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G = C2×Dic14order 112 = 24·7

Direct product of C2 and Dic14

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic14, C14⋊Q8, C4.11D14, C14.1C23, C22.8D14, C28.11C22, Dic7.1C22, C71(C2×Q8), (C2×C4).4D7, (C2×C28).4C2, C2.3(C22×D7), (C2×C14).8C22, (C2×Dic7).3C2, SmallGroup(112,27)

Series: Derived Chief Lower central Upper central

C1C14 — C2×Dic14
C1C7C14Dic7C2×Dic7 — C2×Dic14
C7C14 — C2×Dic14
C1C22C2×C4

Generators and relations for C2×Dic14
 G = < a,b,c | a2=b28=1, c2=b14, ab=ba, ac=ca, cbc-1=b-1 >

7C4
7C4
7C4
7C4
7C2×C4
7Q8
7C2×C4
7Q8
7Q8
7Q8
7C2×Q8

Smallest permutation representation of C2×Dic14
Regular action on 112 points
Generators in S112
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(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 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 57 15 71)(2 84 16 70)(3 83 17 69)(4 82 18 68)(5 81 19 67)(6 80 20 66)(7 79 21 65)(8 78 22 64)(9 77 23 63)(10 76 24 62)(11 75 25 61)(12 74 26 60)(13 73 27 59)(14 72 28 58)(29 105 43 91)(30 104 44 90)(31 103 45 89)(32 102 46 88)(33 101 47 87)(34 100 48 86)(35 99 49 85)(36 98 50 112)(37 97 51 111)(38 96 52 110)(39 95 53 109)(40 94 54 108)(41 93 55 107)(42 92 56 106)

G:=sub<Sym(112)| (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,57,15,71)(2,84,16,70)(3,83,17,69)(4,82,18,68)(5,81,19,67)(6,80,20,66)(7,79,21,65)(8,78,22,64)(9,77,23,63)(10,76,24,62)(11,75,25,61)(12,74,26,60)(13,73,27,59)(14,72,28,58)(29,105,43,91)(30,104,44,90)(31,103,45,89)(32,102,46,88)(33,101,47,87)(34,100,48,86)(35,99,49,85)(36,98,50,112)(37,97,51,111)(38,96,52,110)(39,95,53,109)(40,94,54,108)(41,93,55,107)(42,92,56,106)>;

G:=Group( (1,49)(2,50)(3,51)(4,52)(5,53)(6,54)(7,55)(8,56)(9,29)(10,30)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44)(25,45)(26,46)(27,47)(28,48)(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), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(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), (1,57,15,71)(2,84,16,70)(3,83,17,69)(4,82,18,68)(5,81,19,67)(6,80,20,66)(7,79,21,65)(8,78,22,64)(9,77,23,63)(10,76,24,62)(11,75,25,61)(12,74,26,60)(13,73,27,59)(14,72,28,58)(29,105,43,91)(30,104,44,90)(31,103,45,89)(32,102,46,88)(33,101,47,87)(34,100,48,86)(35,99,49,85)(36,98,50,112)(37,97,51,111)(38,96,52,110)(39,95,53,109)(40,94,54,108)(41,93,55,107)(42,92,56,106) );

G=PermutationGroup([(1,49),(2,50),(3,51),(4,52),(5,53),(6,54),(7,55),(8,56),(9,29),(10,30),(11,31),(12,32),(13,33),(14,34),(15,35),(16,36),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44),(25,45),(26,46),(27,47),(28,48),(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)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28),(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)], [(1,57,15,71),(2,84,16,70),(3,83,17,69),(4,82,18,68),(5,81,19,67),(6,80,20,66),(7,79,21,65),(8,78,22,64),(9,77,23,63),(10,76,24,62),(11,75,25,61),(12,74,26,60),(13,73,27,59),(14,72,28,58),(29,105,43,91),(30,104,44,90),(31,103,45,89),(32,102,46,88),(33,101,47,87),(34,100,48,86),(35,99,49,85),(36,98,50,112),(37,97,51,111),(38,96,52,110),(39,95,53,109),(40,94,54,108),(41,93,55,107),(42,92,56,106)])

C2×Dic14 is a maximal subgroup of
C14.Q16  C28.44D4  C4.12D28  C282Q8  C4.D28  C22⋊Dic14  Dic7.D4  Dic73Q8  C28⋊Q8  D14⋊Q8  D142Q8  C8.D14  C28.48D4  C28.17D4  Dic7⋊Q8  D4.9D14  C2×Q8×D7  D4.10D14
C2×Dic14 is a maximal quotient of
C282Q8  C28.6Q8  C22⋊Dic14  C28⋊Q8  C28.3Q8  C28.48D4

34 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F7A7B7C14A···14I28A···28L
order122244444477714···1428···28
size111122141414142222···22···2

34 irreducible representations

dim111122222
type++++-+++-
imageC1C2C2C2Q8D7D14D14Dic14
kernelC2×Dic14Dic14C2×Dic7C2×C28C14C2×C4C4C22C2
# reps1421236312

Matrix representation of C2×Dic14 in GL3(𝔽29) generated by

2800
010
001
,
2800
068
0214
,
2800
0012
0120
G:=sub<GL(3,GF(29))| [28,0,0,0,1,0,0,0,1],[28,0,0,0,6,21,0,8,4],[28,0,0,0,0,12,0,12,0] >;

C2×Dic14 in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_{14}
% in TeX

G:=Group("C2xDic14");
// GroupNames label

G:=SmallGroup(112,27);
// by ID

G=gap.SmallGroup(112,27);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,40,182,42,2404]);
// Polycyclic

G:=Group<a,b,c|a^2=b^28=1,c^2=b^14,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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Subgroup lattice of C2×Dic14 in TeX

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