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## G = C2×C7⋊C8order 112 = 24·7

### Direct product of C2 and C7⋊C8

Aliases: C2×C7⋊C8, C14⋊C8, C28.3C4, C4.14D14, C4.3Dic7, C28.14C22, C22.2Dic7, C4(C7⋊C8), C72(C2×C8), (C2×C4).5D7, C14.6(C2×C4), (C2×C14).2C4, (C2×C28).6C2, C2.1(C2×Dic7), SmallGroup(112,8)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C2×C7⋊C8
 Chief series C1 — C7 — C14 — C28 — C7⋊C8 — C2×C7⋊C8
 Lower central C7 — C2×C7⋊C8
 Upper central C1 — C2×C4

Generators and relations for C2×C7⋊C8
G = < a,b,c | a2=b7=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C2×C7⋊C8
Regular action on 112 points
Generators in S112
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 31)(8 32)(9 46)(10 47)(11 48)(12 41)(13 42)(14 43)(15 44)(16 45)(17 54)(18 55)(19 56)(20 49)(21 50)(22 51)(23 52)(24 53)(33 103)(34 104)(35 97)(36 98)(37 99)(38 100)(39 101)(40 102)(57 84)(58 85)(59 86)(60 87)(61 88)(62 81)(63 82)(64 83)(65 92)(66 93)(67 94)(68 95)(69 96)(70 89)(71 90)(72 91)(73 108)(74 109)(75 110)(76 111)(77 112)(78 105)(79 106)(80 107)
(1 61 43 100 89 17 105)(2 106 18 90 101 44 62)(3 63 45 102 91 19 107)(4 108 20 92 103 46 64)(5 57 47 104 93 21 109)(6 110 22 94 97 48 58)(7 59 41 98 95 23 111)(8 112 24 96 99 42 60)(9 83 28 73 49 65 33)(10 34 66 50 74 29 84)(11 85 30 75 51 67 35)(12 36 68 52 76 31 86)(13 87 32 77 53 69 37)(14 38 70 54 78 25 88)(15 81 26 79 55 71 39)(16 40 72 56 80 27 82)
(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)

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

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

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

C2×C7⋊C8 is a maximal subgroup of
C42.D7  C28⋊C8  C28.Q8  C4.Dic14  C14.D8  C14.Q16  C8×Dic7  Dic7⋊C8  C56⋊C4  D14⋊C8  C28.53D4  C28.55D4  D4⋊Dic7  Q8⋊Dic7  D7×C2×C8  D28.C4  Q8.Dic7  D4.8D14
C2×C7⋊C8 is a maximal quotient of
C28⋊C8  C28.C8  C28.55D4

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 7A 7B 7C 8A ··· 8H 14A ··· 14I 28A ··· 28L order 1 2 2 2 4 4 4 4 7 7 7 8 ··· 8 14 ··· 14 28 ··· 28 size 1 1 1 1 1 1 1 1 2 2 2 7 ··· 7 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C8 D7 Dic7 D14 Dic7 C7⋊C8 kernel C2×C7⋊C8 C7⋊C8 C2×C28 C28 C2×C14 C14 C2×C4 C4 C4 C22 C2 # reps 1 2 1 2 2 8 3 3 3 3 12

Matrix representation of C2×C7⋊C8 in GL3(𝔽113) generated by

 112 0 0 0 1 0 0 0 1
,
 1 0 0 0 10 112 0 11 112
,
 112 0 0 0 48 7 0 40 65
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,1],[1,0,0,0,10,11,0,112,112],[112,0,0,0,48,40,0,7,65] >;

C2×C7⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes C_8
% in TeX

G:=Group("C2xC7:C8");
// GroupNames label

G:=SmallGroup(112,8);
// by ID

G=gap.SmallGroup(112,8);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,20,42,2404]);
// Polycyclic

G:=Group<a,b,c|a^2=b^7=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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