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## G = C2×C7⋊D7order 196 = 22·72

### Direct product of C2 and C7⋊D7

Aliases: C2×C7⋊D7, C14⋊D7, C72D14, C723C22, (C7×C14)⋊2C2, SmallGroup(196,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C72 — C2×C7⋊D7
 Chief series C1 — C7 — C72 — C7⋊D7 — C2×C7⋊D7
 Lower central C72 — C2×C7⋊D7
 Upper central C1 — C2

Generators and relations for C2×C7⋊D7
G = < a,b,c,d | a2=b7=c7=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Smallest permutation representation of C2×C7⋊D7
On 98 points
Generators in S98
(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 15)(8 56)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(22 70)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 77)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 84)(37 78)(38 79)(39 80)(40 81)(41 82)(42 83)(43 91)(44 85)(45 86)(46 87)(47 88)(48 89)(49 90)(57 94)(58 95)(59 96)(60 97)(61 98)(62 92)(63 93)
(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)
(1 61 83 68 53 87 72)(2 62 84 69 54 88 73)(3 63 78 70 55 89 74)(4 57 79 64 56 90 75)(5 58 80 65 50 91 76)(6 59 81 66 51 85 77)(7 60 82 67 52 86 71)(8 49 34 19 94 38 23)(9 43 35 20 95 39 24)(10 44 29 21 96 40 25)(11 45 30 15 97 41 26)(12 46 31 16 98 42 27)(13 47 32 17 92 36 28)(14 48 33 18 93 37 22)
(1 72)(2 71)(3 77)(4 76)(5 75)(6 74)(7 73)(8 39)(9 38)(10 37)(11 36)(12 42)(13 41)(14 40)(15 32)(16 31)(17 30)(18 29)(19 35)(20 34)(21 33)(22 25)(23 24)(26 28)(43 94)(44 93)(45 92)(46 98)(47 97)(48 96)(49 95)(50 79)(51 78)(52 84)(53 83)(54 82)(55 81)(56 80)(57 91)(58 90)(59 89)(60 88)(61 87)(62 86)(63 85)(64 65)(66 70)(67 69)

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

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

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

C2×C7⋊D7 is a maximal subgroup of   Dic72D7  C7⋊D28  C28⋊D7  C727D4  C2×D72
C2×C7⋊D7 is a maximal quotient of   C724Q8  C28⋊D7  C727D4

52 conjugacy classes

 class 1 2A 2B 2C 7A ··· 7X 14A ··· 14X order 1 2 2 2 7 ··· 7 14 ··· 14 size 1 1 49 49 2 ··· 2 2 ··· 2

52 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D7 D14 kernel C2×C7⋊D7 C7⋊D7 C7×C14 C14 C7 # reps 1 2 1 24 24

Matrix representation of C2×C7⋊D7 in GL4(𝔽29) generated by

 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 22 19 0 0 10 10
,
 0 1 0 0 28 18 0 0 0 0 0 1 0 0 28 7
,
 0 1 0 0 1 0 0 0 0 0 19 19 0 0 7 10
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,22,10,0,0,19,10],[0,28,0,0,1,18,0,0,0,0,0,28,0,0,1,7],[0,1,0,0,1,0,0,0,0,0,19,7,0,0,19,10] >;

C2×C7⋊D7 in GAP, Magma, Sage, TeX

C_2\times C_7\rtimes D_7
% in TeX

G:=Group("C2xC7:D7");
// GroupNames label

G:=SmallGroup(196,11);
// by ID

G=gap.SmallGroup(196,11);
# by ID

G:=PCGroup([4,-2,-2,-7,-7,290,2691]);
// Polycyclic

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

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