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G = D5611C4order 448 = 26·7

The semidirect product of D56 and C4 acting through Inn(D56)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5611C4, C8.14D28, C56.67D4, Dic2811C4, C42.267D14, (C4×C8)⋊10D7, C71(C8○D8), (C4×C56)⋊15C2, C56⋊C27C4, C8.22(C4×D7), C14.9(C4×D4), C56.52(C2×C4), C4.76(C2×D28), C2.12(C4×D28), D28.13(C2×C4), (C2×C8).324D14, C28.296(C2×D4), C56.C417C2, Dic14⋊C415C2, D567C2.10C2, D28.2C411C2, (C4×C28).328C22, C28.104(C22×C4), (C2×C28).791C23, (C2×C56).407C22, Dic14.13(C2×C4), C4○D28.35C22, C22.20(C4○D28), C4.Dic7.33C22, C4.62(C2×C4×D7), (C2×C14).62(C4○D4), (C2×C4).672(C22×D7), SmallGroup(448,234)

Series: Derived Chief Lower central Upper central

C1C28 — D5611C4
C1C7C14C28C2×C28C4○D28D567C2 — D5611C4
C7C14C28 — D5611C4
C1C8C2×C8C4×C8

Generators and relations for D5611C4
 G = < a,b,c | a56=b2=c4=1, bab=a-1, ac=ca, cbc-1=a42b >

Subgroups: 484 in 106 conjugacy classes, 47 normal (33 characteristic)
C1, C2, C2 [×3], C4 [×2], C4 [×4], C22, C22 [×2], C7, C8 [×4], C8 [×2], C2×C4, C2×C4 [×3], D4 [×4], Q8 [×2], D7 [×2], C14, C14, C42, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8, SD16 [×2], Q16, C4○D4 [×2], Dic7 [×2], C28 [×2], C28 [×2], D14 [×2], C2×C14, C4×C8, C4≀C2 [×2], C8.C4, C8○D4 [×2], C4○D8, C7⋊C8 [×2], C56 [×4], Dic14 [×2], C4×D7 [×2], D28 [×2], C7⋊D4 [×2], C2×C28, C2×C28, C8○D8, C8×D7 [×2], C8⋊D7 [×2], C56⋊C2 [×2], D56, Dic28, C4.Dic7 [×2], C4×C28, C2×C56 [×2], C4○D28 [×2], Dic14⋊C4 [×2], C56.C4, C4×C56, D28.2C4 [×2], D567C2, D5611C4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D7, C22×C4, C2×D4, C4○D4, D14 [×3], C4×D4, C4×D7 [×2], D28 [×2], C22×D7, C8○D8, C2×C4×D7, C2×D28, C4○D28, C4×D28, D5611C4

Smallest permutation representation of D5611C4
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 76)(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 112)(22 111)(23 110)(24 109)(25 108)(26 107)(27 106)(28 105)(29 104)(30 103)(31 102)(32 101)(33 100)(34 99)(35 98)(36 97)(37 96)(38 95)(39 94)(40 93)(41 92)(42 91)(43 90)(44 89)(45 88)(46 87)(47 86)(48 85)(49 84)(50 83)(51 82)(52 81)(53 80)(54 79)(55 78)(56 77)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 71 85 99)(58 72 86 100)(59 73 87 101)(60 74 88 102)(61 75 89 103)(62 76 90 104)(63 77 91 105)(64 78 92 106)(65 79 93 107)(66 80 94 108)(67 81 95 109)(68 82 96 110)(69 83 97 111)(70 84 98 112)

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

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

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

124 conjugacy classes

class 1 2A2B2C2D4A4B4C···4G4H4I7A7B7C8A8B8C8D8E···8J8K8L8M8N14A···14I28A···28AJ56A···56AV
order12222444···44477788888···8888814···1428···2856···56
size1122828112···2282822211112···2282828282···22···22···2

124 irreducible representations

dim1111111112222222222
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D7C4○D4D14D14C4×D7D28C8○D8C4○D28D5611C4
kernelD5611C4Dic14⋊C4C56.C4C4×C56D28.2C4D567C2C56⋊C2D56Dic28C56C4×C8C2×C14C42C2×C8C8C8C7C22C1
# reps12112142223236121281248

Matrix representation of D5611C4 in GL2(𝔽113) generated by

1040
025
,
025
1040
,
1120
015
G:=sub<GL(2,GF(113))| [104,0,0,25],[0,104,25,0],[112,0,0,15] >;

D5611C4 in GAP, Magma, Sage, TeX

D_{56}\rtimes_{11}C_4
% in TeX

G:=Group("D56:11C4");
// GroupNames label

G:=SmallGroup(448,234);
// by ID

G=gap.SmallGroup(448,234);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,58,136,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^42*b>;
// generators/relations

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