C2^3.5D28 - GroupNames

G = C₂³.₅D₂₈ order 448 = 2⁶·7

5^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₇=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₅D₂₈, C₂³⋊C₄⋊₄D₇, (C₂×C₂₈).₇D₄, (C₂×C₄).₅D₂₈, C₂²⋊C₄⋊₂D₁₄, (C₂×D₄).₁₂D₁₄, (C₂×Dic₇).₁D₄, (C₂²×D₇).₁D₄, C₂².₂₅(D₄×D₇), C₂².₉(C₂×D₂₈), C₁₄.₁₄C₂²≀C₂, D₄⋊₆D₁₄.₁C₂, (D₄×C₁₄).₉C₂², (C₂²×C₁₄).₁₈D₄, C₇⋊₁(C₂³.₇D₄), C₂³.D₇⋊₂C₂², C₂³.₃(C₂²×D₇), C₂³.₁D₁₄⋊₂C₂, C₂².D₂₈⋊₁C₂, (C₂²×C₁₄).₃C₂³, C₂.₁₇(C₂²⋊D₂₈), C₂³.₁₈D₁₄⋊₁C₂, (C₂²×Dic₇)⋊₁C₂², (C₇×C₂³⋊C₄)⋊₅C₂, (C₂×C₁₄).₁₈(C₂×D₄), (C₇×C₂²⋊C₄)⋊₂C₂², (C₂×C₇⋊D₄).₃C₂², SmallGroup(448,276)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — C₂³.₅D₂₈

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂×C₇⋊D₄ — D₄⋊₆D₁₄ — C₂³.₅D₂₈

Lower central

C₇ — C₁₄ — C₂²×C₁₄ — C₂³.₅D₂₈

Upper central

C₁ — C₂ — C₂³ — C₂³⋊C₄

Generators and relations for C₂³.₅D₂₈
G = < a,b,c,d,e | a²=b²=c²=d²⁸=1, e²=c, ab=ba, ac=ca, dad^-1=eae^-1=abc, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=cd^-1 >

Subgroups: 988 in 160 conjugacy classes, 39 normal (21 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₄○D₄, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₂³⋊C₄, C₂².D₄, 2₊¹⁺⁴, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂³.₇D₄, Dic₇⋊C₄, C₄⋊Dic₇, D₁₄⋊C₄, C₂³.D₇, C₂³.D₇, C₇×C₂²⋊C₄, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, C₂²×Dic₇, C₂×C₇⋊D₄, C₂×C₇⋊D₄, D₄×C₁₄, C₂³.₁D₁₄, C₇×C₂³⋊C₄, C₂².D₂₈, C₂³.₁₈D₁₄, D₄⋊₆D₁₄, C₂³.₅D₂₈
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, D₁₄, C₂²≀C₂, D₂₈, C₂²×D₇, C₂³.₇D₄, C₂×D₂₈, D₄×D₇, C₂²⋊D₂₈, C₂³.₅D₂₈

Smallest permutation representation of C₂³.₅D₂₈
►On 112 points

Generators in S₁₁₂

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

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

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

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

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

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

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

49 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	14A	14B	14C	14D	···	14L	14M	14N	14O	28A	···	28O
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	7	7	7	14	14	14	14	···	14	14	14	14	28	···	28
size	1	1	2	2	2	4	28	28	4	8	8	28	28	28	28	56	2	2	2	2	2	2	4	···	4	8	8	8	8	···	8

49 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

D₂₈

C₂³.₇D₄

D₄×D₇

C₂³.₅D₂₈

kernel

C₂³.₅D₂₈

C₂³.₁D₁₄

C₇×C₂³⋊C₄

C₂².D₂₈

C₂³.₁₈D₁₄

D₄⋊₆D₁₄

C₂×Dic₇

C₂×C₂₈

C₂²×D₇

C₂²×C₁₄

C₂³⋊C₄

C₂²⋊C₄

C₂×D₄

C₂×C₄

C₂³

C₇

C₂²

C₁

# reps

Matrix representation of C₂³.₅D₂₈ ►in GL₆(𝔽₂₉)

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	27	0
0	0	1	0	28	28
0	0	0	0	28	0
0	0	1	28	28	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	27	0	0
0	0	0	28	0	0
0	0	1	28	0	28
0	0	1	28	28	0

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

16	18	0	0	0	0
22	23	0	0	0	0
0	0	1	0	27	0
0	0	0	0	28	1
0	0	1	28	28	0
0	0	1	0	28	0

6	18	0	0	0	0
19	23	0	0	0	0
0	0	17	0	0	0
0	0	0	17	0	0
0	0	0	0	0	17
0	0	0	0	17	0

G:=sub<GL(6,GF(29))| [28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,28,0,0,27,28,28,28,0,0,0,28,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,1,0,0,27,28,28,28,0,0,0,0,0,28,0,0,0,0,28,0],[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],[16,22,0,0,0,0,18,23,0,0,0,0,0,0,1,0,1,1,0,0,0,0,28,0,0,0,27,28,28,28,0,0,0,1,0,0],[6,19,0,0,0,0,18,23,0,0,0,0,0,0,17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,17,0] >;

C₂³.₅D₂₈ in GAP, Magma, Sage, TeX

C_2^3._5D_{28}

% in TeX

G:=Group("C2^3.5D28");

// GroupNames label

G:=SmallGroup(448,276);

// by ID

G=gap.SmallGroup(448,276);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,254,219,226,570,1684,438,18822]);

// Polycyclic

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

// generators/relations

G = C23.5D28 order 448 = 26·7

5th non-split extension by C23 of D28 acting via D28/C7=D4

G = C₂³.₅D₂₈ order 448 = 2⁶·7

5^th non-split extension by C₂³ of D₂₈ acting via D₂₈/C₇=D₄