Copied to
clipboard

## G = C7×C4.A4order 336 = 24·3·7

### Direct product of C7 and C4.A4

Aliases: C7×C4.A4, Q8.C42, C28.2A4, SL2(𝔽3)⋊2C14, C4○D4⋊C21, C4.(C7×A4), C2.3(A4×C14), (C7×Q8).5C6, C14.12(C2×A4), (C7×SL2(𝔽3))⋊5C2, (C7×C4○D4)⋊1C3, SmallGroup(336,170)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — C7×C4.A4
 Chief series C1 — C2 — Q8 — C7×Q8 — C7×SL2(𝔽3) — C7×C4.A4
 Lower central Q8 — C7×C4.A4
 Upper central C1 — C28

Generators and relations for C7×C4.A4
G = < a,b,c,d,e | a7=b4=e3=1, c2=d2=b2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=b2c, ece-1=b2cd, ede-1=c >

Smallest permutation representation of C7×C4.A4
On 112 points
Generators in S112
(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 39 77 8)(2 40 71 9)(3 41 72 10)(4 42 73 11)(5 36 74 12)(6 37 75 13)(7 38 76 14)(15 86 30 84)(16 87 31 78)(17 88 32 79)(18 89 33 80)(19 90 34 81)(20 91 35 82)(21 85 29 83)(22 61 108 52)(23 62 109 53)(24 63 110 54)(25 57 111 55)(26 58 112 56)(27 59 106 50)(28 60 107 51)(43 65 99 97)(44 66 100 98)(45 67 101 92)(46 68 102 93)(47 69 103 94)(48 70 104 95)(49 64 105 96)
(1 70 77 95)(2 64 71 96)(3 65 72 97)(4 66 73 98)(5 67 74 92)(6 68 75 93)(7 69 76 94)(8 48 39 104)(9 49 40 105)(10 43 41 99)(11 44 42 100)(12 45 36 101)(13 46 37 102)(14 47 38 103)(15 28 30 107)(16 22 31 108)(17 23 32 109)(18 24 33 110)(19 25 34 111)(20 26 35 112)(21 27 29 106)(50 85 59 83)(51 86 60 84)(52 87 61 78)(53 88 62 79)(54 89 63 80)(55 90 57 81)(56 91 58 82)
(1 83 77 85)(2 84 71 86)(3 78 72 87)(4 79 73 88)(5 80 74 89)(6 81 75 90)(7 82 76 91)(8 29 39 21)(9 30 40 15)(10 31 41 16)(11 32 42 17)(12 33 36 18)(13 34 37 19)(14 35 38 20)(22 99 108 43)(23 100 109 44)(24 101 110 45)(25 102 111 46)(26 103 112 47)(27 104 106 48)(28 105 107 49)(50 70 59 95)(51 64 60 96)(52 65 61 97)(53 66 62 98)(54 67 63 92)(55 68 57 93)(56 69 58 94)
(15 28 105)(16 22 99)(17 23 100)(18 24 101)(19 25 102)(20 26 103)(21 27 104)(29 106 48)(30 107 49)(31 108 43)(32 109 44)(33 110 45)(34 111 46)(35 112 47)(50 70 83)(51 64 84)(52 65 78)(53 66 79)(54 67 80)(55 68 81)(56 69 82)(57 93 90)(58 94 91)(59 95 85)(60 96 86)(61 97 87)(62 98 88)(63 92 89)

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

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

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

98 conjugacy classes

 class 1 2A 2B 3A 3B 4A 4B 4C 6A 6B 7A ··· 7F 12A 12B 12C 12D 14A ··· 14F 14G ··· 14L 21A ··· 21L 28A ··· 28L 28M ··· 28R 42A ··· 42L 84A ··· 84X order 1 2 2 3 3 4 4 4 6 6 7 ··· 7 12 12 12 12 14 ··· 14 14 ··· 14 21 ··· 21 28 ··· 28 28 ··· 28 42 ··· 42 84 ··· 84 size 1 1 6 4 4 1 1 6 4 4 1 ··· 1 4 4 4 4 1 ··· 1 6 ··· 6 4 ··· 4 1 ··· 1 6 ··· 6 4 ··· 4 4 ··· 4

98 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 3 3 3 3 type + + + + image C1 C2 C3 C6 C7 C14 C21 C42 C4.A4 C7×C4.A4 A4 C2×A4 C7×A4 A4×C14 kernel C7×C4.A4 C7×SL2(𝔽3) C7×C4○D4 C7×Q8 C4.A4 SL2(𝔽3) C4○D4 Q8 C7 C1 C28 C14 C4 C2 # reps 1 1 2 2 6 6 12 12 6 36 1 1 6 6

Matrix representation of C7×C4.A4 in GL2(𝔽29) generated by

 20 0 0 20
,
 12 0 0 12
,
 17 0 16 12
,
 12 18 0 17
,
 28 12 12 0
G:=sub<GL(2,GF(29))| [20,0,0,20],[12,0,0,12],[17,16,0,12],[12,0,18,17],[28,12,12,0] >;

C7×C4.A4 in GAP, Magma, Sage, TeX

C_7\times C_4.A_4
% in TeX

G:=Group("C7xC4.A4");
// GroupNames label

G:=SmallGroup(336,170);
// by ID

G=gap.SmallGroup(336,170);
# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,-2,1008,1017,117,1900,202,88]);
// Polycyclic

G:=Group<a,b,c,d,e|a^7=b^4=e^3=1,c^2=d^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=b^2*c,e*c*e^-1=b^2*c*d,e*d*e^-1=c>;
// generators/relations

Export

׿
×
𝔽