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## G = C2×D56⋊C2order 448 = 26·7

### Direct product of C2 and D56⋊C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×D56⋊C2
 Chief series C1 — C7 — C14 — C28 — C4×D7 — C2×C4×D7 — C2×D4×D7 — C2×D56⋊C2
 Lower central C7 — C14 — C28 — C2×D56⋊C2
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for C2×D56⋊C2
G = < a,b,c,d | a2=b56=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b43, cd=dc >

Subgroups: 1796 in 298 conjugacy classes, 103 normal (33 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D7, C14, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C24, Dic7, C28, C28, D14, D14, C2×C14, C2×C14, C2×M4(2), C2×D8, C2×SD16, C2×SD16, C8⋊C22, C22×D4, C2×C4○D4, C7⋊C8, C56, C4×D7, C4×D7, D28, D28, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×D4, C7×Q8, C7×Q8, C22×D7, C22×D7, C22×C14, C2×C8⋊C22, C8⋊D7, D56, C2×C7⋊C8, D4⋊D7, Q8⋊D7, C2×C56, C7×SD16, C2×C4×D7, C2×C4×D7, C2×D28, C2×D28, D4×D7, D4×D7, Q82D7, Q82D7, C2×C7⋊D4, D4×C14, Q8×C14, C23×D7, C2×C8⋊D7, C2×D56, D56⋊C2, C2×D4⋊D7, C2×Q8⋊D7, C14×SD16, C2×D4×D7, C2×Q82D7, C2×D56⋊C2
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C8⋊C22, C22×D4, C22×D7, C2×C8⋊C22, D4×D7, C23×D7, D56⋊C2, C2×D4×D7, C2×D56⋊C2

Smallest permutation representation of C2×D56⋊C2
On 112 points
Generators in S112
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 85)(13 86)(14 87)(15 88)(16 89)(17 90)(18 91)(19 92)(20 93)(21 94)(22 95)(23 96)(24 97)(25 98)(26 99)(27 100)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 57)(41 58)(42 59)(43 60)(44 61)(45 62)(46 63)(47 64)(48 65)(49 66)(50 67)(51 68)(52 69)(53 70)(54 71)(55 72)(56 73)
(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 35)(2 34)(3 33)(4 32)(5 31)(6 30)(7 29)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(70 112)(71 111)(72 110)(73 109)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)
(1 60)(2 103)(3 90)(4 77)(5 64)(6 107)(7 94)(8 81)(9 68)(10 111)(11 98)(12 85)(13 72)(14 59)(15 102)(16 89)(17 76)(18 63)(19 106)(20 93)(21 80)(22 67)(23 110)(24 97)(25 84)(26 71)(27 58)(28 101)(29 88)(30 75)(31 62)(32 105)(33 92)(34 79)(35 66)(36 109)(37 96)(38 83)(39 70)(40 57)(41 100)(42 87)(43 74)(44 61)(45 104)(46 91)(47 78)(48 65)(49 108)(50 95)(51 82)(52 69)(53 112)(54 99)(55 86)(56 73)

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

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

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

64 conjugacy classes

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

64 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D7 D14 D14 D14 D14 C8⋊C22 D4×D7 D4×D7 D56⋊C2 kernel C2×D56⋊C2 C2×C8⋊D7 C2×D56 D56⋊C2 C2×D4⋊D7 C2×Q8⋊D7 C14×SD16 C2×D4×D7 C2×Q8⋊2D7 C4×D7 C2×Dic7 C22×D7 C2×SD16 C2×C8 SD16 C2×D4 C2×Q8 C14 C4 C22 C2 # reps 1 1 1 8 1 1 1 1 1 2 1 1 3 3 12 3 3 2 3 3 12

Matrix representation of C2×D56⋊C2 in GL8(𝔽113)

 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112
,
 100 1 22 92 0 0 0 0 23 38 86 98 0 0 0 0 93 84 66 112 0 0 0 0 42 15 65 22 0 0 0 0 0 0 0 0 0 0 44 37 0 0 0 0 0 0 76 28 0 0 0 0 91 38 69 76 0 0 0 0 75 99 37 85
,
 9 89 0 0 0 0 0 0 41 104 0 0 0 0 0 0 69 0 0 1 0 0 0 0 57 74 1 0 0 0 0 0 0 0 0 0 0 112 0 111 0 0 0 0 112 0 111 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 83 112 0 0 0 0 0 101 44 0 112 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 1 0 2 0 0 0 0 0 0 112 0 0 0 0 0 0 0 0 112

G:=sub<GL(8,GF(113))| [112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112],[100,23,93,42,0,0,0,0,1,38,84,15,0,0,0,0,22,86,66,65,0,0,0,0,92,98,112,22,0,0,0,0,0,0,0,0,0,0,91,75,0,0,0,0,0,0,38,99,0,0,0,0,44,76,69,37,0,0,0,0,37,28,76,85],[9,41,69,57,0,0,0,0,89,104,0,74,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,111,0,1,0,0,0,0,111,0,1,0],[1,0,0,101,0,0,0,0,0,1,83,44,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,112,0,0,0,0,0,0,2,0,112] >;

C2×D56⋊C2 in GAP, Magma, Sage, TeX

C_2\times D_{56}\rtimes C_2
% in TeX

G:=Group("C2xD56:C2");
// GroupNames label

G:=SmallGroup(448,1212);
// by ID

G=gap.SmallGroup(448,1212);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,1123,185,136,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^56=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^43,c*d=d*c>;
// generators/relations

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