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## G = C2×C23⋊Dic7order 448 = 26·7

### Direct product of C2 and C23⋊Dic7

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C23⋊Dic7
 Chief series C1 — C7 — C14 — C2×C14 — C22×C14 — C23.D7 — C2×C23.D7 — C2×C23⋊Dic7
 Lower central C7 — C14 — C2×C14 — C2×C23⋊Dic7
 Upper central C1 — C22 — C24 — C22×D4

Generators and relations for C2×C23⋊Dic7
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=1, f2=e7, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 788 in 210 conjugacy classes, 71 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, Dic7, C28, C2×C14, C2×C14, C2×C14, C23⋊C4, C2×C22⋊C4, C22×D4, C2×Dic7, C2×C28, C2×C28, C7×D4, C22×C14, C22×C14, C22×C14, C2×C23⋊C4, C23.D7, C23.D7, C22×Dic7, C22×C28, D4×C14, D4×C14, C23×C14, C23⋊Dic7, C2×C23.D7, D4×C2×C14, C2×C23⋊Dic7
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, D7, C22⋊C4, C22×C4, C2×D4, Dic7, D14, C23⋊C4, C2×C22⋊C4, C2×Dic7, C7⋊D4, C22×D7, C2×C23⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, C23⋊Dic7, C2×C23.D7, C2×C23⋊Dic7

Smallest permutation representation of C2×C23⋊Dic7
On 112 points
Generators in S112
(1 26)(2 27)(3 28)(4 22)(5 23)(6 24)(7 25)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 43)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 56)(29 39)(30 40)(31 41)(32 42)(33 36)(34 37)(35 38)(57 109)(58 110)(59 111)(60 112)(61 99)(62 100)(63 101)(64 102)(65 103)(66 104)(67 105)(68 106)(69 107)(70 108)(71 94)(72 95)(73 96)(74 97)(75 98)(76 85)(77 86)(78 87)(79 88)(80 89)(81 90)(82 91)(83 92)(84 93)
(1 79)(2 73)(3 81)(4 75)(5 83)(6 77)(7 71)(8 72)(9 80)(10 74)(11 82)(12 76)(13 84)(14 78)(15 59)(16 67)(17 61)(18 69)(19 63)(20 57)(21 65)(22 98)(23 92)(24 86)(25 94)(26 88)(27 96)(28 90)(29 100)(30 108)(31 102)(32 110)(33 104)(34 112)(35 106)(36 66)(37 60)(38 68)(39 62)(40 70)(41 64)(42 58)(43 87)(44 95)(45 89)(46 97)(47 91)(48 85)(49 93)(50 111)(51 105)(52 99)(53 107)(54 101)(55 109)(56 103)
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(22 53)(23 54)(24 55)(25 56)(26 50)(27 51)(28 52)(29 47)(30 48)(31 49)(32 43)(33 44)(34 45)(35 46)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)(63 83)(64 84)(65 71)(66 72)(67 73)(68 74)(69 75)(70 76)(85 108)(86 109)(87 110)(88 111)(89 112)(90 99)(91 100)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 36)(16 37)(17 38)(18 39)(19 40)(20 41)(21 42)(22 47)(23 48)(24 49)(25 43)(26 44)(27 45)(28 46)(29 53)(30 54)(31 55)(32 56)(33 50)(34 51)(35 52)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(71 78)(72 79)(73 80)(74 81)(75 82)(76 83)(77 84)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(99 106)(100 107)(101 108)(102 109)(103 110)(104 111)(105 112)
(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)
(2 7)(3 6)(4 5)(9 14)(10 13)(11 12)(15 36)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 23)(24 28)(25 27)(29 54)(30 53)(31 52)(32 51)(33 50)(34 56)(35 55)(43 45)(46 49)(47 48)(57 74 64 81)(58 73 65 80)(59 72 66 79)(60 71 67 78)(61 84 68 77)(62 83 69 76)(63 82 70 75)(85 100 92 107)(86 99 93 106)(87 112 94 105)(88 111 95 104)(89 110 96 103)(90 109 97 102)(91 108 98 101)

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

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

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

82 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 2K 4A 4B 4C ··· 4J 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 2 2 2 ··· 2 2 2 4 4 4 ··· 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 1 1 2 ··· 2 4 4 4 4 28 ··· 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

82 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + - - + - + + image C1 C2 C2 C2 C4 C4 C4 D4 D7 Dic7 Dic7 D14 Dic7 D14 C7⋊D4 C23⋊C4 C23⋊Dic7 kernel C2×C23⋊Dic7 C23⋊Dic7 C2×C23.D7 D4×C2×C14 C22×C28 D4×C14 C23×C14 C22×C14 C22×D4 C22×C4 C2×D4 C2×D4 C24 C24 C23 C14 C2 # reps 1 4 2 1 2 4 2 4 3 3 6 6 3 3 24 2 12

Matrix representation of C2×C23⋊Dic7 in GL6(𝔽29)

 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 0 0 0 0 0 0 28 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 2 0 0 0 0 1 0 2 0 0 0 0 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 5 16 0 0 0 0 13 24 0 0 0 0 0 0 5 16 0 0 0 0 13 24
,
 1 0 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 28 0 0 0 0 0 0 28
,
 28 0 0 0 0 0 0 28 0 0 0 0 0 0 3 28 0 0 0 0 1 0 0 0 0 0 26 1 26 1 0 0 28 0 28 0
,
 12 0 0 0 0 0 12 17 0 0 0 0 0 0 1 0 0 0 0 0 3 28 0 0 0 0 26 21 24 13 0 0 12 3 27 5

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,27,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,2,0,28,0,0,0,0,2,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,5,13,0,0,0,0,16,24,0,0,0,0,0,0,5,13,0,0,0,0,16,24],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,3,1,26,28,0,0,28,0,1,0,0,0,0,0,26,28,0,0,0,0,1,0],[12,12,0,0,0,0,0,17,0,0,0,0,0,0,1,3,26,12,0,0,0,28,21,3,0,0,0,0,24,27,0,0,0,0,13,5] >;

C2×C23⋊Dic7 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes {\rm Dic}_7
% in TeX

G:=Group("C2xC2^3:Dic7");
// GroupNames label

G:=SmallGroup(448,753);
// by ID

G=gap.SmallGroup(448,753);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,56,422,297,1684,18822]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^14=1,f^2=e^7,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^-1>;
// generators/relations

׿
×
𝔽