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G = D564C4order 448 = 26·7

4th semidirect product of D56 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D564C4, C56.49D4, C8.27D28, Dic284C4, C42.23D14, C8.8(C4×D7), C8⋊C45D7, C56⋊C22C4, C56.5(C2×C4), C71(C8.26D4), C4.78(C2×D28), C14.14(C4×D4), C2.17(C4×D28), Dic14⋊C41C2, D28.15(C2×C4), C28.298(C2×D4), (C2×C8).161D14, C56.C412C2, D567C2.8C2, (C4×C28).17C22, D28.2C412C2, (C2×C56).229C22, C28.108(C22×C4), (C2×C28).792C23, Dic14.15(C2×C4), C4○D28.36C22, C22.21(C4○D28), C4.Dic7.34C22, C4.66(C2×C4×D7), (C7×C8⋊C4)⋊1C2, (C2×C14).63(C4○D4), (C2×C4).682(C22×D7), SmallGroup(448,251)

Series: Derived Chief Lower central Upper central

C1C28 — D564C4
C1C7C14C28C2×C28C4○D28D567C2 — D564C4
C7C14C28 — D564C4
C1C4C2×C8C8⋊C4

Generators and relations for D564C4
 G = < a,b,c | a56=b2=c4=1, bab=a-1, cac-1=a29, cbc-1=a14b >

Subgroups: 484 in 104 conjugacy classes, 47 normal (29 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C8, C2×C4, C2×C4, D4, Q8, D7, C14, C14, C42, C2×C8, C2×C8, M4(2), D8, SD16, Q16, C4○D4, Dic7, C28, C28, D14, C2×C14, C8⋊C4, C4≀C2, C8.C4, C8○D4, C4○D8, C7⋊C8, C56, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C2×C28, C8.26D4, C8×D7, C8⋊D7, C56⋊C2, D56, Dic28, C4.Dic7, C4×C28, C2×C56, C4○D28, Dic14⋊C4, C56.C4, C7×C8⋊C4, D28.2C4, D567C2, D564C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22×C4, C2×D4, C4○D4, D14, C4×D4, C4×D7, D28, C22×D7, C8.26D4, C2×C4×D7, C2×D28, C4○D28, C4×D28, D564C4

Smallest permutation representation of D564C4
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 71)(2 70)(3 69)(4 68)(5 67)(6 66)(7 65)(8 64)(9 63)(10 62)(11 61)(12 60)(13 59)(14 58)(15 57)(16 112)(17 111)(18 110)(19 109)(20 108)(21 107)(22 106)(23 105)(24 104)(25 103)(26 102)(27 101)(28 100)(29 99)(30 98)(31 97)(32 96)(33 95)(34 94)(35 93)(36 92)(37 91)(38 90)(39 89)(40 88)(41 87)(42 86)(43 85)(44 84)(45 83)(46 82)(47 81)(48 80)(49 79)(50 78)(51 77)(52 76)(53 75)(54 74)(55 73)(56 72)
(1 15 29 43)(2 44 30 16)(3 17 31 45)(4 46 32 18)(5 19 33 47)(6 48 34 20)(7 21 35 49)(8 50 36 22)(9 23 37 51)(10 52 38 24)(11 25 39 53)(12 54 40 26)(13 27 41 55)(14 56 42 28)(58 86)(60 88)(62 90)(64 92)(66 94)(68 96)(70 98)(72 100)(74 102)(76 104)(78 106)(80 108)(82 110)(84 112)

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

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

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

82 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G7A7B7C8A8B8C8D8E8F8G8H8I8J14A···14I28A···28L28M···28X56A···56X
order122224444444777888888888814···1428···2828···2856···56
size1122828112442828222222244282828282···22···24···44···4

82 irreducible representations

dim1111111112222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4C4D4D7C4○D4D14D14C4×D7D28C4○D28C8.26D4D564C4
kernelD564C4Dic14⋊C4C56.C4C7×C8⋊C4D28.2C4D567C2C56⋊C2D56Dic28C56C8⋊C4C2×C14C42C2×C8C8C8C22C7C1
# reps12112142223236121212212

Matrix representation of D564C4 in GL4(𝔽113) generated by

1029700
431100
10827112
143106106
,
510578
7803951
10827112
92146755
,
98000
631500
3001120
860991
G:=sub<GL(4,GF(113))| [102,43,108,14,97,11,2,3,0,0,7,106,0,0,112,106],[51,78,108,92,0,0,2,14,57,39,7,67,8,51,112,55],[98,63,30,86,0,15,0,0,0,0,112,99,0,0,0,1] >;

D564C4 in GAP, Magma, Sage, TeX

D_{56}\rtimes_4C_4
% in TeX

G:=Group("D56:4C4");
// GroupNames label

G:=SmallGroup(448,251);
// by ID

G=gap.SmallGroup(448,251);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,120,387,58,136,1684,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^14*b>;
// generators/relations

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