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G = D567C2order 224 = 25·7

The semidirect product of D56 and C2 acting through Inn(D56)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D567C2, C4.20D28, C28.35D4, C8.17D14, Dic287C2, C22.1D28, C56.17C22, C28.30C23, D28.7C22, Dic14.6C22, (C2×C8)⋊4D7, (C2×C56)⋊6C2, C71(C4○D8), C4○D281C2, C56⋊C27C2, C2.13(C2×D28), C14.11(C2×D4), (C2×C14).18D4, (C2×C4).81D14, C4.28(C22×D7), (C2×C28).99C22, SmallGroup(224,99)

Series: Derived Chief Lower central Upper central

C1C28 — D567C2
C1C7C14C28D28C4○D28 — D567C2
C7C14C28 — D567C2
C1C4C2×C4C2×C8

Generators and relations for D567C2
 G = < a,b,c | a56=b2=c2=1, bab=a-1, ac=ca, cbc=a28b >

Subgroups: 302 in 62 conjugacy classes, 29 normal (21 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×2], C22, C22 [×2], C7, C8 [×2], C2×C4, C2×C4 [×2], D4 [×4], Q8 [×2], D7 [×2], C14, C14, C2×C8, D8, SD16 [×2], Q16, C4○D4 [×2], Dic7 [×2], C28 [×2], D14 [×2], C2×C14, C4○D8, C56 [×2], Dic14 [×2], C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C2×C28, C56⋊C2 [×2], D56, Dic28, C2×C56, C4○D28 [×2], D567C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C4○D8, D28 [×2], C22×D7, C2×D28, D567C2

Smallest permutation representation of D567C2
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 7)(2 6)(3 5)(8 56)(9 55)(10 54)(11 53)(12 52)(13 51)(14 50)(15 49)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)(25 39)(26 38)(27 37)(28 36)(29 35)(30 34)(31 33)(57 73)(58 72)(59 71)(60 70)(61 69)(62 68)(63 67)(64 66)(74 112)(75 111)(76 110)(77 109)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)
(1 104)(2 105)(3 106)(4 107)(5 108)(6 109)(7 110)(8 111)(9 112)(10 57)(11 58)(12 59)(13 60)(14 61)(15 62)(16 63)(17 64)(18 65)(19 66)(20 67)(21 68)(22 69)(23 70)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 99)(53 100)(54 101)(55 102)(56 103)

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

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

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

D567C2 is a maximal subgroup of
D568C4  Dic28.C4  D56.1C4  D562C4  D5611C4  D564C4  D5610C4  D567C4  D1127C2  C16⋊D14  C16.D14  D8.D14  Q16.D14  C56.30C23  C56.9C23  D4.11D28  D4.12D28  D4.13D28  D813D14  D28.29D4  D28.30D4  D7×C4○D8  D810D14
D567C2 is a maximal quotient of
C56.13Q8  C4×C56⋊C2  C4×D56  C8.8D28  C42.264D14  C4×Dic28  C23.10D28  D28.32D4  D2814D4  C23.13D28  Dic14.3Q8  D28.19D4  C42.36D14  D28.3Q8  C23.22D28  C23.23D28  C5630D4  C5629D4  C56.82D4

62 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E7A7B7C8A8B8C8D14A···14I28A···28L56A···56X
order1222244444777888814···1428···2856···56
size1122828112282822222222···22···22···2

62 irreducible representations

dim111111222222222
type+++++++++++++
imageC1C2C2C2C2C2D4D4D7D14D14C4○D8D28D28D567C2
kernelD567C2C56⋊C2D56Dic28C2×C56C4○D28C28C2×C14C2×C8C8C2×C4C7C4C22C1
# reps1211121136346624

Matrix representation of D567C2 in GL2(𝔽113) generated by

8521
9248
,
3255
481
,
79108
534
G:=sub<GL(2,GF(113))| [85,92,21,48],[32,4,55,81],[79,5,108,34] >;

D567C2 in GAP, Magma, Sage, TeX

D_{56}\rtimes_7C_2
% in TeX

G:=Group("D56:7C2");
// GroupNames label

G:=SmallGroup(224,99);
// by ID

G=gap.SmallGroup(224,99);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,50,579,69,6917]);
// Polycyclic

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

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