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G = C8.D14order 224 = 25·7

1st non-split extension by C8 of D14 acting via D14/C7=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8.1D14, C28.13D4, C4.15D28, Dic282C2, M4(2)⋊2D7, C56.1C22, C22.6D28, C28.33C23, D28.8C22, Dic14.8C22, C56⋊C22C2, (C2×C14).6D4, C4○D28.4C2, (C2×C4).16D14, C14.14(C2×D4), C2.16(C2×D28), C71(C8.C22), (C2×Dic14)⋊8C2, (C7×M4(2))⋊2C2, C4.31(C22×D7), (C2×C28).28C22, SmallGroup(224,104)

Series: Derived Chief Lower central Upper central

C1C28 — C8.D14
C1C7C14C28D28C4○D28 — C8.D14
C7C14C28 — C8.D14
C1C2C2×C4M4(2)

Generators and relations for C8.D14
 G = < a,b,c | a8=1, b14=c2=a4, bab-1=a5, cac-1=a-1, cbc-1=b13 >

Subgroups: 270 in 60 conjugacy classes, 29 normal (19 characteristic)
C1, C2, C2 [×2], C4 [×2], C4 [×3], C22, C22, C7, C8 [×2], C2×C4, C2×C4 [×2], D4 [×2], Q8 [×4], D7, C14, C14, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic7 [×3], C28 [×2], D14, C2×C14, C8.C22, C56 [×2], Dic14, Dic14 [×2], Dic14, C4×D7, D28, C2×Dic7, C7⋊D4, C2×C28, C56⋊C2 [×2], Dic28 [×2], C7×M4(2), C2×Dic14, C4○D28, C8.D14
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D7, C2×D4, D14 [×3], C8.C22, D28 [×2], C22×D7, C2×D28, C8.D14

Smallest permutation representation of C8.D14
On 112 points
Generators in S112
(1 29 84 98 15 43 70 112)(2 44 57 85 16 30 71 99)(3 31 58 100 17 45 72 86)(4 46 59 87 18 32 73 101)(5 33 60 102 19 47 74 88)(6 48 61 89 20 34 75 103)(7 35 62 104 21 49 76 90)(8 50 63 91 22 36 77 105)(9 37 64 106 23 51 78 92)(10 52 65 93 24 38 79 107)(11 39 66 108 25 53 80 94)(12 54 67 95 26 40 81 109)(13 41 68 110 27 55 82 96)(14 56 69 97 28 42 83 111)
(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 14 15 28)(2 27 16 13)(3 12 17 26)(4 25 18 11)(5 10 19 24)(6 23 20 9)(7 8 21 22)(29 111 43 97)(30 96 44 110)(31 109 45 95)(32 94 46 108)(33 107 47 93)(34 92 48 106)(35 105 49 91)(36 90 50 104)(37 103 51 89)(38 88 52 102)(39 101 53 87)(40 86 54 100)(41 99 55 85)(42 112 56 98)(57 68 71 82)(58 81 72 67)(59 66 73 80)(60 79 74 65)(61 64 75 78)(62 77 76 63)(69 84 83 70)

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

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

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

C8.D14 is a maximal subgroup of
D28.1D4  D28.2D4  D28.4D4  D28.7D4  M4(2)⋊D14  D4.9D28  C8.20D28  C8.24D28  C56.9C23  D4.11D28  D4.13D28  SD16⋊D14  D86D14  D7×C8.C22  D28.44D4
C8.D14 is a maximal quotient of
C8⋊Dic14  C42.14D14  C42.16D14  C42.20D14  C8.D28  Dic28⋊C4  C23.34D28  C23.10D28  D28.32D4  C22.D56  Dic1414D4  C22⋊Dic28  Dic14.3Q8  C28⋊SD16  C42.36D14  D284Q8  C4⋊Dic28  Dic144Q8  C23.46D28  C23.47D28  C23.49D28  C562D4  C56.4D4

41 conjugacy classes

class 1 2A2B2C4A4B4C4D4E7A7B7C8A8B14A14B14C14D14E14F28A···28F28G28H28I56A···56L
order1222444447778814141414141428···2828282856···56
size1122822282828222442224442···24444···4

41 irreducible representations

dim111111222222244
type+++++++++++++--
imageC1C2C2C2C2C2D4D4D7D14D14D28D28C8.C22C8.D14
kernelC8.D14C56⋊C2Dic28C7×M4(2)C2×Dic14C4○D28C28C2×C14M4(2)C8C2×C4C4C22C7C1
# reps122111113636616

Matrix representation of C8.D14 in GL4(𝔽113) generated by

006274
00390
223500
411100
,
212170111
929810290
13833192
3876676
,
212170111
9892491
83132484
8733589
G:=sub<GL(4,GF(113))| [0,0,22,41,0,0,35,11,62,39,0,0,74,0,0,0],[21,92,13,3,21,98,83,87,70,102,31,66,111,90,92,76],[21,98,83,87,21,92,13,3,70,4,24,35,111,91,84,89] >;

C8.D14 in GAP, Magma, Sage, TeX

C_8.D_{14}
% in TeX

G:=Group("C8.D14");
// GroupNames label

G:=SmallGroup(224,104);
// by ID

G=gap.SmallGroup(224,104);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,103,218,188,50,579,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^8=1,b^14=c^2=a^4,b*a*b^-1=a^5,c*a*c^-1=a^-1,c*b*c^-1=b^13>;
// generators/relations

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