Copied to
clipboard

?

G = C2×C23⋊D14  order 448 = 26·7

Direct product of C2 and C23⋊D14

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

Aliases: C2×C23⋊D14, C24⋊5D14, C14⋊2C22≀C2, D14⋊12(C2×D4), (C2×D4)⋊35D14, (C2×C28)⋊9C23, (D7×C24)⋊3C2, (C22×D4)⋊5D7, (C22×D7)⋊15D4, (C22×C14)⋊11D4, (C22×C4)⋊26D14, C23⋊6(C7⋊D4), C23⋊3(C22×D7), D14⋊C4⋊71C22, (D4×C14)⋊55C22, (C22×C14)⋊5C23, (C2×Dic7)⋊3C23, C22.146(D4×D7), (C2×C14).294C24, (C22×C28)⋊43C22, (C23×C14)⋊12C22, C14.141(C22×D4), (C23×D7)⋊21C22, C23.D7⋊60C22, C22.307(C23×D7), (C22×Dic7)⋊32C22, (C22×D7).238C23, C7⋊3(C2×C22≀C2), (D4×C2×C14)⋊15C2, (C2×C14)⋊7(C2×D4), C2.101(C2×D4×D7), (C2×C4)⋊4(C22×D7), C22⋊3(C2×C7⋊D4), (C2×D14⋊C4)⋊41C2, (C2×C7⋊D4)⋊43C22, (C22×C7⋊D4)⋊12C2, (C2×C23.D7)⋊27C2, C2.14(C22×C7⋊D4), SmallGroup(448,1252)

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

Subgroups: 3604 in 662 conjugacy classes, 143 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×14], C4 [×6], C22, C22 [×10], C22 [×86], C7, C2×C4 [×2], C2×C4 [×10], D4 [×24], C23, C23 [×8], C23 [×86], D7 [×8], C14, C14 [×6], C14 [×6], C22⋊C4 [×12], C22×C4, C22×C4 [×2], C2×D4 [×4], C2×D4 [×20], C24 [×2], C24 [×18], Dic7 [×4], C28 [×2], D14 [×8], D14 [×56], C2×C14, C2×C14 [×10], C2×C14 [×22], C2×C22⋊C4 [×3], C22≀C2 [×8], C22×D4, C22×D4 [×2], C25, C2×Dic7 [×4], C2×Dic7 [×4], C7⋊D4 [×16], C2×C28 [×2], C2×C28 [×2], C7×D4 [×8], C22×D7 [×12], C22×D7 [×64], C22×C14, C22×C14 [×8], C22×C14 [×10], C2×C22≀C2, D14⋊C4 [×8], C23.D7 [×4], C22×Dic7 [×2], C2×C7⋊D4 [×8], C2×C7⋊D4 [×8], C22×C28, D4×C14 [×4], D4×C14 [×4], C23×D7 [×6], C23×D7 [×12], C23×C14 [×2], C2×D14⋊C4 [×2], C23⋊D14 [×8], C2×C23.D7, C22×C7⋊D4 [×2], D4×C2×C14, D7×C24, C2×C23⋊D14

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], D7, C2×D4 [×18], C24, D14 [×7], C22≀C2 [×4], C22×D4 [×3], C7⋊D4 [×4], C22×D7 [×7], C2×C22≀C2, D4×D7 [×4], C2×C7⋊D4 [×6], C23×D7, C23⋊D14 [×4], C2×D4×D7 [×2], C22×C7⋊D4, C2×C23⋊D14

Generators and relations
 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 >

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

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

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

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

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

Matrix representation ►G ⊆ GL5(𝔽29)

280000
028000
002800
000280
000028
,
280000
051600
0132400
00001
00010
,
10000
028000
002800
00010
00001
,
10000
01000
00100
000280
000028
,
280000
032100
08800
000280
00001
,
280000
028000
026100
00010
000028

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] >;
 

88 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N···2U4A4B4C4D4E4F7A7B7C14A···14U14V···14AS28A···28L
order12···22222222···244444477714···1414···1428···28
size11···122224414···1444282828282222···24···44···4

88 irreducible representations

dim111111122222224
type++++++++++++++
imageC1C2C2C2C2C2C2D4D4D7D14D14D14C7⋊D4D4×D7
kernelC2×C23⋊D14C2×D14⋊C4C23⋊D14C2×C23.D7C22×C7⋊D4D4×C2×C14D7×C24C22×D7C22×C14C22×D4C22×C4C2×D4C24C23C22
# reps128121184331262412

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
 

׿
×
⋊
ℤ
𝔽
○
≀