direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C7×C22.D4, C4⋊C4⋊4C14, C2.7(D4×C14), C22⋊C4⋊4C14, (C22×C4)⋊3C14, (C22×C28)⋊5C2, (C2×D4).4C14, C14.70(C2×D4), (C2×C14).23D4, C22.4(C7×D4), (D4×C14).11C2, C14.43(C4○D4), (C2×C28).65C22, (C2×C14).78C23, C23.10(C2×C14), C22.13(C22×C14), (C22×C14).29C22, (C7×C4⋊C4)⋊13C2, C2.6(C7×C4○D4), (C2×C4).5(C2×C14), (C7×C22⋊C4)⋊12C2, SmallGroup(224,158)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7×C22.D4
G = < a,b,c,d,e | a7=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=ebe=bc=cb, cd=dc, ce=ec, ede=cd-1 >
Subgroups: 116 in 78 conjugacy classes, 44 normal (20 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C14, C14, C14, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C28, C2×C14, C2×C14, C2×C14, C22.D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×C14, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C22×C28, D4×C14, C7×C22.D4
Quotients: C1, C2, C22, C7, D4, C23, C14, C2×D4, C4○D4, C2×C14, C22.D4, C7×D4, C22×C14, D4×C14, C7×C4○D4, C7×C22.D4
►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 91)(2 85)(3 86)(4 87)(5 88)(6 89)(7 90)(8 65)(9 66)(10 67)(11 68)(12 69)(13 70)(14 64)(15 59)(16 60)(17 61)(18 62)(19 63)(20 57)(21 58)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)(29 95)(30 96)(31 97)(32 98)(33 92)(34 93)(35 94)(36 100)(37 101)(38 102)(39 103)(40 104)(41 105)(42 99)(43 81)(44 82)(45 83)(46 84)(47 78)(48 79)(49 80)(71 109)(72 110)(73 111)(74 112)(75 106)(76 107)(77 108)
(1 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 109)(16 110)(17 111)(18 112)(19 106)(20 107)(21 108)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
(1 50 47 63)(2 51 48 57)(3 52 49 58)(4 53 43 59)(5 54 44 60)(6 55 45 61)(7 56 46 62)(8 84 112 90)(9 78 106 91)(10 79 107 85)(11 80 108 86)(12 81 109 87)(13 82 110 88)(14 83 111 89)(15 99 27 97)(16 100 28 98)(17 101 22 92)(18 102 23 93)(19 103 24 94)(20 104 25 95)(21 105 26 96)(29 76 40 67)(30 77 41 68)(31 71 42 69)(32 72 36 70)(33 73 37 64)(34 74 38 65)(35 75 39 66)
(8 112)(9 106)(10 107)(11 108)(12 109)(13 110)(14 111)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)(50 75)(51 76)(52 77)(53 71)(54 72)(55 73)(56 74)(57 67)(58 68)(59 69)(60 70)(61 64)(62 65)(63 66)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)
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), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,64)(15,59)(16,60)(17,61)(18,62)(19,63)(20,57)(21,58)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(43,81)(44,82)(45,83)(46,84)(47,78)(48,79)(49,80)(71,109)(72,110)(73,111)(74,112)(75,106)(76,107)(77,108), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,50,47,63)(2,51,48,57)(3,52,49,58)(4,53,43,59)(5,54,44,60)(6,55,45,61)(7,56,46,62)(8,84,112,90)(9,78,106,91)(10,79,107,85)(11,80,108,86)(12,81,109,87)(13,82,110,88)(14,83,111,89)(15,99,27,97)(16,100,28,98)(17,101,22,92)(18,102,23,93)(19,103,24,94)(20,104,25,95)(21,105,26,96)(29,76,40,67)(30,77,41,68)(31,71,42,69)(32,72,36,70)(33,73,37,64)(34,74,38,65)(35,75,39,66), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103)>;
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), (1,91)(2,85)(3,86)(4,87)(5,88)(6,89)(7,90)(8,65)(9,66)(10,67)(11,68)(12,69)(13,70)(14,64)(15,59)(16,60)(17,61)(18,62)(19,63)(20,57)(21,58)(22,55)(23,56)(24,50)(25,51)(26,52)(27,53)(28,54)(29,95)(30,96)(31,97)(32,98)(33,92)(34,93)(35,94)(36,100)(37,101)(38,102)(39,103)(40,104)(41,105)(42,99)(43,81)(44,82)(45,83)(46,84)(47,78)(48,79)(49,80)(71,109)(72,110)(73,111)(74,112)(75,106)(76,107)(77,108), (1,39)(2,40)(3,41)(4,42)(5,36)(6,37)(7,38)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,22)(15,109)(16,110)(17,111)(18,112)(19,106)(20,107)(21,108)(29,48)(30,49)(31,43)(32,44)(33,45)(34,46)(35,47)(50,66)(51,67)(52,68)(53,69)(54,70)(55,64)(56,65)(57,76)(58,77)(59,71)(60,72)(61,73)(62,74)(63,75)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103), (1,50,47,63)(2,51,48,57)(3,52,49,58)(4,53,43,59)(5,54,44,60)(6,55,45,61)(7,56,46,62)(8,84,112,90)(9,78,106,91)(10,79,107,85)(11,80,108,86)(12,81,109,87)(13,82,110,88)(14,83,111,89)(15,99,27,97)(16,100,28,98)(17,101,22,92)(18,102,23,93)(19,103,24,94)(20,104,25,95)(21,105,26,96)(29,76,40,67)(30,77,41,68)(31,71,42,69)(32,72,36,70)(33,73,37,64)(34,74,38,65)(35,75,39,66), (8,112)(9,106)(10,107)(11,108)(12,109)(13,110)(14,111)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26)(50,75)(51,76)(52,77)(53,71)(54,72)(55,73)(56,74)(57,67)(58,68)(59,69)(60,70)(61,64)(62,65)(63,66)(78,94)(79,95)(80,96)(81,97)(82,98)(83,92)(84,93)(85,104)(86,105)(87,99)(88,100)(89,101)(90,102)(91,103) );
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)], [(1,91),(2,85),(3,86),(4,87),(5,88),(6,89),(7,90),(8,65),(9,66),(10,67),(11,68),(12,69),(13,70),(14,64),(15,59),(16,60),(17,61),(18,62),(19,63),(20,57),(21,58),(22,55),(23,56),(24,50),(25,51),(26,52),(27,53),(28,54),(29,95),(30,96),(31,97),(32,98),(33,92),(34,93),(35,94),(36,100),(37,101),(38,102),(39,103),(40,104),(41,105),(42,99),(43,81),(44,82),(45,83),(46,84),(47,78),(48,79),(49,80),(71,109),(72,110),(73,111),(74,112),(75,106),(76,107),(77,108)], [(1,39),(2,40),(3,41),(4,42),(5,36),(6,37),(7,38),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,22),(15,109),(16,110),(17,111),(18,112),(19,106),(20,107),(21,108),(29,48),(30,49),(31,43),(32,44),(33,45),(34,46),(35,47),(50,66),(51,67),(52,68),(53,69),(54,70),(55,64),(56,65),(57,76),(58,77),(59,71),(60,72),(61,73),(62,74),(63,75),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103)], [(1,50,47,63),(2,51,48,57),(3,52,49,58),(4,53,43,59),(5,54,44,60),(6,55,45,61),(7,56,46,62),(8,84,112,90),(9,78,106,91),(10,79,107,85),(11,80,108,86),(12,81,109,87),(13,82,110,88),(14,83,111,89),(15,99,27,97),(16,100,28,98),(17,101,22,92),(18,102,23,93),(19,103,24,94),(20,104,25,95),(21,105,26,96),(29,76,40,67),(30,77,41,68),(31,71,42,69),(32,72,36,70),(33,73,37,64),(34,74,38,65),(35,75,39,66)], [(8,112),(9,106),(10,107),(11,108),(12,109),(13,110),(14,111),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26),(50,75),(51,76),(52,77),(53,71),(54,72),(55,73),(56,74),(57,67),(58,68),(59,69),(60,70),(61,64),(62,65),(63,66),(78,94),(79,95),(80,96),(81,97),(82,98),(83,92),(84,93),(85,104),(86,105),(87,99),(88,100),(89,101),(90,102),(91,103)]])
C7×C22.D4 is a maximal subgroup of
(C22×C28)⋊C4 C22⋊C4⋊D14 C14.792- 1+4 C4⋊C4.197D14 C14.802- 1+4 C14.602+ 1+4 C14.1202+ 1+4 C14.1212+ 1+4 C14.822- 1+4 C4⋊C4⋊28D14 C14.612+ 1+4 C14.1222+ 1+4 C14.622+ 1+4 C14.832- 1+4 C14.642+ 1+4 C14.842- 1+4 C14.662+ 1+4 C14.672+ 1+4 C14.852- 1+4 C14.682+ 1+4 C14.862- 1+4
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 7A | ··· | 7F | 14A | ··· | 14R | 14S | ··· | 14AD | 14AE | ··· | 14AJ | 28A | ··· | 28X | 28Y | ··· | 28AP |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | ··· | 7 | 14 | ··· | 14 | 14 | ··· | 14 | 14 | ··· | 14 | 28 | ··· | 28 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C7 | C14 | C14 | C14 | C14 | D4 | C4○D4 | C7×D4 | C7×C4○D4 |
| kernel | C7×C22.D4 | C7×C22⋊C4 | C7×C4⋊C4 | C22×C28 | D4×C14 | C22.D4 | C22⋊C4 | C4⋊C4 | C22×C4 | C2×D4 | C2×C14 | C14 | C22 | C2 |
| # reps | 1 | 3 | 2 | 1 | 1 | 6 | 18 | 12 | 6 | 6 | 2 | 4 | 12 | 24 |
Matrix representation of C7×C22.D4 ►in GL4(𝔽29) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 24 | 0 |
| 0 | 0 | 0 | 24 |
| 0 | 12 | 0 | 0 |
| 17 | 0 | 0 | 0 |
| 0 | 0 | 1 | 27 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 0 | 28 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 12 | 17 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 28 |
G:=sub<GL(4,GF(29))| [16,0,0,0,0,16,0,0,0,0,24,0,0,0,0,24],[0,17,0,0,12,0,0,0,0,0,1,0,0,0,27,28],[28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[0,28,0,0,28,0,0,0,0,0,12,12,0,0,0,17],[1,0,0,0,0,28,0,0,0,0,1,1,0,0,0,28] >;
C7×C22.D4 in GAP, Magma, Sage, TeX
C_7\times C_2^2.D_4
% in TeX
G:=Group("C7xC2^2.D4"); // GroupNames label
G:=SmallGroup(224,158);
// by ID
G=gap.SmallGroup(224,158);
# by ID
G:=PCGroup([6,-2,-2,-2,-7,-2,-2,697,2090,266]);
// Polycyclic
G:=Group<a,b,c,d,e|a^7=b^2=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=c*d^-1>;
// generators/relations