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G = C2xQ8xD7order 224 = 25·7

Direct product of C2, Q8 and D7

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

Aliases: C2xQ8xD7, C14.8C24, C28.22C23, Dic14:9C22, D14.10C23, Dic7.5C23, C14:2(C2xQ8), C7:2(C22xQ8), (Q8xC14):5C2, (C2xC4).61D14, (C7xQ8):5C22, C2.9(C23xD7), C4.22(C22xD7), (C2xDic14):13C2, (C2xC28).46C22, (C2xC14).66C23, (C4xD7).13C22, C22.31(C22xD7), (C2xDic7).44C22, (C22xD7).36C22, (C2xC4xD7).6C2, SmallGroup(224,181)

Series: Derived Chief Lower central Upper central

C1C14 — C2xQ8xD7
C1C7C14D14C22xD7C2xC4xD7 — C2xQ8xD7
C7C14 — C2xQ8xD7
C1C22C2xQ8

Generators and relations for C2xQ8xD7
 G = < a,b,c,d,e | a2=b4=d7=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 510 in 156 conjugacy classes, 97 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C2xC4, C2xC4, Q8, Q8, C23, D7, C14, C14, C22xC4, C2xQ8, C2xQ8, Dic7, C28, D14, C2xC14, C22xQ8, Dic14, C4xD7, C2xDic7, C2xC28, C7xQ8, C22xD7, C2xDic14, C2xC4xD7, Q8xD7, Q8xC14, C2xQ8xD7
Quotients: C1, C2, C22, Q8, C23, D7, C2xQ8, C24, D14, C22xQ8, C22xD7, Q8xD7, C23xD7, C2xQ8xD7

Smallest permutation representation of C2xQ8xD7
On 112 points
Generators in S112
(1 62)(2 63)(3 57)(4 58)(5 59)(6 60)(7 61)(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 41 13 34)(2 42 14 35)(3 36 8 29)(4 37 9 30)(5 38 10 31)(6 39 11 32)(7 40 12 33)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)(57 92 64 85)(58 93 65 86)(59 94 66 87)(60 95 67 88)(61 96 68 89)(62 97 69 90)(63 98 70 91)(71 106 78 99)(72 107 79 100)(73 108 80 101)(74 109 81 102)(75 110 82 103)(76 111 83 104)(77 112 84 105)
(1 83 13 76)(2 84 14 77)(3 78 8 71)(4 79 9 72)(5 80 10 73)(6 81 11 74)(7 82 12 75)(15 57 22 64)(16 58 23 65)(17 59 24 66)(18 60 25 67)(19 61 26 68)(20 62 27 69)(21 63 28 70)(29 99 36 106)(30 100 37 107)(31 101 38 108)(32 102 39 109)(33 103 40 110)(34 104 41 111)(35 105 42 112)(43 92 50 85)(44 93 51 86)(45 94 52 87)(46 95 53 88)(47 96 54 89)(48 97 55 90)(49 98 56 91)
(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 61)(2 60)(3 59)(4 58)(5 57)(6 63)(7 62)(8 66)(9 65)(10 64)(11 70)(12 69)(13 68)(14 67)(15 73)(16 72)(17 71)(18 77)(19 76)(20 75)(21 74)(22 80)(23 79)(24 78)(25 84)(26 83)(27 82)(28 81)(29 87)(30 86)(31 85)(32 91)(33 90)(34 89)(35 88)(36 94)(37 93)(38 92)(39 98)(40 97)(41 96)(42 95)(43 101)(44 100)(45 99)(46 105)(47 104)(48 103)(49 102)(50 108)(51 107)(52 106)(53 112)(54 111)(55 110)(56 109)

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

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

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

C2xQ8xD7 is a maximal subgroup of
(Q8xD7):C4  Q8:2D28  D14:4Q16  Dic14:7D4  D14:5Q16  C42.125D14  Q8:5D28  C14.162- 1+4  Dic14:21D4  Dic14:22D4  C42.141D14  Dic14:10D4  C42.171D14  D28:8Q8  C14.1072- 1+4
C2xQ8xD7 is a maximal quotient of
C14.102+ 1+4  Dic14:10Q8  C42.232D14  D28:10Q8  (Q8xDic7):C2  C14.752- 1+4  Dic14:21D4  C14.512+ 1+4  C14.1182+ 1+4  C14.522+ 1+4  Dic14:7Q8  C42.236D14  C42.148D14  D28:7Q8  Dic14:8Q8  Dic14:9Q8  D28:8Q8  C42.241D14  C42.174D14  D28:9Q8

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4F4G···4L7A7B7C14A···14I28A···28R
order122222224···44···477714···1428···28
size111177772···214···142222···24···4

50 irreducible representations

dim1111122224
type+++++-+++-
imageC1C2C2C2C2Q8D7D14D14Q8xD7
kernelC2xQ8xD7C2xDic14C2xC4xD7Q8xD7Q8xC14D14C2xQ8C2xC4Q8C2
# reps13381439126

Matrix representation of C2xQ8xD7 in GL4(F29) generated by

28000
02800
00280
00028
,
28000
02800
0001
00280
,
28000
02800
00120
00017
,
0100
281800
0010
0001
,
02800
28000
0010
0001
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,0,28,0,0,1,0],[28,0,0,0,0,28,0,0,0,0,12,0,0,0,0,17],[0,28,0,0,1,18,0,0,0,0,1,0,0,0,0,1],[0,28,0,0,28,0,0,0,0,0,1,0,0,0,0,1] >;

C2xQ8xD7 in GAP, Magma, Sage, TeX

C_2\times Q_8\times D_7
% in TeX

G:=Group("C2xQ8xD7");
// GroupNames label

G:=SmallGroup(224,181);
// by ID

G=gap.SmallGroup(224,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,86,159,69,6917]);
// 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=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations

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