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G = D7×C22.D4order 448 = 26·7

Direct product of D7 and C22.D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — D7×C22.D4
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C22×C4 — D7×C22.D4
 Lower central C7 — C2×C14 — D7×C22.D4
 Upper central C1 — C22 — C22.D4

Generators and relations for D7×C22.D4
G = < a,b,c,d,e,f | a7=b2=c2=d2=e4=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=de-1 >

Subgroups: 1772 in 342 conjugacy classes, 111 normal (39 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, C23, C23, D7, D7, C14, C14, C14, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C2×C4⋊C4, C22.D4, C22.D4, C23×C4, C22×D4, C4×D7, D28, C2×Dic7, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×D7, C22×C14, C2×C22.D4, Dic7⋊C4, C4⋊Dic7, D14⋊C4, C23.D7, C23.D7, C7×C22⋊C4, C7×C22⋊C4, C7×C4⋊C4, C2×C4×D7, C2×C4×D7, C2×C4×D7, C2×D28, D4×D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, C23×D7, D7×C22⋊C4, D7×C22⋊C4, D14.D4, C22.D28, D7×C4⋊C4, D14.5D4, C23.23D14, C23.18D14, C7×C22.D4, D7×C22×C4, C2×D4×D7, D7×C22.D4
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C4○D4, C24, D14, C22.D4, C22×D4, C2×C4○D4, C22×D7, C2×C22.D4, D4×D7, C23×D7, C2×D4×D7, D7×C4○D4, D7×C22.D4

Smallest permutation representation of D7×C22.D4
On 112 points
Generators in S112
(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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)(57 80)(58 79)(59 78)(60 84)(61 83)(62 82)(63 81)(64 73)(65 72)(66 71)(67 77)(68 76)(69 75)(70 74)(85 108)(86 107)(87 106)(88 112)(89 111)(90 110)(91 109)(92 101)(93 100)(94 99)(95 105)(96 104)(97 103)(98 102)
(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 20)(2 21)(3 15)(4 16)(5 17)(6 18)(7 19)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)(29 43)(30 44)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 51)(38 52)(39 53)(40 54)(41 55)(42 56)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(67 81)(68 82)(69 83)(70 84)(85 99)(86 100)(87 101)(88 102)(89 103)(90 104)(91 105)(92 106)(93 107)(94 108)(95 109)(96 110)(97 111)(98 112)
(1 104 27 97)(2 105 28 98)(3 99 22 92)(4 100 23 93)(5 101 24 94)(6 102 25 95)(7 103 26 96)(8 106 15 85)(9 107 16 86)(10 108 17 87)(11 109 18 88)(12 110 19 89)(13 111 20 90)(14 112 21 91)(29 71 50 64)(30 72 51 65)(31 73 52 66)(32 74 53 67)(33 75 54 68)(34 76 55 69)(35 77 56 70)(36 78 43 57)(37 79 44 58)(38 80 45 59)(39 81 46 60)(40 82 47 61)(41 83 48 62)(42 84 49 63)
(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 53)(26 54)(27 55)(28 56)(57 92)(58 93)(59 94)(60 95)(61 96)(62 97)(63 98)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 106)(72 107)(73 108)(74 109)(75 110)(76 111)(77 112)(78 99)(79 100)(80 101)(81 102)(82 103)(83 104)(84 105)

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

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 7A 7B 7C 14A ··· 14I 14J ··· 14O 14P 14Q 14R 28A ··· 28L 28M ··· 28U order 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 14 14 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 2 4 7 7 7 7 14 14 28 2 2 2 2 4 4 4 14 14 14 14 28 28 28 2 2 2 2 ··· 2 4 ··· 4 8 8 8 4 ··· 4 8 ··· 8

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D7 C4○D4 D14 D14 D14 D14 D4×D7 D7×C4○D4 kernel D7×C22.D4 D7×C22⋊C4 D14.D4 C22.D28 D7×C4⋊C4 D14.5D4 C23.23D14 C23.18D14 C7×C22.D4 D7×C22×C4 C2×D4×D7 C22×D7 C22.D4 D14 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C22 C2 # reps 1 3 2 1 2 2 1 1 1 1 1 4 3 8 9 6 3 3 6 12

Matrix representation of D7×C22.D4 in GL6(𝔽29)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 28 18 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 28 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 27 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 17 17 0 0 0 0 24 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 22 11 0 0 0 0 6 7
,
 28 28 0 0 0 0 0 1 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 1 12 0 0 0 0 0 28

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,27,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[17,24,0,0,0,0,17,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,22,6,0,0,0,0,11,7],[28,0,0,0,0,0,28,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,12,28] >;

D7×C22.D4 in GAP, Magma, Sage, TeX

D_7\times C_2^2.D_4
% in TeX

G:=Group("D7xC2^2.D4");
// GroupNames label

G:=SmallGroup(448,1105);
// by ID

G=gap.SmallGroup(448,1105);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,100,346,297,18822]);
// Polycyclic

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

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