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## G = (C2×C56)⋊C4order 448 = 26·7

### 1st semidirect product of C2×C56 and C4 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — (C2×C56)⋊C4
 Chief series C1 — C7 — C14 — C2×C14 — C2×C28 — C22×C28 — C23.21D14 — (C2×C56)⋊C4
 Lower central C7 — C14 — C2×C14 — (C2×C56)⋊C4
 Upper central C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for (C2×C56)⋊C4
G = < a,b,c | a2=b56=c4=1, ab=ba, cac-1=ab28, cbc-1=ab13 >

Subgroups: 324 in 90 conjugacy classes, 47 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, C23, C14, C14, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, Dic7, C28, C2×C14, C2×C14, C42⋊C2, C2×M4(2), C2×M4(2), C7⋊C8, C56, C2×Dic7, C2×C28, C22×C14, M4(2)⋊4C4, C2×C7⋊C8, C4.Dic7, C4.Dic7, C4×Dic7, C4⋊Dic7, C23.D7, C2×C56, C7×M4(2), C22×C28, C2×C4.Dic7, C23.21D14, C14×M4(2), (C2×C56)⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D7, C42, C22⋊C4, C4⋊C4, Dic7, D14, C2.C42, Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, M4(2)⋊4C4, C4×Dic7, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C14.C42, (C2×C56)⋊C4

Smallest permutation representation of (C2×C56)⋊C4
On 112 points
Generators in S112
(1 59)(2 60)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 79)(22 80)(23 81)(24 82)(25 83)(26 84)(27 85)(28 86)(29 87)(30 88)(31 89)(32 90)(33 91)(34 92)(35 93)(36 94)(37 95)(38 96)(39 97)(40 98)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 105)(48 106)(49 107)(50 108)(51 109)(52 110)(53 111)(54 112)(55 57)(56 58)
(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 43 29 15)(2 86)(3 13 31 41)(4 112)(5 39 33 11)(6 82)(7 9 35 37)(8 108)(10 78)(12 104)(14 74)(16 100)(17 27 45 55)(18 70)(19 53 47 25)(20 96)(21 23 49 51)(22 66)(24 92)(26 62)(28 88)(30 58)(32 84)(34 110)(36 80)(38 106)(40 76)(42 102)(44 72)(46 98)(48 68)(50 94)(52 64)(54 90)(56 60)(57 103 85 75)(59 73 87 101)(61 99 89 71)(63 69 91 97)(65 95 93 67)(77 83 105 111)(79 109 107 81)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 8E 8F 8G 8H 14A ··· 14I 14J ··· 14O 28A ··· 28L 28M ··· 28R 56A ··· 56X order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 7 7 7 8 8 8 8 8 8 8 8 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 56 ··· 56 size 1 1 2 2 2 1 1 2 2 2 28 28 28 28 2 2 2 4 4 4 4 28 28 28 28 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + - + - + - + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D7 Dic7 D14 Dic14 C4×D7 D28 C7⋊D4 C4×D7 M4(2)⋊4C4 (C2×C56)⋊C4 kernel (C2×C56)⋊C4 C2×C4.Dic7 C23.21D14 C14×M4(2) C4.Dic7 C23.D7 C2×C56 C2×C28 C2×C28 C2×M4(2) C2×C8 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C23 C7 C1 # reps 1 1 1 1 4 4 4 3 1 3 6 3 6 6 6 12 6 2 12

Matrix representation of (C2×C56)⋊C4 in GL4(𝔽113) generated by

 91 12 0 0 101 22 0 0 0 0 22 12 0 0 101 91
,
 79 100 51 23 13 69 51 51 111 73 44 13 85 111 100 34
,
 98 0 0 0 58 15 0 0 88 72 94 100 72 100 19 19
G:=sub<GL(4,GF(113))| [91,101,0,0,12,22,0,0,0,0,22,101,0,0,12,91],[79,13,111,85,100,69,73,111,51,51,44,100,23,51,13,34],[98,58,88,72,0,15,72,100,0,0,94,19,0,0,100,19] >;

(C2×C56)⋊C4 in GAP, Magma, Sage, TeX

(C_2\times C_{56})\rtimes C_4
% in TeX

G:=Group("(C2xC56):C4");
// GroupNames label

G:=SmallGroup(448,113);
// by ID

G=gap.SmallGroup(448,113);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,253,64,387,136,1684,18822]);
// Polycyclic

G:=Group<a,b,c|a^2=b^56=c^4=1,a*b=b*a,c*a*c^-1=a*b^28,c*b*c^-1=a*b^13>;
// generators/relations

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