Copied to
clipboard

## G = C2×Q8⋊D7order 224 = 25·7

### Direct product of C2 and Q8⋊D7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C28 — C2×Q8⋊D7
 Chief series C1 — C7 — C14 — C28 — D28 — C2×D28 — C2×Q8⋊D7
 Lower central C7 — C14 — C28 — C2×Q8⋊D7
 Upper central C1 — C22 — C2×C4 — C2×Q8

Generators and relations for C2×Q8⋊D7
G = < a,b,c,d,e | a2=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 318 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D7, C14, C14, C2×C8, SD16, C2×D4, C2×Q8, C28, C28, D14, C2×C14, C2×SD16, C7⋊C8, D28, D28, C2×C28, C2×C28, C7×Q8, C7×Q8, C22×D7, C2×C7⋊C8, Q8⋊D7, C2×D28, Q8×C14, C2×Q8⋊D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2×D4, D14, C2×SD16, C7⋊D4, C22×D7, Q8⋊D7, C2×C7⋊D4, C2×Q8⋊D7

Smallest permutation representation of C2×Q8⋊D7
On 112 points
Generators in S112
(1 57)(2 58)(3 59)(4 60)(5 61)(6 62)(7 63)(8 64)(9 65)(10 66)(11 67)(12 68)(13 69)(14 70)(15 71)(16 72)(17 73)(18 74)(19 75)(20 76)(21 77)(22 78)(23 79)(24 80)(25 81)(26 82)(27 83)(28 84)(29 85)(30 86)(31 87)(32 88)(33 89)(34 90)(35 91)(36 92)(37 93)(38 94)(39 95)(40 96)(41 97)(42 98)(43 99)(44 100)(45 101)(46 102)(47 103)(48 104)(49 105)(50 106)(51 107)(52 108)(53 109)(54 110)(55 111)(56 112)
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)(29 43 36 50)(30 44 37 51)(31 45 38 52)(32 46 39 53)(33 47 40 54)(34 48 41 55)(35 49 42 56)(57 78 64 71)(58 79 65 72)(59 80 66 73)(60 81 67 74)(61 82 68 75)(62 83 69 76)(63 84 70 77)(85 99 92 106)(86 100 93 107)(87 101 94 108)(88 102 95 109)(89 103 96 110)(90 104 97 111)(91 105 98 112)
(1 92 8 85)(2 93 9 86)(3 94 10 87)(4 95 11 88)(5 96 12 89)(6 97 13 90)(7 98 14 91)(15 106 22 99)(16 107 23 100)(17 108 24 101)(18 109 25 102)(19 110 26 103)(20 111 27 104)(21 112 28 105)(29 57 36 64)(30 58 37 65)(31 59 38 66)(32 60 39 67)(33 61 40 68)(34 62 41 69)(35 63 42 70)(43 71 50 78)(44 72 51 79)(45 73 52 80)(46 74 53 81)(47 75 54 82)(48 76 55 83)(49 77 56 84)
(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 7)(2 6)(3 5)(8 14)(9 13)(10 12)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 56)(37 55)(38 54)(39 53)(40 52)(41 51)(42 50)(57 63)(58 62)(59 61)(64 70)(65 69)(66 68)(71 84)(72 83)(73 82)(74 81)(75 80)(76 79)(77 78)(85 105)(86 104)(87 103)(88 102)(89 101)(90 100)(91 99)(92 112)(93 111)(94 110)(95 109)(96 108)(97 107)(98 106)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 7A 7B 7C 8A 8B 8C 8D 14A ··· 14I 28A ··· 28R order 1 2 2 2 2 2 4 4 4 4 7 7 7 8 8 8 8 14 ··· 14 28 ··· 28 size 1 1 1 1 28 28 2 2 4 4 2 2 2 14 14 14 14 2 ··· 2 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D4 D7 SD16 D14 D14 C7⋊D4 C7⋊D4 Q8⋊D7 kernel C2×Q8⋊D7 C2×C7⋊C8 Q8⋊D7 C2×D28 Q8×C14 C28 C2×C14 C2×Q8 C14 C2×C4 Q8 C4 C22 C2 # reps 1 1 4 1 1 1 1 3 4 3 6 6 6 6

Matrix representation of C2×Q8⋊D7 in GL4(𝔽113) generated by

 112 0 0 0 0 112 0 0 0 0 112 0 0 0 0 112
,
 0 1 0 0 112 0 0 0 0 0 1 0 0 0 0 1
,
 13 13 0 0 13 100 0 0 0 0 112 0 0 0 0 112
,
 1 0 0 0 0 1 0 0 0 0 80 112 0 0 81 112
,
 1 0 0 0 0 112 0 0 0 0 9 79 0 0 9 104
G:=sub<GL(4,GF(113))| [112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,112,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[13,13,0,0,13,100,0,0,0,0,112,0,0,0,0,112],[1,0,0,0,0,1,0,0,0,0,80,81,0,0,112,112],[1,0,0,0,0,112,0,0,0,0,9,9,0,0,79,104] >;

C2×Q8⋊D7 in GAP, Magma, Sage, TeX

C_2\times Q_8\rtimes D_7
% in TeX

G:=Group("C2xQ8:D7");
// GroupNames label

G:=SmallGroup(224,136);
// by ID

G=gap.SmallGroup(224,136);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,218,86,579,159,69,6917]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=d^7=e^2=1,c^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,e*c*e=b^-1*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽