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## G = C2×C23⋊D14order 448 = 26·7

### Direct product of C2 and C23⋊D14

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C14 — C2×C23⋊D14
 Chief series C1 — C7 — C14 — C2×C14 — C22×D7 — C23×D7 — D7×C24 — C2×C23⋊D14
 Lower central C7 — C2×C14 — C2×C23⋊D14
 Upper central C1 — C23 — C22×D4

Generators and relations for C2×C23⋊D14
G = < a,b,c,d,e,f | a2=b2=c2=d2=e14=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 3604 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C7, C2×C4, C2×C4, D4, C23, C23, C23, D7, C14, C14, C14, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, Dic7, C28, D14, D14, C2×C14, C2×C14, C2×C14, C2×C22⋊C4, C22≀C2, C22×D4, C22×D4, C25, C2×Dic7, C2×Dic7, C7⋊D4, C2×C28, C2×C28, C7×D4, C22×D7, C22×D7, C22×C14, C22×C14, C22×C14, C2×C22≀C2, D14⋊C4, C23.D7, C22×Dic7, C2×C7⋊D4, C2×C7⋊D4, C22×C28, D4×C14, D4×C14, C23×D7, C23×D7, C23×C14, C2×D14⋊C4, C23⋊D14, C2×C23.D7, C22×C7⋊D4, D4×C2×C14, D7×C24, C2×C23⋊D14
Quotients: C1, C2, C22, D4, C23, D7, C2×D4, C24, D14, C22≀C2, C22×D4, C7⋊D4, C22×D7, C2×C22≀C2, D4×D7, C2×C7⋊D4, C23×D7, C23⋊D14, C2×D4×D7, C22×C7⋊D4, C2×C23⋊D14

Smallest permutation representation of C2×C23⋊D14
On 112 points
Generators in S112
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 111)(16 112)(17 99)(18 100)(19 101)(20 102)(21 103)(22 104)(23 105)(24 106)(25 107)(26 108)(27 109)(28 110)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 71)(39 72)(40 73)(41 74)(42 75)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)(49 96)(50 97)(51 98)(52 85)(53 86)(54 87)(55 88)(56 89)
(1 20)(2 89)(3 22)(4 91)(5 24)(6 93)(7 26)(8 95)(9 28)(10 97)(11 16)(12 85)(13 18)(14 87)(15 76)(17 78)(19 80)(21 82)(23 84)(25 72)(27 74)(29 111)(30 51)(31 99)(32 53)(33 101)(34 55)(35 103)(36 43)(37 105)(38 45)(39 107)(40 47)(41 109)(42 49)(44 66)(46 68)(48 70)(50 58)(52 60)(54 62)(56 64)(57 110)(59 112)(61 100)(63 102)(65 104)(67 106)(69 108)(71 92)(73 94)(75 96)(77 98)(79 86)(81 88)(83 90)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 79)(7 80)(8 81)(9 82)(10 83)(11 84)(12 71)(13 72)(14 73)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)(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 81)(2 82)(3 83)(4 84)(5 71)(6 72)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 79)(14 80)(15 97)(16 98)(17 85)(18 86)(19 87)(20 88)(21 89)(22 90)(23 91)(24 92)(25 93)(26 94)(27 95)(28 96)(29 58)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 57)(43 104)(44 105)(45 106)(46 107)(47 108)(48 109)(49 110)(50 111)(51 112)(52 99)(53 100)(54 101)(55 102)(56 103)
(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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 42)(14 41)(15 106)(16 105)(17 104)(18 103)(19 102)(20 101)(21 100)(22 99)(23 112)(24 111)(25 110)(26 109)(27 108)(28 107)(43 85)(44 98)(45 97)(46 96)(47 95)(48 94)(49 93)(50 92)(51 91)(52 90)(53 89)(54 88)(55 87)(56 86)(57 79)(58 78)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 84)(67 83)(68 82)(69 81)(70 80)

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

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

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

88 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N ··· 2U 4A 4B 4C 4D 4E 4F 7A 7B 7C 14A ··· 14U 14V ··· 14AS 28A ··· 28L order 1 2 ··· 2 2 2 2 2 2 2 2 ··· 2 4 4 4 4 4 4 7 7 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 ··· 1 2 2 2 2 4 4 14 ··· 14 4 4 28 28 28 28 2 2 2 2 ··· 2 4 ··· 4 4 ··· 4

88 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 D4 D4 D7 D14 D14 D14 C7⋊D4 D4×D7 kernel C2×C23⋊D14 C2×D14⋊C4 C23⋊D14 C2×C23.D7 C22×C7⋊D4 D4×C2×C14 D7×C24 C22×D7 C22×C14 C22×D4 C22×C4 C2×D4 C24 C23 C22 # reps 1 2 8 1 2 1 1 8 4 3 3 12 6 24 12

Matrix representation of C2×C23⋊D14 in GL5(𝔽29)

 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 28
,
 28 0 0 0 0 0 5 16 0 0 0 13 24 0 0 0 0 0 0 1 0 0 0 1 0
,
 1 0 0 0 0 0 28 0 0 0 0 0 28 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 28 0 0 0 0 0 28
,
 28 0 0 0 0 0 3 21 0 0 0 8 8 0 0 0 0 0 28 0 0 0 0 0 1
,
 28 0 0 0 0 0 28 0 0 0 0 26 1 0 0 0 0 0 1 0 0 0 0 0 28

G:=sub<GL(5,GF(29))| [28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,5,13,0,0,0,16,24,0,0,0,0,0,0,1,0,0,0,1,0],[1,0,0,0,0,0,28,0,0,0,0,0,28,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28,0,0,0,0,0,28],[28,0,0,0,0,0,3,8,0,0,0,21,8,0,0,0,0,0,28,0,0,0,0,0,1],[28,0,0,0,0,0,28,26,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,28] >;

C2×C23⋊D14 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes D_{14}
% in TeX

G:=Group("C2xC2^3:D14");
// GroupNames label

G:=SmallGroup(448,1252);
// by ID

G=gap.SmallGroup(448,1252);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,297,18822]);
// Polycyclic

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

׿
×
𝔽