direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C22×Dic7, C23.2D7, C14.9C23, C22.11D14, C14⋊2(C2×C4), (C2×C14)⋊3C4, C7⋊2(C22×C4), C2.2(C22×D7), (C22×C14).3C2, (C2×C14).12C22, SmallGroup(112,35)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C22×Dic7 |
Generators and relations for C22×Dic7
G = < a,b,c,d | a2=b2=c14=1, d2=c7, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 120 in 54 conjugacy classes, 43 normal (7 characteristic)
C1, C2, C2, C4, C22, C7, C2×C4, C23, C14, C14, C22×C4, Dic7, C2×C14, C2×Dic7, C22×C14, C22×Dic7
Quotients: C1, C2, C4, C22, C2×C4, C23, D7, C22×C4, Dic7, D14, C2×Dic7, C22×D7, C22×Dic7
►Regular action on 112 points
Generators in S112
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 29)(15 54)(16 55)(17 56)(18 43)(19 44)(20 45)(21 46)(22 47)(23 48)(24 49)(25 50)(26 51)(27 52)(28 53)(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)(78 106)(79 107)(80 108)(81 109)(82 110)(83 111)(84 112)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 15)(29 54)(30 55)(31 56)(32 43)(33 44)(34 45)(35 46)(36 47)(37 48)(38 49)(39 50)(40 51)(41 52)(42 53)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(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 71 8 78)(2 84 9 77)(3 83 10 76)(4 82 11 75)(5 81 12 74)(6 80 13 73)(7 79 14 72)(15 58 22 65)(16 57 23 64)(17 70 24 63)(18 69 25 62)(19 68 26 61)(20 67 27 60)(21 66 28 59)(29 100 36 107)(30 99 37 106)(31 112 38 105)(32 111 39 104)(33 110 40 103)(34 109 41 102)(35 108 42 101)(43 97 50 90)(44 96 51 89)(45 95 52 88)(46 94 53 87)(47 93 54 86)(48 92 55 85)(49 91 56 98)
G:=sub<Sym(112)| (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(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)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,15)(29,54)(30,55)(31,56)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,71,8,78)(2,84,9,77)(3,83,10,76)(4,82,11,75)(5,81,12,74)(6,80,13,73)(7,79,14,72)(15,58,22,65)(16,57,23,64)(17,70,24,63)(18,69,25,62)(19,68,26,61)(20,67,27,60)(21,66,28,59)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,97,50,90)(44,96,51,89)(45,95,52,88)(46,94,53,87)(47,93,54,86)(48,92,55,85)(49,91,56,98)>;
G:=Group( (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,29)(15,54)(16,55)(17,56)(18,43)(19,44)(20,45)(21,46)(22,47)(23,48)(24,49)(25,50)(26,51)(27,52)(28,53)(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)(78,106)(79,107)(80,108)(81,109)(82,110)(83,111)(84,112), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,25)(11,26)(12,27)(13,28)(14,15)(29,54)(30,55)(31,56)(32,43)(33,44)(34,45)(35,46)(36,47)(37,48)(38,49)(39,50)(40,51)(41,52)(42,53)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(67,81)(68,82)(69,83)(70,84)(85,99)(86,100)(87,101)(88,102)(89,103)(90,104)(91,105)(92,106)(93,107)(94,108)(95,109)(96,110)(97,111)(98,112), (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), (1,71,8,78)(2,84,9,77)(3,83,10,76)(4,82,11,75)(5,81,12,74)(6,80,13,73)(7,79,14,72)(15,58,22,65)(16,57,23,64)(17,70,24,63)(18,69,25,62)(19,68,26,61)(20,67,27,60)(21,66,28,59)(29,100,36,107)(30,99,37,106)(31,112,38,105)(32,111,39,104)(33,110,40,103)(34,109,41,102)(35,108,42,101)(43,97,50,90)(44,96,51,89)(45,95,52,88)(46,94,53,87)(47,93,54,86)(48,92,55,85)(49,91,56,98) );
G=PermutationGroup([[(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,29),(15,54),(16,55),(17,56),(18,43),(19,44),(20,45),(21,46),(22,47),(23,48),(24,49),(25,50),(26,51),(27,52),(28,53),(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),(78,106),(79,107),(80,108),(81,109),(82,110),(83,111),(84,112)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,25),(11,26),(12,27),(13,28),(14,15),(29,54),(30,55),(31,56),(32,43),(33,44),(34,45),(35,46),(36,47),(37,48),(38,49),(39,50),(40,51),(41,52),(42,53),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(67,81),(68,82),(69,83),(70,84),(85,99),(86,100),(87,101),(88,102),(89,103),(90,104),(91,105),(92,106),(93,107),(94,108),(95,109),(96,110),(97,111),(98,112)], [(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)], [(1,71,8,78),(2,84,9,77),(3,83,10,76),(4,82,11,75),(5,81,12,74),(6,80,13,73),(7,79,14,72),(15,58,22,65),(16,57,23,64),(17,70,24,63),(18,69,25,62),(19,68,26,61),(20,67,27,60),(21,66,28,59),(29,100,36,107),(30,99,37,106),(31,112,38,105),(32,111,39,104),(33,110,40,103),(34,109,41,102),(35,108,42,101),(43,97,50,90),(44,96,51,89),(45,95,52,88),(46,94,53,87),(47,93,54,86),(48,92,55,85),(49,91,56,98)]])
C22×Dic7 is a maximal subgroup of
C14.C42 C23.11D14 C22⋊Dic14 Dic7⋊4D4 C22.D28 C23.18D14 Dic7⋊D4 D7×C22×C4 Dic7⋊A4
C22×Dic7 is a maximal quotient of
C23.21D14 Q8.Dic7
40 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 7A | 7B | 7C | 14A | ··· | 14U |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 7 | 7 | 7 | 14 | ··· | 14 |
| size | 1 | 1 | ··· | 1 | 7 | ··· | 7 | 2 | 2 | 2 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C4 | D7 | Dic7 | D14 |
| kernel | C22×Dic7 | C2×Dic7 | C22×C14 | C2×C14 | C23 | C22 | C22 |
| # reps | 1 | 6 | 1 | 8 | 3 | 12 | 9 |
Matrix representation of C22×Dic7 ►in GL4(𝔽29) generated by
| 28 | 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 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 28 |
| 0 | 0 | 9 | 21 |
| 1 | 0 | 0 | 0 |
| 0 | 28 | 0 | 0 |
| 0 | 0 | 2 | 8 |
| 0 | 0 | 3 | 27 |
G:=sub<GL(4,GF(29))| [28,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,9,0,0,28,21],[1,0,0,0,0,28,0,0,0,0,2,3,0,0,8,27] >;
C22×Dic7 in GAP, Magma, Sage, TeX
C_2^2\times {\rm Dic}_7 % in TeX
G:=Group("C2^2xDic7"); // GroupNames label
G:=SmallGroup(112,35);
// by ID
G=gap.SmallGroup(112,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-7,40,2404]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^14=1,d^2=c^7,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations