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G = C56.23D4order 448 = 26·7

23rd non-split extension by C56 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C56.23D4, D28.25D4, Dic14.25D4, (C14×D8)⋊2C2, (C2×D8)⋊10D7, C4.61(D4×D7), (C2×C8).87D14, C56.C43C2, (C2×D4).67D14, C28.D47C2, C28.169(C2×D4), C74(D4.4D4), C8.27(C7⋊D4), D28.2C41C2, D4.D144C2, (C2×C56).32C22, C2.18(C282D4), (C2×C28).437C23, C4○D28.46C22, (D4×C14).86C22, C14.111(C4⋊D4), C4.Dic7.19C22, C22.20(D42D7), C4.81(C2×C7⋊D4), (C2×C4).126(C22×D7), (C2×C14).158(C4○D4), SmallGroup(448,694)

Series: Derived Chief Lower central Upper central

C1C2×C28 — C56.23D4
C1C7C14C28C2×C28C4○D28D28.2C4 — C56.23D4
C7C14C2×C28 — C56.23D4
C1C2C2×C4C2×D8

Generators and relations for C56.23D4
 G = < a,b,c | a56=c2=1, b4=a28, bab-1=a-1, cac=a41, cbc=a28b3 >

Subgroups: 484 in 108 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D7, C14, C14, C2×C8, C2×C8, M4(2), D8, SD16, C2×D4, C4○D4, Dic7, C28, D14, C2×C14, C2×C14, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, C7⋊C8, C56, Dic14, C4×D7, D28, C7⋊D4, C2×C28, C7×D4, C22×C14, D4.4D4, C8×D7, C8⋊D7, C4.Dic7, C4.Dic7, D4⋊D7, D4.D7, C2×C56, C7×D8, C4○D28, D4×C14, C56.C4, C28.D4, D28.2C4, D4.D14, C14×D8, C56.23D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, D14, C4⋊D4, C7⋊D4, C22×D7, D4.4D4, D4×D7, D42D7, C2×C7⋊D4, C282D4, C56.23D4

Smallest permutation representation of C56.23D4
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 57 43 71 29 85 15 99)(2 112 44 70 30 84 16 98)(3 111 45 69 31 83 17 97)(4 110 46 68 32 82 18 96)(5 109 47 67 33 81 19 95)(6 108 48 66 34 80 20 94)(7 107 49 65 35 79 21 93)(8 106 50 64 36 78 22 92)(9 105 51 63 37 77 23 91)(10 104 52 62 38 76 24 90)(11 103 53 61 39 75 25 89)(12 102 54 60 40 74 26 88)(13 101 55 59 41 73 27 87)(14 100 56 58 42 72 28 86)
(1 57)(2 98)(3 83)(4 68)(5 109)(6 94)(7 79)(8 64)(9 105)(10 90)(11 75)(12 60)(13 101)(14 86)(15 71)(16 112)(17 97)(18 82)(19 67)(20 108)(21 93)(22 78)(23 63)(24 104)(25 89)(26 74)(27 59)(28 100)(29 85)(30 70)(31 111)(32 96)(33 81)(34 66)(35 107)(36 92)(37 77)(38 62)(39 103)(40 88)(41 73)(42 58)(43 99)(44 84)(45 69)(46 110)(47 95)(48 80)(49 65)(50 106)(51 91)(52 76)(53 61)(54 102)(55 87)(56 72)

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

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

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

58 conjugacy classes

class 1 2A2B2C2D2E4A4B4C7A7B7C8A8B8C8D8E8F8G14A···14I14J···14U28A···28F56A···56L
order122222444777888888814···1414···1428···2856···56
size11288282228222224282856562···28···84···44···4

58 irreducible representations

dim111111222222224444
type++++++++++++++-
imageC1C2C2C2C2C2D4D4D4D7C4○D4D14D14C7⋊D4D4.4D4D4×D7D42D7C56.23D4
kernelC56.23D4C56.C4C28.D4D28.2C4D4.D14C14×D8C56Dic14D28C2×D8C2×C14C2×C8C2×D4C8C7C4C22C1
# reps11212121132361223312

Matrix representation of C56.23D4 in GL4(𝔽113) generated by

444400
694400
515799
61041049
,
1313016
4040160
44377373
7770100100
,
7373970
100100097
83764040
36431313
G:=sub<GL(4,GF(113))| [44,69,51,6,44,44,57,104,0,0,9,104,0,0,9,9],[13,40,44,77,13,40,37,70,0,16,73,100,16,0,73,100],[73,100,83,36,73,100,76,43,97,0,40,13,0,97,40,13] >;

C56.23D4 in GAP, Magma, Sage, TeX

C_{56}._{23}D_4
% in TeX

G:=Group("C56.23D4");
// GroupNames label

G:=SmallGroup(448,694);
// by ID

G=gap.SmallGroup(448,694);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,120,254,219,1123,297,136,438,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^56=c^2=1,b^4=a^28,b*a*b^-1=a^-1,c*a*c=a^41,c*b*c=a^28*b^3>;
// generators/relations

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