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G = C2xQ8:D7order 224 = 25·7

Direct product of C2 and Q8:D7

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2xQ8:D7, Q8:3D14, C14:3SD16, C28.18D4, C28.14C23, D28.9C22, C7:C8:9C22, (C2xQ8):1D7, C7:4(C2xSD16), (Q8xC14):1C2, (C2xD28).8C2, (C2xC4).53D14, (C2xC14).41D4, C14.53(C2xD4), C4.8(C7:D4), (C7xQ8):3C22, C4.14(C22xD7), (C2xC28).36C22, C22.23(C7:D4), (C2xC7:C8):6C2, C2.17(C2xC7:D4), SmallGroup(224,136)

Series: Derived Chief Lower central Upper central

C1C28 — C2xQ8:D7
C1C7C14C28D28C2xD28 — C2xQ8:D7
C7C14C28 — C2xQ8:D7
C1C22C2xC4C2xQ8

Generators and relations for C2xQ8: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, C2xC4, C2xC4, D4, Q8, Q8, C23, D7, C14, C14, C2xC8, SD16, C2xD4, C2xQ8, C28, C28, D14, C2xC14, C2xSD16, C7:C8, D28, D28, C2xC28, C2xC28, C7xQ8, C7xQ8, C22xD7, C2xC7:C8, Q8:D7, C2xD28, Q8xC14, C2xQ8:D7
Quotients: C1, C2, C22, D4, C23, D7, SD16, C2xD4, D14, C2xSD16, C7:D4, C22xD7, Q8:D7, C2xC7:D4, C2xQ8:D7

Smallest permutation representation of C2xQ8: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)]])

C2xQ8:D7 is a maximal subgroup of
D28.6D4  Dic7:7SD16  Q8:2D28  D14:2SD16  D28:4D4  C7:(C8:D4)  Q8:D7:C4  Dic7:SD16  D28.12D4  C42.56D14  Q8:D28  Q8.1D28  D28.36D4  D28.37D4  C7:C8:24D4  C7:C8:6D4  D28.23D4  C42.64D14  C42.214D14  C28:5SD16  C28:6SD16  C42.80D14  Dic7:5SD16  D14:6SD16  C56:15D4  C56:9D4  (C2xQ16):D7  D28.17D4  C56.37D4  C56.28D4  M4(2).15D14  (C7xQ8):13D4  (C7xD4):14D4  C2xD7xSD16  C56.C23  D28.34C23
C2xQ8:D7 is a maximal quotient of
C4:C4.228D14  C28.48SD16  Q8:D28  C22:Q8.D7  D28.36D4  C7:C8:24D4  C28.SD16  C28.Q16  C28:5SD16  D28:5Q8  C28:6SD16  C28.D8  (C7xQ8):13D4

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C8A8B8C8D14A···14I28A···28R
order1222224444777888814···1428···28
size111128282244222141414142···24···4

44 irreducible representations

dim11111222222224
type+++++++++++
imageC1C2C2C2C2D4D4D7SD16D14D14C7:D4C7:D4Q8:D7
kernelC2xQ8:D7C2xC7:C8Q8:D7C2xD28Q8xC14C28C2xC14C2xQ8C14C2xC4Q8C4C22C2
# reps11411113436666

Matrix representation of C2xQ8:D7 in GL4(F113) generated by

112000
011200
001120
000112
,
0100
112000
0010
0001
,
131300
1310000
001120
000112
,
1000
0100
0080112
0081112
,
1000
011200
00979
009104
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] >;

C2xQ8: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

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