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## G = C2×Dic14order 112 = 24·7

### Direct product of C2 and Dic14

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

 Derived series C1 — C14 — C2×Dic14
 Chief series C1 — C7 — C14 — Dic7 — C2×Dic7 — C2×Dic14
 Lower central C7 — C14 — C2×Dic14
 Upper central C1 — C22 — C2×C4

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

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 2A 2B 2C 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A ··· 14I 28A ··· 28L order 1 2 2 2 4 4 4 4 4 4 7 7 7 14 ··· 14 28 ··· 28 size 1 1 1 1 2 2 14 14 14 14 2 2 2 2 ··· 2 2 ··· 2

34 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + - + + + - image C1 C2 C2 C2 Q8 D7 D14 D14 Dic14 kernel C2×Dic14 Dic14 C2×Dic7 C2×C28 C14 C2×C4 C4 C22 C2 # reps 1 4 2 1 2 3 6 3 12

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

 28 0 0 0 1 0 0 0 1
,
 28 0 0 0 6 8 0 21 4
,
 28 0 0 0 0 12 0 12 0
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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