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G = C2×C56⋊C2order 224 = 25·7

Direct product of C2 and C56⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C56⋊C2, C88D14, C4.6D28, C569C22, C141SD16, C28.29D4, C28.28C23, D28.6C22, C22.12D28, Dic143C22, (C2×C8)⋊5D7, (C2×C56)⋊7C2, C71(C2×SD16), C14.9(C2×D4), (C2×D28).4C2, C2.11(C2×D28), (C2×C14).16D4, (C2×C4).79D14, (C2×Dic14)⋊5C2, C4.26(C22×D7), (C2×C28).88C22, SmallGroup(224,97)

Series: Derived Chief Lower central Upper central

C1C28 — C2×C56⋊C2
C1C7C14C28D28C2×D28 — C2×C56⋊C2
C7C14C28 — C2×C56⋊C2
C1C22C2×C4C2×C8

Generators and relations for C2×C56⋊C2
 G = < a,b,c | a2=b56=c2=1, ab=ba, ac=ca, cbc=b27 >

Subgroups: 366 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, Dic7, C28, D14, C2×C14, C2×SD16, C56, Dic14, Dic14, D28, D28, C2×Dic7, C2×C28, C22×D7, C56⋊C2, C2×C56, C2×Dic14, C2×D28, C2×C56⋊C2
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C2×SD16, D28, C22×D7, C56⋊C2, C2×D28, C2×C56⋊C2

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

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

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

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

C2×C56⋊C2 is a maximal subgroup of
C85D28  C8.8D28  C42.16D14  C8⋊D28  C8.D28  D28.31D4  D28.32D4  D2814D4  Dic1414D4  Dic142D4  D4.6D28  D43D28  D28.D4  Dic14.11D4  Q82D28  Q8.D28  Dic7⋊SD16  C28⋊SD16  D28.19D4  C42.36D14  Dic148D4  Dic78SD16  C88D28  C83D28  C56⋊C2⋊C4  C8.24D28  C5630D4  C562D4  D4.3D28  C5611D4  C56.43D4  C5615D4  C56.37D4  D4.11D28  C2×D7×SD16  D811D14
C2×C56⋊C2 is a maximal quotient of
C569Q8  C28.14Q16  C85D28  C4.5D56  C23.34D28  D28.31D4  C23.38D28  Dic1414D4  C28⋊SD16  D283Q8  Dic148D4  Dic144Q8  C5630D4

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C8A8B8C8D14A···14I28A···28L56A···56X
order1222224444777888814···1428···2856···56
size1111282822282822222222···22···22···2

62 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2D4D4D7SD16D14D14D28D28C56⋊C2
kernelC2×C56⋊C2C56⋊C2C2×C56C2×Dic14C2×D28C28C2×C14C2×C8C14C8C2×C4C4C22C2
# reps141111134636624

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

112000
011200
001120
000112
,
13900
1044600
00898
009776
,
112000
34100
001524
006698
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[13,104,0,0,9,46,0,0,0,0,89,97,0,0,8,76],[112,34,0,0,0,1,0,0,0,0,15,66,0,0,24,98] >;

C2×C56⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC56:C2");
// GroupNames label

G:=SmallGroup(224,97);
// by ID

G=gap.SmallGroup(224,97);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,50,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^2=b^56=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^27>;
// generators/relations

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