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## G = C2×D7×C4○D4order 448 = 26·7

### Direct product of C2, D7 and C4○D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C14 — C2×D7×C4○D4
 Chief series C1 — C7 — C14 — D14 — C22×D7 — C23×D7 — D7×C22×C4 — C2×D7×C4○D4
 Lower central C7 — C14 — C2×D7×C4○D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for C2×D7×C4○D4
G = < a,b,c,d,e,f | a2=b7=c2=d4=f2=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, cbc=b-1, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=d2e >

Subgroups: 3476 in 890 conjugacy classes, 455 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C7, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, D7, D7, C14, C14, C14, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, Dic14, C4×D7, D28, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C7×Q8, C22×D7, C22×D7, C22×D7, C22×C14, C22×C4○D4, C2×Dic14, C2×C4×D7, C2×C4×D7, C2×D28, C4○D28, D4×D7, D42D7, Q8×D7, Q82D7, C22×Dic7, C2×C7⋊D4, C22×C28, D4×C14, Q8×C14, C7×C4○D4, C23×D7, D7×C22×C4, C2×C4○D28, C2×D4×D7, C2×D42D7, C2×Q8×D7, C2×Q82D7, D7×C4○D4, C14×C4○D4, C2×D7×C4○D4
Quotients: C1, C2, C22, C23, D7, C4○D4, C24, D14, C2×C4○D4, C25, C22×D7, C22×C4○D4, C23×D7, D7×C4○D4, D7×C24, C2×D7×C4○D4

Smallest permutation representation of C2×D7×C4○D4
On 112 points
Generators in S112
(1 69)(2 70)(3 64)(4 65)(5 66)(6 67)(7 68)(8 57)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(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 92)(30 93)(31 94)(32 95)(33 96)(34 97)(35 98)(36 85)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(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 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 80)(16 79)(17 78)(18 84)(19 83)(20 82)(21 81)(22 73)(23 72)(24 71)(25 77)(26 76)(27 75)(28 74)(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)(43 108)(44 107)(45 106)(46 112)(47 111)(48 110)(49 109)(50 101)(51 100)(52 99)(53 105)(54 104)(55 103)(56 102)
(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 20 13 27)(2 21 14 28)(3 15 8 22)(4 16 9 23)(5 17 10 24)(6 18 11 25)(7 19 12 26)(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)(85 106 92 99)(86 107 93 100)(87 108 94 101)(88 109 95 102)(89 110 96 103)(90 111 97 104)(91 112 98 105)
(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 78)(58 79)(59 80)(60 81)(61 82)(62 83)(63 84)(64 71)(65 72)(66 73)(67 74)(68 75)(69 76)(70 77)(85 106)(86 107)(87 108)(88 109)(89 110)(90 111)(91 112)(92 99)(93 100)(94 101)(95 102)(96 103)(97 104)(98 105)

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

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

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

100 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 2L 2M 2N ··· 2S 4A 4B 4C 4D 4E ··· 4J 4K 4L 4M 4N 4O ··· 4T 7A 7B 7C 14A ··· 14I 14J ··· 14AA 28A ··· 28L 28M ··· 28AD order 1 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 4 4 4 4 4 ··· 4 4 4 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 1 1 1 1 2 ··· 2 7 7 7 7 14 ··· 14 2 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

100 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D7 C4○D4 D14 D14 D14 D14 D7×C4○D4 kernel C2×D7×C4○D4 D7×C22×C4 C2×C4○D28 C2×D4×D7 C2×D4⋊2D7 C2×Q8×D7 C2×Q8⋊2D7 D7×C4○D4 C14×C4○D4 C2×C4○D4 D14 C22×C4 C2×D4 C2×Q8 C4○D4 C2 # reps 1 3 3 3 3 1 1 16 1 3 8 9 9 3 24 12

Matrix representation of C2×D7×C4○D4 in GL4(𝔽29) generated by

 28 0 0 0 0 28 0 0 0 0 28 0 0 0 0 28
,
 11 1 0 0 13 25 0 0 0 0 1 0 0 0 0 1
,
 4 1 0 0 14 25 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 17 0 0 0 0 17
,
 1 0 0 0 0 1 0 0 0 0 0 28 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(29))| [28,0,0,0,0,28,0,0,0,0,28,0,0,0,0,28],[11,13,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[4,14,0,0,1,25,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,17,0,0,0,0,17],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,28,0],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C2×D7×C4○D4 in GAP, Magma, Sage, TeX

C_2\times D_7\times C_4\circ D_4
% in TeX

G:=Group("C2xD7xC4oD4");
// GroupNames label

G:=SmallGroup(448,1375);
// by ID

G=gap.SmallGroup(448,1375);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,136,438,18822]);
// Polycyclic

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

׿
×
𝔽