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## G = D56⋊11C4order 448 = 26·7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — D56⋊11C4
 Chief series C1 — C7 — C14 — C28 — C2×C28 — C4○D28 — D56⋊7C2 — D56⋊11C4
 Lower central C7 — C14 — C28 — D56⋊11C4
 Upper central C1 — C8 — C2×C8 — C4×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 2A 2B 2C 2D 4A 4B 4C ··· 4G 4H 4I 7A 7B 7C 8A 8B 8C 8D 8E ··· 8J 8K 8L 8M 8N 14A ··· 14I 28A ··· 28AJ 56A ··· 56AV order 1 2 2 2 2 4 4 4 ··· 4 4 4 7 7 7 8 8 8 8 8 ··· 8 8 8 8 8 14 ··· 14 28 ··· 28 56 ··· 56 size 1 1 2 28 28 1 1 2 ··· 2 28 28 2 2 2 1 1 1 1 2 ··· 2 28 28 28 28 2 ··· 2 2 ··· 2 2 ··· 2

124 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 D7 C4○D4 D14 D14 C4×D7 D28 C8○D8 C4○D28 D56⋊11C4 kernel D56⋊11C4 Dic14⋊C4 C56.C4 C4×C56 D28.2C4 D56⋊7C2 C56⋊C2 D56 Dic28 C56 C4×C8 C2×C14 C42 C2×C8 C8 C8 C7 C22 C1 # reps 1 2 1 1 2 1 4 2 2 2 3 2 3 6 12 12 8 12 48

Matrix representation of D5611C4 in GL2(𝔽113) generated by

 104 0 0 25
,
 0 25 104 0
,
 112 0 0 15
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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