Copied to
clipboard

## G = C8.D14order 224 = 25·7

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C8.D14
 Chief series C1 — C7 — C14 — C28 — D28 — C4○D28 — C8.D14
 Lower central C7 — C14 — C28 — C8.D14
 Upper central C1 — C2 — C2×C4 — M4(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)])

41 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 7A 7B 7C 8A 8B 14A 14B 14C 14D 14E 14F 28A ··· 28F 28G 28H 28I 56A ··· 56L order 1 2 2 2 4 4 4 4 4 7 7 7 8 8 14 14 14 14 14 14 28 ··· 28 28 28 28 56 ··· 56 size 1 1 2 28 2 2 28 28 28 2 2 2 4 4 2 2 2 4 4 4 2 ··· 2 4 4 4 4 ··· 4

41 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D28 D28 C8.C22 C8.D14 kernel C8.D14 C56⋊C2 Dic28 C7×M4(2) C2×Dic14 C4○D28 C28 C2×C14 M4(2) C8 C2×C4 C4 C22 C7 C1 # reps 1 2 2 1 1 1 1 1 3 6 3 6 6 1 6

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

 0 0 62 74 0 0 39 0 22 35 0 0 41 11 0 0
,
 21 21 70 111 92 98 102 90 13 83 31 92 3 87 66 76
,
 21 21 70 111 98 92 4 91 83 13 24 84 87 3 35 89
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

׿
×
𝔽