Copied to
clipboard

G = C7⋊SD32order 224 = 25·7

The semidirect product of C7 and SD32 acting via SD32/Q16=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C73SD32, Q161D7, C8.6D14, C28.5D4, D56.2C2, C14.10D8, C56.4C22, C7⋊C163C2, (C7×Q16)⋊1C2, C2.6(D4⋊D7), C4.3(C7⋊D4), SmallGroup(224,34)

Series: Derived Chief Lower central Upper central

C1C56 — C7⋊SD32
C1C7C14C28C56D56 — C7⋊SD32
C7C14C28C56 — C7⋊SD32
C1C2C4C8Q16

Generators and relations for C7⋊SD32
 G = < a,b,c | a7=b16=c2=1, bab-1=cac=a-1, cbc=b7 >

56C2
4C4
28C22
8D7
2Q8
14D4
4D14
4C28
7C16
7D8
2D28
2C7×Q8
7SD32

Smallest permutation representation of C7⋊SD32
On 112 points
Generators in S112
(1 44 52 81 101 28 68)(2 69 29 102 82 53 45)(3 46 54 83 103 30 70)(4 71 31 104 84 55 47)(5 48 56 85 105 32 72)(6 73 17 106 86 57 33)(7 34 58 87 107 18 74)(8 75 19 108 88 59 35)(9 36 60 89 109 20 76)(10 77 21 110 90 61 37)(11 38 62 91 111 22 78)(12 79 23 112 92 63 39)(13 40 64 93 97 24 80)(14 65 25 98 94 49 41)(15 42 50 95 99 26 66)(16 67 27 100 96 51 43)
(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)
(2 8)(3 15)(4 6)(5 13)(7 11)(10 16)(12 14)(17 55)(18 62)(19 53)(20 60)(21 51)(22 58)(23 49)(24 56)(25 63)(26 54)(27 61)(28 52)(29 59)(30 50)(31 57)(32 64)(33 71)(34 78)(35 69)(36 76)(37 67)(38 74)(39 65)(40 72)(41 79)(42 70)(43 77)(44 68)(45 75)(46 66)(47 73)(48 80)(81 101)(82 108)(83 99)(84 106)(85 97)(86 104)(87 111)(88 102)(89 109)(90 100)(91 107)(92 98)(93 105)(94 112)(95 103)(96 110)

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

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

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

C7⋊SD32 is a maximal subgroup of   D7×SD32  D112⋊C2  Q32⋊D7  Q323D7  Q16.D14  Q16⋊D14  C56.30C23
C7⋊SD32 is a maximal quotient of   C8.5Dic14  C14.D16  C14.Q32

32 conjugacy classes

class 1 2A2B4A4B7A7B7C8A8B14A14B14C16A16B16C16D28A28B28C28D···28I56A···56F
order12244777881414141616161628282828···2856···56
size11562822222222141414144448···84···4

32 irreducible representations

dim111122222244
type++++++++++
imageC1C2C2C2D4D7D8D14SD32C7⋊D4D4⋊D7C7⋊SD32
kernelC7⋊SD32C7⋊C16D56C7×Q16C28Q16C14C8C7C4C2C1
# reps111113234636

Matrix representation of C7⋊SD32 in GL4(𝔽113) generated by

1000
0100
0001
001129
,
232200
686500
0010
009112
,
1000
4311200
0010
009112
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,0,112,0,0,1,9],[23,68,0,0,22,65,0,0,0,0,1,9,0,0,0,112],[1,43,0,0,0,112,0,0,0,0,1,9,0,0,0,112] >;

C7⋊SD32 in GAP, Magma, Sage, TeX

C_7\rtimes {\rm SD}_{32}
% in TeX

G:=Group("C7:SD32");
// GroupNames label

G:=SmallGroup(224,34);
// by ID

G=gap.SmallGroup(224,34);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,73,103,218,116,122,579,297,69,6917]);
// Polycyclic

G:=Group<a,b,c|a^7=b^16=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^7>;
// generators/relations

Export

Subgroup lattice of C7⋊SD32 in TeX

׿
×
𝔽