C2^4.56D14 - GroupNames

G = C₂⁴.₅₆D₁₄ order 448 = 2⁶·7

14^th non-split extension by C₂⁴ of D₁₄ acting via D₁₄/D₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₅₆D₁₄, C₂²wrC₂:₉D₇, D₁₄:C₄:₈C₂², (C₂xDic₇):₂₀D₄, (D₄xDic₇):₁₀C₂, C₂⁴:D₇:₄C₂, C₂².₃₉(D₄xD₇), Dic₇:D₄:₁C₂, Dic₇:₄D₄:₁C₂, (C₂xD₄).₁₄₈D₁₄, (C₂xC₂₈).₂₅C₂³, Dic₇:C₄:₆C₂², C₄:Dic₇:₂₃C₂², C₂²:C₄.₄₄D₁₄, Dic₇.₄₄(C₂xD₄), (C₂³xDic₇):₅C₂, C₁₄.₅₃(C₂²xD₄), C₂²:₃(D₄:₂D₇), (C₂xC₁₄).₁₃₀C₂⁴, C₇:₃(C₂².₁₉C₂⁴), (C₄xDic₇):₁₂C₂², C₂².D₂₈:₈C₂, (C₂²xC₁₄).₇C₂³, C₂³.D₇:₁₁C₂², C₂²:Dic₁₄:₁₁C₂, (C₂xDic₁₄):₁₈C₂², (D₄xC₁₄).₁₀₉C₂², C₂³.₁₈D₁₄:₂C₂, C₂³.₁₁D₁₄:₁C₂, (C₂³xC₁₄).₆₆C₂², (C₂²xD₇).₅₂C₂³, C₂³.₁₇₅(C₂²xD₇), C₂².₁₅₁(C₂³xD₇), (C₂xDic₇).₂₁₉C₂³, (C₂²xDic₇):₁₀C₂², C₂.₂₆(C₂xD₄xD₇), (C₂xC₄xD₇):₄C₂², (C₇xC₂²wrC₂):₂C₂, (C₂xD₄:₂D₇):₅C₂, (C₂xC₁₄):₉(C₄oD₄), C₁₄.₇₅(C₂xC₄oD₄), (C₂xC₁₄).₅₂(C₂xD₄), (C₂xC₇:D₄):₆C₂², C₂.₂₆(C₂xD₄:₂D₇), (C₂xC₄).₂₅(C₂²xD₇), (C₇xC₂²:C₄).₁C₂², SmallGroup(448,1039)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₄ — C₂⁴.₅₆D₁₄

Chief series

C₁ — C₇ — C₁₄ — C₂xC₁₄ — C₂xDic₇ — C₂²xDic₇ — C₂³xDic₇ — C₂⁴.₅₆D₁₄

Lower central

C₇ — C₂xC₁₄ — C₂⁴.₅₆D₁₄

Upper central

C₁ — C₂² — C₂²wrC₂

Generators and relations for C₂⁴.₅₆D₁₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e¹⁴=1, f²=d, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, ebe^-1=fbf^-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 1356 in 330 conjugacy classes, 111 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₄², C₂²:C₄, C₂²:C₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, Dic₇, Dic₇, C₂₈, D₁₄, C₂xC₁₄, C₂xC₁₄, C₂xC₁₄, C₄²:C₂, C₄xD₄, C₂²wrC₂, C₂²wrC₂, C₄:D₄, C₂²:Q₈, C₂².D₄, C₂³xC₄, C₂xC₄oD₄, Dic₁₄, C₄xD₇, C₂xDic₇, C₂xDic₇, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₂xC₂₈, C₇xD₄, C₂²xD₇, C₂²xC₁₄, C₂²xC₁₄, C₂²xC₁₄, C₂².₁₉C₂⁴, C₄xDic₇, Dic₇:C₄, C₄:Dic₇, D₁₄:C₄, C₂³.D₇, C₂³.D₇, C₇xC₂²:C₄, C₇xC₂²:C₄, C₂xDic₁₄, C₂xC₄xD₇, D₄:₂D₇, C₂²xDic₇, C₂²xDic₇, C₂²xDic₇, C₂xC₇:D₄, C₂xC₇:D₄, D₄xC₁₄, D₄xC₁₄, C₂³xC₁₄, C₂³.₁₁D₁₄, C₂²:Dic₁₄, Dic₇:₄D₄, C₂².D₂₈, D₄xDic₇, C₂³.₁₈D₁₄, Dic₇:D₄, C₂⁴:D₇, C₇xC₂²wrC₂, C₂xD₄:₂D₇, C₂³xDic₇, C₂⁴.₅₆D₁₄
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂xD₄, C₄oD₄, C₂⁴, D₁₄, C₂²xD₄, C₂xC₄oD₄, C₂²xD₇, C₂².₁₉C₂⁴, D₄xD₇, D₄:₂D₇, C₂³xD₇, C₂xD₄xD₇, C₂xD₄:₂D₇, C₂⁴.₅₆D₁₄

Smallest permutation representation of C₂⁴.₅₆D₁₄
►On 112 points

Generators in S₁₁₂

(2 34)(4 36)(6 38)(8 40)(10 42)(12 30)(14 32)(16 88)(18 90)(20 92)(22 94)(24 96)(26 98)(28 86)(44 57)(46 59)(48 61)(50 63)(52 65)(54 67)(56 69)(71 112)(73 100)(75 102)(77 104)(79 106)(81 108)(83 110)

(2 61)(4 63)(6 65)(8 67)(10 69)(12 57)(14 59)(16 108)(18 110)(20 112)(22 100)(24 102)(26 104)(28 106)(30 44)(32 46)(34 48)(36 50)(38 52)(40 54)(42 56)(71 92)(73 94)(75 96)(77 98)(79 86)(81 88)(83 90)

(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)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 99)(22 100)(23 101)(24 102)(25 103)(26 104)(27 105)(28 106)(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)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 85)(79 86)(80 87)(81 88)(82 89)(83 90)(84 91)

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

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

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	4H	···	4M	4N	4O	4P	7A	7B	7C	14A	···	14I	14J	···	14AA	14AB	14AC	14AD	28A	···	28I
order	1	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	14	14	14	28	···	28
size	1	1	1	1	2	···	2	4	28	4	4	4	7	7	7	7	14	···	14	28	28	28	2	2	2	2	···	2	4	···	4	8	8	8	8	···	8

70 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

C₄oD₄

D₁₄

D₄xD₇

D₄:₂D₇

kernel

C₂⁴.₅₆D₁₄

C₂³.₁₁D₁₄

C₂²:Dic₁₄

Dic₇:₄D₄

C₂².D₂₈

D₄xDic₇

C₂³.₁₈D₁₄

Dic₇:D₄

C₂⁴:D₇

C₇xC₂²wrC₂

C₂xD₄:₂D₇

C₂³xDic₇

C₂xDic₇

C₂²wrC₂

C₂xC₁₄

C₂²:C₄

C₂xD₄

C₂⁴

C₂²

# reps

Matrix representation of C₂⁴.₅₆D₁₄ ►in GL₆(F₂₉)

1	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	0
0	0	0	0	0	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	0	28

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

1	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	28	0
0	0	0	0	0	28

0	8	0	0	0	0
18	11	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

11	21	0	0	0	0
15	18	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	17
0	0	0	0	17	0

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[0,18,0,0,0,0,8,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,15,0,0,0,0,21,18,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;

C₂⁴.₅₆D₁₄ in GAP, Magma, Sage, TeX

C_2^4._{56}D_{14}

% in TeX

G:=Group("C2^4.56D14");

// GroupNames label

G:=SmallGroup(448,1039);

// by ID

G=gap.SmallGroup(448,1039);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,570,185,18822]);

// Polycyclic

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

// generators/relations

G = C24.56D14 order 448 = 26·7

14th non-split extension by C24 of D14 acting via D14/D7=C2

G = C₂⁴.₅₆D₁₄ order 448 = 2⁶·7

14^th non-split extension by C₂⁴ of D₁₄ acting via D₁₄/D₇=C₂