C2xC2^2:D28 - GroupNames

G = C₂xC₂²:D₂₈ order 448 = 2⁶·7

Direct product of C₂ and C₂²:D₂₈

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

Aliases: C₂xC₂²:D₂₈, C₂³:₅D₂₈, C₂⁴.₅₄D₁₄, C₁₄:₁C₂²wrC₂, D₁₄:₁₀(C₂xD₄), (C₂xC₂₈):₃C₂³, (D₇xC₂⁴):₁C₂, C₂²:₃(C₂xD₂₈), (C₂²xC₁₄):₉D₄, (C₂²xC₄):₇D₁₄, C₂²:C₄:₃₇D₁₄, (C₂²xD₇):₁₃D₄, (C₂²xD₂₈):₅C₂, C₁₄.₆(C₂²xD₄), C₂.₈(C₂²xD₂₈), D₁₄:C₄:₄₄C₂², (C₂xD₂₈):₄₃C₂², (C₂xC₁₄).₃₁C₂⁴, (C₂²xC₂₈):₇C₂², (C₂xDic₇):₁C₂³, C₂².₁₂₅(D₄xD₇), (C₂²xD₇):₁C₂³, (C₂³xD₇):₃C₂², C₂².₇₀(C₂³xD₇), (C₂³xC₁₄).₅₇C₂², (C₂²xDic₇):₆C₂², C₂³.₁₄₆(C₂²xD₇), (C₂²xC₁₄).₁₂₃C₂³, C₂.₈(C₂xD₄xD₇), C₇:₁(C₂xC₂²wrC₂), (C₂xC₁₄):₄(C₂xD₄), (C₂xC₄):₃(C₂²xD₇), (C₂xD₁₄:C₄):₁₆C₂, (C₂xC₂²:C₄):₁₀D₇, (C₂²xC₇:D₄):₃C₂, (C₁₄xC₂²:C₄):₁₃C₂, (C₂xC₇:D₄):₃₄C₂², (C₇xC₂²:C₄):₄₆C₂², SmallGroup(448,940)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₄ — C₂xC₂²:D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂xC₁₄ — C₂²xD₇ — C₂³xD₇ — D₇xC₂⁴ — C₂xC₂²:D₂₈

Lower central

C₇ — C₂xC₁₄ — C₂xC₂²:D₂₈

Upper central

C₁ — C₂³ — C₂xC₂²:C₄

Generators and relations for C₂xC₂²:D₂₈
G = < a,b,c,d,e | a²=b²=c²=d²⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=ebe=bc=cb, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 3988 in 662 conjugacy classes, 143 normal (17 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₁₄, C₂²:C₄, C₂²:C₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂⁴, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂xC₁₄, C₂xC₁₄, C₂xC₁₄, C₂xC₂²:C₄, C₂xC₂²:C₄, C₂²wrC₂, C₂²xD₄, C₂⁵, D₂₈, C₂xDic₇, C₂xDic₇, C₇:D₄, C₂xC₂₈, C₂xC₂₈, C₂²xD₇, C₂²xD₇, C₂²xC₁₄, C₂²xC₁₄, C₂²xC₁₄, C₂xC₂²wrC₂, D₁₄:C₄, C₇xC₂²:C₄, C₂xD₂₈, C₂xD₂₈, C₂²xDic₇, C₂xC₇:D₄, C₂xC₇:D₄, C₂²xC₂₈, C₂³xD₇, C₂³xD₇, C₂³xD₇, C₂³xC₁₄, C₂²:D₂₈, C₂xD₁₄:C₄, C₁₄xC₂²:C₄, C₂²xD₂₈, C₂²xC₇:D₄, D₇xC₂⁴, C₂xC₂²:D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂xD₄, C₂⁴, D₁₄, C₂²wrC₂, C₂²xD₄, D₂₈, C₂²xD₇, C₂xC₂²wrC₂, C₂xD₂₈, D₄xD₇, C₂³xD₇, C₂²:D₂₈, C₂²xD₂₈, C₂xD₄xD₇, C₂xC₂²:D₂₈

Smallest permutation representation of C₂xC₂²:D₂₈
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

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

88 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	2L	···	2S	2T	2U	4A	4B	4C	4D	4E	4F	7A	7B	7C	14A	···	14U	14V	···	14AG	28A	···	28X
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	14	···	14	28	28	4	4	4	4	28	28	2	2	2	2	···	2	4	···	4	4	···	4

88 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

D₂₈

D₄xD₇

kernel

C₂xC₂²:D₂₈

C₂²:D₂₈

C₂xD₁₄:C₄

C₁₄xC₂²:C₄

C₂²xD₂₈

C₂²xC₇:D₄

D₇xC₂⁴

C₂²xD₇

C₂²xC₁₄

C₂xC₂²:C₄

C₂²:C₄

C₂²xC₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂xC₂²:D₂₈ ►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	1	0
0	0	0	0	0	1

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

15	11	0	0	0	0
7	11	0	0	0	0
0	0	28	1	0	0
0	0	27	1	0	0
0	0	0	0	0	28
0	0	0	0	1	0

7	1	0	0	0	0
10	22	0	0	0	0
0	0	1	28	0	0
0	0	0	28	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[15,7,0,0,0,0,11,11,0,0,0,0,0,0,28,27,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,28,0],[7,10,0,0,0,0,1,22,0,0,0,0,0,0,1,0,0,0,0,0,28,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C₂xC₂²:D₂₈ in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes D_{28}

% in TeX

G:=Group("C2xC2^2:D28");

// GroupNames label

G:=SmallGroup(448,940);

// by ID

G=gap.SmallGroup(448,940);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,675,297,80,18822]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^28=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=e*b*e=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C2xC22:D28 order 448 = 26·7

Direct product of C2 and C22:D28

G = C₂xC₂²:D₂₈ order 448 = 2⁶·7

Direct product of C₂ and C₂²:D₂₈