C2xD4:8D14 - GroupNames

G = C₂×D₄⋊₈D₁₄ order 448 = 2⁶·7

Direct product of C₂ and D₄⋊₈D₁₄

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

Aliases: C₂×D₄⋊₈D₁₄, D₂₈⋊₁₂C₂³, C₂₈.₄₇C₂⁴, C₁₄.₁₂C₂⁵, D₁₄.₆C₂⁴, C₁₄⋊₂2₊¹⁺⁴, Dic₇.₇C₂⁴, Dic₁₄⋊₁₄C₂³, C₄○D₄⋊₁₇D₁₄, (C₂×D₄)⋊₅₀D₁₄, (C₂×C₂₈)⋊₇C₂³, (C₂×Q₈)⋊₃₉D₁₄, (C₄×D₇)⋊₂C₂³, D₄⋊₉(C₂²×D₇), C₇⋊D₄⋊₅C₂³, (C₇×Q₈)⋊₉C₂³, Q₈⋊₈(C₂²×D₇), (D₄×D₇)⋊₁₂C₂², (C₇×D₄)⋊₁₀C₂³, (C₂²×C₄)⋊₃₃D₁₄, (C₂×C₁₄).₃C₂⁴, C₂.₁₃(D₇×C₂⁴), C₄.₆₂(C₂³×D₇), C₇⋊₂(C₂×2₊¹⁺⁴), C₄○D₂₈⋊₂₅C₂², (C₂×D₂₈)⋊₆₄C₂², (C₂²×D₂₈)⋊₂₄C₂, (D₄×C₁₄)⋊₅₃C₂², (Q₈×C₁₄)⋊₄₆C₂², (C₂²×D₇)⋊₅C₂³, (C₂²×C₂₈)⋊₂₈C₂², Q₈⋊₂D₇⋊₁₃C₂², (C₂³×D₇)⋊₁₈C₂², C₂².₅₆(C₂³×D₇), (C₂×Dic₁₄)⋊₇₈C₂², C₂³.₂₁₄(C₂²×D₇), (C₂²×C₁₄).₂₄₈C₂³, (C₂×Dic₇).₂₉₉C₂³, (C₂×D₄×D₇)⋊₂₈C₂, (C₂×C₄○D₄)⋊₁₃D₇, (C₂×C₄×D₇)⋊₃₄C₂², (C₂×C₄)⋊₆(C₂²×D₇), (C₁₄×C₄○D₄)⋊₁₄C₂, (C₂×C₄○D₂₈)⋊₃₇C₂, (C₂×Q₈⋊₂D₇)⋊₂₁C₂, (C₇×C₄○D₄)⋊₂₀C₂², (C₂×C₇⋊D₄)⋊₅₄C₂², SmallGroup(448,1376)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — C₂×D₄⋊₈D₁₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — C₂×D₄×D₇ — C₂×D₄⋊₈D₁₄

Lower central

C₇ — C₁₄ — C₂×D₄⋊₈D₁₄

Upper central

C₁ — C₂² — C₂×C₄○D₄

Generators and relations for C₂×D₄⋊₈D₁₄
G = < a,b,c,d,e | a²=b⁴=c²=d¹⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b^-1, bd=db, dcd^-1=b²c, ce=ec, ede=d^-1 >

Subgroups: 4180 in 898 conjugacy classes, 447 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, 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, C₂²×D₄, C₂×C₄○D₄, C₂×C₄○D₄, 2₊¹⁺⁴, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂×2₊¹⁺⁴, C₂×Dic₁₄, C₂×C₄×D₇, C₂×D₂₈, C₄○D₂₈, D₄×D₇, Q₈⋊₂D₇, C₂×C₇⋊D₄, C₂²×C₂₈, D₄×C₁₄, Q₈×C₁₄, C₇×C₄○D₄, C₂³×D₇, C₂²×D₂₈, C₂×C₄○D₂₈, C₂×D₄×D₇, C₂×Q₈⋊₂D₇, D₄⋊₈D₁₄, C₁₄×C₄○D₄, C₂×D₄⋊₈D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₂⁴, D₁₄, 2₊¹⁺⁴, C₂⁵, C₂²×D₇, C₂×2₊¹⁺⁴, C₂³×D₇, D₄⋊₈D₁₄, D₇×C₂⁴, C₂×D₄⋊₈D₁₄

Smallest permutation representation of C₂×D₄⋊₈D₁₄
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

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

94 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2U	4A	···	4H	4I	4J	4K	4L	7A	7B	7C	14A	···	14I	14J	···	14AA	28A	···	28L	28M	···	28AD
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	4	4	4	7	7	7	14	···	14	14	···	14	28	···	28	28	···	28
size	1	1	1	1	2	···	2	14	···	14	2	···	2	14	14	14	14	2	2	2	2	···	2	4	···	4	2	···	2	4	···	4

94 irreducible representations

dim

type

image

C₁

C₂

D₇

D₁₄

2₊¹⁺⁴

D₄⋊₈D₁₄

kernel

C₂×D₄⋊₈D₁₄

C₂²×D₂₈

C₂×C₄○D₂₈

C₂×D₄×D₇

C₂×Q₈⋊₂D₇

D₄⋊₈D₁₄

C₁₄×C₄○D₄

C₂×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×Q₈

C₄○D₄

C₁₄

C₂

# reps

Matrix representation of C₂×D₄⋊₈D₁₄ ►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	0	0	0	0	0
0	1	0	0	0	0
0	0	27	11	0	0
0	0	18	2	0	0
0	0	0	0	2	18
0	0	0	0	11	27

28	0	0	0	0	0
0	28	0	0	0	0
0	0	0	0	27	11
0	0	0	0	18	2
0	0	2	18	0	0
0	0	11	27	0	0

1	25	0	0	0	0
24	21	0	0	0	0
0	0	26	8	0	0
0	0	21	21	0	0
0	0	0	0	3	21
0	0	0	0	8	8

26	1	0	0	0	0
21	3	0	0	0	0
0	0	26	8	0	0
0	0	28	3	0	0
0	0	0	0	3	21
0	0	0	0	1	26

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,27,18,0,0,0,0,11,2,0,0,0,0,0,0,2,11,0,0,0,0,18,27],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,2,11,0,0,0,0,18,27,0,0,27,18,0,0,0,0,11,2,0,0],[1,24,0,0,0,0,25,21,0,0,0,0,0,0,26,21,0,0,0,0,8,21,0,0,0,0,0,0,3,8,0,0,0,0,21,8],[26,21,0,0,0,0,1,3,0,0,0,0,0,0,26,28,0,0,0,0,8,3,0,0,0,0,0,0,3,1,0,0,0,0,21,26] >;

C₂×D₄⋊₈D₁₄ in GAP, Magma, Sage, TeX

C_2\times D_4\rtimes_8D_{14}

% in TeX

G:=Group("C2xD4:8D14");

// GroupNames label

G:=SmallGroup(448,1376);

// by ID

G=gap.SmallGroup(448,1376);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,1684,102,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^14=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^2*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C2×D4⋊8D14 order 448 = 26·7

Direct product of C2 and D4⋊8D14

G = C₂×D₄⋊₈D₁₄ order 448 = 2⁶·7

Direct product of C₂ and D₄⋊₈D₁₄