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## G = C2×C4.F7order 336 = 24·3·7

### Direct product of C2 and C4.F7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×C4.F7
 Chief series C1 — C7 — C14 — C2×C7⋊C3 — C7⋊C12 — C2×C7⋊C12 — C2×C4.F7
 Lower central C7 — C14 — C2×C4.F7
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×C4.F7
G = < a,b,c,d | a2=b4=c7=1, d6=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 256 in 76 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C7, C2×C4, C2×C4, Q8, C12, C2×C6, C14, C14, C2×Q8, C7⋊C3, C2×C12, C3×Q8, Dic7, C28, C2×C14, C2×C7⋊C3, C2×C7⋊C3, C6×Q8, Dic14, C2×Dic7, C2×C28, C7⋊C12, C4×C7⋊C3, C22×C7⋊C3, C2×Dic14, C4.F7, C2×C7⋊C12, C2×C4×C7⋊C3, C2×C4.F7
Quotients: C1, C2, C3, C22, C6, Q8, C23, C2×C6, C2×Q8, C3×Q8, C22×C6, F7, C6×Q8, C2×F7, C4.F7, C22×F7, C2×C4.F7

Smallest permutation representation of C2×C4.F7
On 112 points
Generators in S112
(1 9)(2 10)(3 11)(4 12)(5 14)(6 15)(7 16)(8 13)(17 50)(18 51)(19 52)(20 41)(21 42)(22 43)(23 44)(24 45)(25 46)(26 47)(27 48)(28 49)(29 71)(30 72)(31 73)(32 74)(33 75)(34 76)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(53 94)(54 95)(55 96)(56 97)(57 98)(58 99)(59 100)(60 89)(61 90)(62 91)(63 92)(64 93)(77 108)(78 109)(79 110)(80 111)(81 112)(82 101)(83 102)(84 103)(85 104)(86 105)(87 106)(88 107)
(1 16 3 14)(2 15 4 13)(5 9 7 11)(6 12 8 10)(17 61 23 55)(18 56 24 62)(19 63 25 57)(20 58 26 64)(21 53 27 59)(22 60 28 54)(29 103 35 109)(30 110 36 104)(31 105 37 111)(32 112 38 106)(33 107 39 101)(34 102 40 108)(41 99 47 93)(42 94 48 100)(43 89 49 95)(44 96 50 90)(45 91 51 97)(46 98 52 92)(65 78 71 84)(66 85 72 79)(67 80 73 86)(68 87 74 81)(69 82 75 88)(70 77 76 83)
(1 37 33 26 29 18 22)(2 19 27 38 23 30 34)(3 31 39 20 35 24 28)(4 25 21 32 17 36 40)(5 86 82 99 78 91 95)(6 92 100 87 96 79 83)(7 80 88 93 84 97 89)(8 98 94 81 90 85 77)(9 67 75 47 71 51 43)(10 52 48 68 44 72 76)(11 73 69 41 65 45 49)(12 46 42 74 50 66 70)(13 57 53 112 61 104 108)(14 105 101 58 109 62 54)(15 63 59 106 55 110 102)(16 111 107 64 103 56 60)
(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)| (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,50)(18,51)(19,52)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,89)(61,90)(62,91)(63,92)(64,93)(77,108)(78,109)(79,110)(80,111)(81,112)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107), (1,16,3,14)(2,15,4,13)(5,9,7,11)(6,12,8,10)(17,61,23,55)(18,56,24,62)(19,63,25,57)(20,58,26,64)(21,53,27,59)(22,60,28,54)(29,103,35,109)(30,110,36,104)(31,105,37,111)(32,112,38,106)(33,107,39,101)(34,102,40,108)(41,99,47,93)(42,94,48,100)(43,89,49,95)(44,96,50,90)(45,91,51,97)(46,98,52,92)(65,78,71,84)(66,85,72,79)(67,80,73,86)(68,87,74,81)(69,82,75,88)(70,77,76,83), (1,37,33,26,29,18,22)(2,19,27,38,23,30,34)(3,31,39,20,35,24,28)(4,25,21,32,17,36,40)(5,86,82,99,78,91,95)(6,92,100,87,96,79,83)(7,80,88,93,84,97,89)(8,98,94,81,90,85,77)(9,67,75,47,71,51,43)(10,52,48,68,44,72,76)(11,73,69,41,65,45,49)(12,46,42,74,50,66,70)(13,57,53,112,61,104,108)(14,105,101,58,109,62,54)(15,63,59,106,55,110,102)(16,111,107,64,103,56,60), (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( (1,9)(2,10)(3,11)(4,12)(5,14)(6,15)(7,16)(8,13)(17,50)(18,51)(19,52)(20,41)(21,42)(22,43)(23,44)(24,45)(25,46)(26,47)(27,48)(28,49)(29,71)(30,72)(31,73)(32,74)(33,75)(34,76)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(53,94)(54,95)(55,96)(56,97)(57,98)(58,99)(59,100)(60,89)(61,90)(62,91)(63,92)(64,93)(77,108)(78,109)(79,110)(80,111)(81,112)(82,101)(83,102)(84,103)(85,104)(86,105)(87,106)(88,107), (1,16,3,14)(2,15,4,13)(5,9,7,11)(6,12,8,10)(17,61,23,55)(18,56,24,62)(19,63,25,57)(20,58,26,64)(21,53,27,59)(22,60,28,54)(29,103,35,109)(30,110,36,104)(31,105,37,111)(32,112,38,106)(33,107,39,101)(34,102,40,108)(41,99,47,93)(42,94,48,100)(43,89,49,95)(44,96,50,90)(45,91,51,97)(46,98,52,92)(65,78,71,84)(66,85,72,79)(67,80,73,86)(68,87,74,81)(69,82,75,88)(70,77,76,83), (1,37,33,26,29,18,22)(2,19,27,38,23,30,34)(3,31,39,20,35,24,28)(4,25,21,32,17,36,40)(5,86,82,99,78,91,95)(6,92,100,87,96,79,83)(7,80,88,93,84,97,89)(8,98,94,81,90,85,77)(9,67,75,47,71,51,43)(10,52,48,68,44,72,76)(11,73,69,41,65,45,49)(12,46,42,74,50,66,70)(13,57,53,112,61,104,108)(14,105,101,58,109,62,54)(15,63,59,106,55,110,102)(16,111,107,64,103,56,60), (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([[(1,9),(2,10),(3,11),(4,12),(5,14),(6,15),(7,16),(8,13),(17,50),(18,51),(19,52),(20,41),(21,42),(22,43),(23,44),(24,45),(25,46),(26,47),(27,48),(28,49),(29,71),(30,72),(31,73),(32,74),(33,75),(34,76),(35,65),(36,66),(37,67),(38,68),(39,69),(40,70),(53,94),(54,95),(55,96),(56,97),(57,98),(58,99),(59,100),(60,89),(61,90),(62,91),(63,92),(64,93),(77,108),(78,109),(79,110),(80,111),(81,112),(82,101),(83,102),(84,103),(85,104),(86,105),(87,106),(88,107)], [(1,16,3,14),(2,15,4,13),(5,9,7,11),(6,12,8,10),(17,61,23,55),(18,56,24,62),(19,63,25,57),(20,58,26,64),(21,53,27,59),(22,60,28,54),(29,103,35,109),(30,110,36,104),(31,105,37,111),(32,112,38,106),(33,107,39,101),(34,102,40,108),(41,99,47,93),(42,94,48,100),(43,89,49,95),(44,96,50,90),(45,91,51,97),(46,98,52,92),(65,78,71,84),(66,85,72,79),(67,80,73,86),(68,87,74,81),(69,82,75,88),(70,77,76,83)], [(1,37,33,26,29,18,22),(2,19,27,38,23,30,34),(3,31,39,20,35,24,28),(4,25,21,32,17,36,40),(5,86,82,99,78,91,95),(6,92,100,87,96,79,83),(7,80,88,93,84,97,89),(8,98,94,81,90,85,77),(9,67,75,47,71,51,43),(10,52,48,68,44,72,76),(11,73,69,41,65,45,49),(12,46,42,74,50,66,70),(13,57,53,112,61,104,108),(14,105,101,58,109,62,54),(15,63,59,106,55,110,102),(16,111,107,64,103,56,60)], [(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)]])

38 conjugacy classes

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 4E 4F 6A ··· 6F 7 12A ··· 12L 14A 14B 14C 28A 28B 28C 28D order 1 2 2 2 3 3 4 4 4 4 4 4 6 ··· 6 7 12 ··· 12 14 14 14 28 28 28 28 size 1 1 1 1 7 7 2 2 14 14 14 14 7 ··· 7 6 14 ··· 14 6 6 6 6 6 6 6

38 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 6 6 6 6 type + + + + - + + + - image C1 C2 C2 C2 C3 C6 C6 C6 Q8 C3×Q8 F7 C2×F7 C2×F7 C4.F7 kernel C2×C4.F7 C4.F7 C2×C7⋊C12 C2×C4×C7⋊C3 C2×Dic14 Dic14 C2×Dic7 C2×C28 C2×C7⋊C3 C14 C2×C4 C4 C22 C2 # reps 1 4 2 1 2 8 4 2 2 4 1 2 1 4

Matrix representation of C2×C4.F7 in GL8(𝔽337)

 336 0 0 0 0 0 0 0 0 336 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 148 0 0 0 0 0 0 0 336 189 0 0 0 0 0 0 0 0 320 0 34 303 34 0 0 0 303 320 34 0 0 34 0 0 303 303 17 0 34 0 0 0 0 303 0 320 34 34 0 0 303 0 0 303 17 34 0 0 0 303 34 303 0 17
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 336 1 0 0 0 0 0 0 336 0 1 0 0 0 0 0 336 0 0 1 0 0 0 0 336 0 0 0 1 0 0 0 336 0 0 0 0 1 0 0 336 0 0 0 0 0
,
 72 81 0 0 0 0 0 0 0 265 0 0 0 0 0 0 0 0 185 0 0 185 294 152 0 0 185 185 294 0 152 0 0 0 142 0 152 185 152 0 0 0 0 185 152 185 0 294 0 0 0 185 0 142 152 152 0 0 185 142 152 0 0 152

G:=sub<GL(8,GF(337))| [336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[148,336,0,0,0,0,0,0,0,189,0,0,0,0,0,0,0,0,320,303,303,0,303,0,0,0,0,320,303,303,0,303,0,0,34,34,17,0,0,34,0,0,303,0,0,320,303,303,0,0,34,0,34,34,17,0,0,0,0,34,0,34,34,17],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,336,336,336,336,336,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0],[72,0,0,0,0,0,0,0,81,265,0,0,0,0,0,0,0,0,185,185,142,0,0,185,0,0,0,185,0,185,185,142,0,0,0,294,152,152,0,152,0,0,185,0,185,185,142,0,0,0,294,152,152,0,152,0,0,0,152,0,0,294,152,152] >;

C2×C4.F7 in GAP, Magma, Sage, TeX

C_2\times C_4.F_7
% in TeX

G:=Group("C2xC4.F7");
// GroupNames label

G:=SmallGroup(336,121);
// by ID

G=gap.SmallGroup(336,121);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-2,-7,144,506,122,10373,887]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^7=1,d^6=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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