direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Q8×D7, C14.8C24, C28.22C23, Dic14⋊9C22, D14.10C23, Dic7.5C23, C14⋊2(C2×Q8), C7⋊2(C22×Q8), (Q8×C14)⋊5C2, (C2×C4).61D14, (C7×Q8)⋊5C22, C2.9(C23×D7), C4.22(C22×D7), (C2×Dic14)⋊13C2, (C2×C28).46C22, (C2×C14).66C23, (C4×D7).13C22, C22.31(C22×D7), (C2×Dic7).44C22, (C22×D7).36C22, (C2×C4×D7).6C2, SmallGroup(224,181)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Q8×D7
G = < a,b,c,d,e | a2=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 510 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2×C4, C2×C4, Q8, Q8, C23, D7, C14, C14, C22×C4, C2×Q8, C2×Q8, Dic7, C28, D14, C2×C14, C22×Q8, Dic14, C4×D7, C2×Dic7, C2×C28, C7×Q8, C22×D7, C2×Dic14, C2×C4×D7, Q8×D7, Q8×C14, C2×Q8×D7
Quotients: C1, C2, C22, Q8, C23, D7, C2×Q8, C24, D14, C22×Q8, C22×D7, Q8×D7, C23×D7, C2×Q8×D7
►On 112 points
Generators in S112
(1 62)(2 63)(3 57)(4 58)(5 59)(6 60)(7 61)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(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 83 13 76)(2 84 14 77)(3 78 8 71)(4 79 9 72)(5 80 10 73)(6 81 11 74)(7 82 12 75)(15 57 22 64)(16 58 23 65)(17 59 24 66)(18 60 25 67)(19 61 26 68)(20 62 27 69)(21 63 28 70)(29 99 36 106)(30 100 37 107)(31 101 38 108)(32 102 39 109)(33 103 40 110)(34 104 41 111)(35 105 42 112)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 63)(7 62)(8 66)(9 65)(10 64)(11 70)(12 69)(13 68)(14 67)(15 73)(16 72)(17 71)(18 77)(19 76)(20 75)(21 74)(22 80)(23 79)(24 78)(25 84)(26 83)(27 82)(28 81)(29 87)(30 86)(31 85)(32 91)(33 90)(34 89)(35 88)(36 94)(37 93)(38 92)(39 98)(40 97)(41 96)(42 95)(43 101)(44 100)(45 99)(46 105)(47 104)(48 103)(49 102)(50 108)(51 107)(52 106)(53 112)(54 111)(55 110)(56 109)
G:=sub<Sym(112)| (1,62)(2,63)(3,57)(4,58)(5,59)(6,60)(7,61)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,99,36,106)(30,100,37,107)(31,101,38,108)(32,102,39,109)(33,103,40,110)(34,104,41,111)(35,105,42,112)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,63)(7,62)(8,66)(9,65)(10,64)(11,70)(12,69)(13,68)(14,67)(15,73)(16,72)(17,71)(18,77)(19,76)(20,75)(21,74)(22,80)(23,79)(24,78)(25,84)(26,83)(27,82)(28,81)(29,87)(30,86)(31,85)(32,91)(33,90)(34,89)(35,88)(36,94)(37,93)(38,92)(39,98)(40,97)(41,96)(42,95)(43,101)(44,100)(45,99)(46,105)(47,104)(48,103)(49,102)(50,108)(51,107)(52,106)(53,112)(54,111)(55,110)(56,109)>;
G:=Group( (1,62)(2,63)(3,57)(4,58)(5,59)(6,60)(7,61)(8,64)(9,65)(10,66)(11,67)(12,68)(13,69)(14,70)(15,71)(16,72)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,81)(26,82)(27,83)(28,84)(29,85)(30,86)(31,87)(32,88)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,97)(42,98)(43,99)(44,100)(45,101)(46,102)(47,103)(48,104)(49,105)(50,106)(51,107)(52,108)(53,109)(54,110)(55,111)(56,112), (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), (1,83,13,76)(2,84,14,77)(3,78,8,71)(4,79,9,72)(5,80,10,73)(6,81,11,74)(7,82,12,75)(15,57,22,64)(16,58,23,65)(17,59,24,66)(18,60,25,67)(19,61,26,68)(20,62,27,69)(21,63,28,70)(29,99,36,106)(30,100,37,107)(31,101,38,108)(32,102,39,109)(33,103,40,110)(34,104,41,111)(35,105,42,112)(43,92,50,85)(44,93,51,86)(45,94,52,87)(46,95,53,88)(47,96,54,89)(48,97,55,90)(49,98,56,91), (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), (1,61)(2,60)(3,59)(4,58)(5,57)(6,63)(7,62)(8,66)(9,65)(10,64)(11,70)(12,69)(13,68)(14,67)(15,73)(16,72)(17,71)(18,77)(19,76)(20,75)(21,74)(22,80)(23,79)(24,78)(25,84)(26,83)(27,82)(28,81)(29,87)(30,86)(31,85)(32,91)(33,90)(34,89)(35,88)(36,94)(37,93)(38,92)(39,98)(40,97)(41,96)(42,95)(43,101)(44,100)(45,99)(46,105)(47,104)(48,103)(49,102)(50,108)(51,107)(52,106)(53,112)(54,111)(55,110)(56,109) );
G=PermutationGroup([[(1,62),(2,63),(3,57),(4,58),(5,59),(6,60),(7,61),(8,64),(9,65),(10,66),(11,67),(12,68),(13,69),(14,70),(15,71),(16,72),(17,73),(18,74),(19,75),(20,76),(21,77),(22,78),(23,79),(24,80),(25,81),(26,82),(27,83),(28,84),(29,85),(30,86),(31,87),(32,88),(33,89),(34,90),(35,91),(36,92),(37,93),(38,94),(39,95),(40,96),(41,97),(42,98),(43,99),(44,100),(45,101),(46,102),(47,103),(48,104),(49,105),(50,106),(51,107),(52,108),(53,109),(54,110),(55,111),(56,112)], [(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)], [(1,83,13,76),(2,84,14,77),(3,78,8,71),(4,79,9,72),(5,80,10,73),(6,81,11,74),(7,82,12,75),(15,57,22,64),(16,58,23,65),(17,59,24,66),(18,60,25,67),(19,61,26,68),(20,62,27,69),(21,63,28,70),(29,99,36,106),(30,100,37,107),(31,101,38,108),(32,102,39,109),(33,103,40,110),(34,104,41,111),(35,105,42,112),(43,92,50,85),(44,93,51,86),(45,94,52,87),(46,95,53,88),(47,96,54,89),(48,97,55,90),(49,98,56,91)], [(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)], [(1,61),(2,60),(3,59),(4,58),(5,57),(6,63),(7,62),(8,66),(9,65),(10,64),(11,70),(12,69),(13,68),(14,67),(15,73),(16,72),(17,71),(18,77),(19,76),(20,75),(21,74),(22,80),(23,79),(24,78),(25,84),(26,83),(27,82),(28,81),(29,87),(30,86),(31,85),(32,91),(33,90),(34,89),(35,88),(36,94),(37,93),(38,92),(39,98),(40,97),(41,96),(42,95),(43,101),(44,100),(45,99),(46,105),(47,104),(48,103),(49,102),(50,108),(51,107),(52,106),(53,112),(54,111),(55,110),(56,109)]])
C2×Q8×D7 is a maximal subgroup of
(Q8×D7)⋊C4 Q8⋊2D28 D14⋊4Q16 Dic14⋊7D4 D14⋊5Q16 C42.125D14 Q8⋊5D28 C14.162- 1+4 Dic14⋊21D4 Dic14⋊22D4 C42.141D14 Dic14⋊10D4 C42.171D14 D28⋊8Q8 C14.1072- 1+4
C2×Q8×D7 is a maximal quotient of
C14.102+ 1+4 Dic14⋊10Q8 C42.232D14 D28⋊10Q8 (Q8×Dic7)⋊C2 C14.752- 1+4 Dic14⋊21D4 C14.512+ 1+4 C14.1182+ 1+4 C14.522+ 1+4 Dic14⋊7Q8 C42.236D14 C42.148D14 D28⋊7Q8 Dic14⋊8Q8 Dic14⋊9Q8 D28⋊8Q8 C42.241D14 C42.174D14 D28⋊9Q8
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | ··· | 4F | 4G | ··· | 4L | 7A | 7B | 7C | 14A | ··· | 14I | 28A | ··· | 28R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 | 28 | ··· | 28 |
| size | 1 | 1 | 1 | 1 | 7 | 7 | 7 | 7 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | Q8 | D7 | D14 | D14 | Q8×D7 |
| kernel | C2×Q8×D7 | C2×Dic14 | C2×C4×D7 | Q8×D7 | Q8×C14 | D14 | C2×Q8 | C2×C4 | Q8 | C2 |
| # reps | 1 | 3 | 3 | 8 | 1 | 4 | 3 | 9 | 12 | 6 |
Matrix representation of C2×Q8×D7 ►in GL4(𝔽29) generated by
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 28 | 0 |
| 0 | 0 | 0 | 28 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 28 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 17 |
| 0 | 1 | 0 | 0 |
| 28 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 28 | 0 | 0 |
| 28 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,17],[0,28,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,28,0,0,28,0,0,0,0,0,1,0,0,0,0,1] >;
C2×Q8×D7 in GAP, Magma, Sage, TeX
C_2\times Q_8\times D_7
% in TeX
G:=Group("C2xQ8xD7"); // GroupNames label
G:=SmallGroup(224,181);
// by ID
G=gap.SmallGroup(224,181);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,159,69,6917]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^4=d^7=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations