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G = C2xD7xD8order 448 = 26·7

Direct product of C2, D7 and D8

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

Aliases: C2xD7xD8, C56:3C23, D28:1C23, C28.1C24, D56:15C22, C14:2(C2xD8), C7:C8:6C23, C7:2(C22xD8), (C14xD8):7C2, (C2xC8):26D14, C4.39(D4xD7), C8:5(C22xD7), (C2xD56):19C2, (C2xD4):27D14, D4:D7:8C22, (C4xD7).26D4, C28.76(C2xD4), D4:1(C22xD7), (C7xD4):1C23, (C7xD8):9C22, (D4xD7):4C22, C4.1(C23xD7), (C2xC56):11C22, D14.63(C2xD4), (C8xD7):13C22, (C2xD28):32C22, (D4xC14):18C22, Dic7.11(C2xD4), (C4xD7).23C23, C22.135(D4xD7), (C2xC28).518C23, (C2xDic7).121D4, (C22xD7).110D4, C14.102(C22xD4), (D7xC2xC8):4C2, (C2xD4xD7):21C2, C2.75(C2xD4xD7), (C2xD4:D7):25C2, (C2xC7:C8):35C22, (C2xC14).391(C2xD4), (C2xC4xD7).255C22, (C2xC4).608(C22xD7), SmallGroup(448,1207)

Series: Derived Chief Lower central Upper central

C1C28 — C2xD7xD8
C1C7C14C28C4xD7C2xC4xD7C2xD4xD7 — C2xD7xD8
C7C14C28 — C2xD7xD8
C1C22C2xC4C2xD8

Generators and relations for C2xD7xD8
 G = < a,b,c,d,e | a2=b7=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 2212 in 338 conjugacy classes, 111 normal (23 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C7, C8, C8, C2xC4, C2xC4, D4, D4, C23, D7, D7, C14, C14, C14, C2xC8, C2xC8, D8, D8, C22xC4, C2xD4, C2xD4, C24, Dic7, C28, D14, D14, C2xC14, C2xC14, C22xC8, C2xD8, C2xD8, C22xD4, C7:C8, C56, C4xD7, D28, D28, C2xDic7, C7:D4, C2xC28, C7xD4, C7xD4, C22xD7, C22xD7, C22xC14, C22xD8, C8xD7, D56, C2xC7:C8, D4:D7, C2xC56, C7xD8, C2xC4xD7, C2xD28, D4xD7, D4xD7, C2xC7:D4, D4xC14, C23xD7, D7xC2xC8, C2xD56, D7xD8, C2xD4:D7, C14xD8, C2xD4xD7, C2xD7xD8
Quotients: C1, C2, C22, D4, C23, D7, D8, C2xD4, C24, D14, C2xD8, C22xD4, C22xD7, C22xD8, D4xD7, C23xD7, D7xD8, C2xD4xD7, C2xD7xD8

Smallest permutation representation of C2xD7xD8
On 112 points
Generators in S112
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 61)(10 62)(11 63)(12 64)(13 57)(14 58)(15 59)(16 60)(17 69)(18 70)(19 71)(20 72)(21 65)(22 66)(23 67)(24 68)(33 102)(34 103)(35 104)(36 97)(37 98)(38 99)(39 100)(40 101)(41 81)(42 82)(43 83)(44 84)(45 85)(46 86)(47 87)(48 88)(49 89)(50 90)(51 91)(52 92)(53 93)(54 94)(55 95)(56 96)(73 107)(74 108)(75 109)(76 110)(77 111)(78 112)(79 105)(80 106)
(1 105 95 23 12 43 40)(2 106 96 24 13 44 33)(3 107 89 17 14 45 34)(4 108 90 18 15 46 35)(5 109 91 19 16 47 36)(6 110 92 20 9 48 37)(7 111 93 21 10 41 38)(8 112 94 22 11 42 39)(25 80 56 68 57 84 102)(26 73 49 69 58 85 103)(27 74 50 70 59 86 104)(28 75 51 71 60 87 97)(29 76 52 72 61 88 98)(30 77 53 65 62 81 99)(31 78 54 66 63 82 100)(32 79 55 67 64 83 101)
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 92)(10 93)(11 94)(12 95)(13 96)(14 89)(15 90)(16 91)(25 102)(26 103)(27 104)(28 97)(29 98)(30 99)(31 100)(32 101)(41 111)(42 112)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 58)(50 59)(51 60)(52 61)(53 62)(54 63)(55 64)(56 57)(73 85)(74 86)(75 87)(76 88)(77 81)(78 82)(79 83)(80 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 4)(2 3)(5 8)(6 7)(9 10)(11 16)(12 15)(13 14)(17 24)(18 23)(19 22)(20 21)(25 26)(27 32)(28 31)(29 30)(33 34)(35 40)(36 39)(37 38)(41 48)(42 47)(43 46)(44 45)(49 56)(50 55)(51 54)(52 53)(57 58)(59 64)(60 63)(61 62)(65 72)(66 71)(67 70)(68 69)(73 80)(74 79)(75 78)(76 77)(81 88)(82 87)(83 86)(84 85)(89 96)(90 95)(91 94)(92 93)(97 100)(98 99)(101 104)(102 103)(105 108)(106 107)(109 112)(110 111)

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

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

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

70 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D7A7B7C8A8B8C8D8E8F8G8H14A···14I14J···14U28A···28F56A···56L
order122222222222222244447778888888814···1414···1428···2856···56
size111144447777282828282214142222222141414142···28···84···44···4

70 irreducible representations

dim111111122222222444
type++++++++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D7D8D14D14D14D4xD7D4xD7D7xD8
kernelC2xD7xD8D7xC2xC8C2xD56D7xD8C2xD4:D7C14xD8C2xD4xD7C4xD7C2xDic7C22xD7C2xD8D14C2xC8D8C2xD4C4C22C2
# reps11182122113831263312

Matrix representation of C2xD7xD8 in GL5(F113)

1120000
01000
00100
0001120
0000112
,
10000
088100
011110400
00010
00001
,
10000
0347900
0247900
00010
00001
,
1120000
01000
00100
000051
0003162
,
10000
01000
00100
000062
000310

G:=sub<GL(5,GF(113))| [112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,112,0,0,0,0,0,112],[1,0,0,0,0,0,88,111,0,0,0,1,104,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,34,24,0,0,0,79,79,0,0,0,0,0,1,0,0,0,0,0,1],[112,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,51,62],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,31,0,0,0,62,0] >;

C2xD7xD8 in GAP, Magma, Sage, TeX

C_2\times D_7\times D_8
% in TeX

G:=Group("C2xD7xD8");
// GroupNames label

G:=SmallGroup(448,1207);
// by ID

G=gap.SmallGroup(448,1207);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,185,438,235,102,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^7=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=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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