,21,35)(3,43,15,29)(4,44,16,30)(5,45,17,31)(6,46,18,32)(7,47,19,33)(8,50,22,36)(9,51,23,37)(10,52,24,38)(11,53,25,39)(12,54,26,40)(13,55,27,41)(14,56,28,42)(57,99,71,85)(58,100,72,86)(59,101,73,87)(60,102,74,88)(61,103,75,89)(62,104,76,90)(63,105,77,91)(64,106,78,92)(65,107,79,93)(66,108,80,94)(67,109,81,95)(68,110,82,96)(69,111,83,97)(70,112,84,98), (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)(29,38)(30,37)(31,36)(32,42)(33,41)(34,40)(35,39)(43,52)(44,51)(45,50)(46,56)(47,55)(48,54)(49,53)(57,59)(60,63)(61,62)(64,66)(67,70)(68,69)(71,73)(74,77)(75,76)(78,80)(81,84)(82,83)(85,94)(86,93)(87,92)(88,98)(89,97)(90,96)(91,95)(99,108)(100,107)(101,106)(102,112)(103,111)(104,110)(105,109) );
G=PermutationGroup([[(1,69,13,62),(2,70,14,63),(3,64,8,57),(4,65,9,58),(5,66,10,59),(6,67,11,60),(7,68,12,61),(15,78,22,71),(16,79,23,72),(17,80,24,73),(18,81,25,74),(19,82,26,75),(20,83,27,76),(21,84,28,77),(29,92,36,85),(30,93,37,86),(31,94,38,87),(32,95,39,88),(33,96,40,89),(34,97,41,90),(35,98,42,91),(43,106,50,99),(44,107,51,100),(45,108,52,101),(46,109,53,102),(47,110,54,103),(48,111,55,104),(49,112,56,105)], [(1,48,20,34),(2,49,21,35),(3,43,15,29),(4,44,16,30),(5,45,17,31),(6,46,18,32),(7,47,19,33),(8,50,22,36),(9,51,23,37),(10,52,24,38),(11,53,25,39),(12,54,26,40),(13,55,27,41),(14,56,28,42),(57,99,71,85),(58,100,72,86),(59,101,73,87),(60,102,74,88),(61,103,75,89),(62,104,76,90),(63,105,77,91),(64,106,78,92),(65,107,79,93),(66,108,80,94),(67,109,81,95),(68,110,82,96),(69,111,83,97),(70,112,84,98)], [(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),(29,38),(30,37),(31,36),(32,42),(33,41),(34,40),(35,39),(43,52),(44,51),(45,50),(46,56),(47,55),(48,54),(49,53),(57,59),(60,63),(61,62),(64,66),(67,70),(68,69),(71,73),(74,77),(75,76),(78,80),(81,84),(82,83),(85,94),(86,93),(87,92),(88,98),(89,97),(90,96),(91,95),(99,108),(100,107),(101,106),(102,112),(103,111),(104,110),(105,109)]])
C42⋊D7 is a maximal subgroup of
D14.C42 C42.243D14 D14.4C42 C42.185D14 C42⋊D14 C42.30D14 C42.31D14 C4×C4○D28 C42.277D14 D7×C42⋊C2 C42.188D14 C42⋊10D14 C42.93D14 C42.94D14 C42.96D14 C42.97D14 C42.102D14 C42.104D14 C42⋊11D14 C42.108D14 C42⋊12D14 C42⋊16D14 C42.115D14 C42.116D14 C42.122D14 C42.125D14 C42.126D14 C42.232D14 C42.131D14 C42.132D14 C42.133D14 C42.134D14 C42.138D14 C42⋊18D14 C42.141D14 C42⋊21D14 C42.144D14 C42.148D14 C42.151D14 C42.155D14 C42.156D14 C42.160D14 C42⋊24D14 C42.162D14 C42.163D14 C42⋊26D14 C42.168D14 C42.171D14 C42.174D14 C42.176D14 C42.178D14
C42⋊D7 is a maximal quotient of
C7⋊(C42⋊8C4) C7⋊(C42⋊5C4) Dic7⋊C4⋊C4 C22.58(D4×D7) D14⋊(C4⋊C4) D14⋊C4⋊5C4 C42.282D14 C42.243D14 C42.182D14 C42.185D14 C4×Dic7⋊C4 C42⋊4Dic7 (C2×C42).D7 C4×D14⋊C4 (C2×C42)⋊D7
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28AJ |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 14 | 14 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D7 | C4○D4 | D14 | C4×D7 | C4○D28 |
| kernel | C42⋊D7 | C4×Dic7 | Dic7⋊C4 | D14⋊C4 | C4×C28 | C2×C4×D7 | C4×D7 | C42 | C14 | C2×C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 8 | 3 | 4 | 9 | 12 | 24 |
Matrix representation of C42⋊D7 ►in GL3(𝔽29) generated by
| 1 | 0 | 0 |
| 0 | 17 | 0 |
| 0 | 0 | 17 |
| 17 | 0 | 0 |
| 0 | 16 | 5 |
| 0 | 24 | 13 |
| 1 | 0 | 0 |
| 0 | 18 | 1 |
| 0 | 28 | 0 |
| 28 | 0 | 0 |
| 0 | 4 | 25 |
| 0 | 11 | 25 |
G:=sub<GL(3,GF(29))| [1,0,0,0,17,0,0,0,17],[17,0,0,0,16,24,0,5,13],[1,0,0,0,18,28,0,1,0],[28,0,0,0,4,11,0,25,25] >;
C42⋊D7 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_7
% in TeX
G:=Group("C4^2:D7"); // GroupNames label
G:=SmallGroup(224,67);
// by ID
G=gap.SmallGroup(224,67);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,362,50,6917]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^7=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations