(C2xC28).D4 - GroupNames

G = (C₂×C₂₈).D₄ order 448 = 2⁶·7

2^nd non-split extension by C₂×C₂₈ of D₄ acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C₂×C₂₈).₂D₄, (C₂×C₄).₂D₂₈, (C₂×D₄).₂D₁₄, C₂³.D₇⋊₂C₄, C₂³⋊C₄.₁D₇, C₂³.₂(C₄×D₇), (C₂²×C₁₄).₉D₄, C₁₄.₈(C₂³⋊C₄), C₂₈.D₄.₁C₂, (C₂²×Dic₇)⋊₁C₄, (D₄×C₁₄).₂C₂², C₂³.₂(C₇⋊D₄), C₇⋊₁(C₂³.D₄), C₂².₉(D₁₄⋊C₄), C₂³.₁₈D₁₄.₁C₂, C₂.₉(C₂³.₁D₁₄), (C₇×C₂³⋊C₄).₁C₂, (C₂²×C₁₄).₂(C₂×C₄), (C₂×C₁₄).₂(C₂²⋊C₄), SmallGroup(448,29)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — (C₂×C₂₈).D₄

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — D₄×C₁₄ — C₂³.₁₈D₁₄ — (C₂×C₂₈).D₄

Lower central

C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — (C₂×C₂₈).D₄

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂³⋊C₄

Generators and relations for (C₂×C₂₈).D₄
G = < a,b,c,d | a²=b⁴=c²⁸=1, d²=b^-1, ab=ba, cac^-1=dad^-1=ab², cbc^-1=ab^-1, bd=db, dcd^-1=bc^-1 >

Subgroups: 348 in 68 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₄⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, Dic₇, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₄.D₄, C₂².D₄, C₇⋊C₈, C₂×Dic₇, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂³.D₄, C₄.Dic₇, Dic₇⋊C₄, C₂³.D₇, C₂³.D₇, C₇×C₂²⋊C₄, C₂²×Dic₇, D₄×C₁₄, C₂₈.D₄, C₇×C₂³⋊C₄, C₂³.₁₈D₁₄, (C₂×C₂₈).D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₇, C₂²⋊C₄, D₁₄, C₂³⋊C₄, C₄×D₇, D₂₈, C₇⋊D₄, C₂³.D₄, D₁₄⋊C₄, C₂³.₁D₁₄, (C₂×C₂₈).D₄

Smallest permutation representation of (C₂×C₂₈).D₄
►On 112 points

Generators in S₁₁₂

(1 59)(3 61)(5 63)(7 65)(9 67)(11 69)(13 71)(15 73)(17 75)(19 77)(21 79)(23 81)(25 83)(27 57)(29 103)(31 105)(33 107)(35 109)(37 111)(39 85)(41 87)(43 89)(45 91)(47 93)(49 95)(51 97)(53 99)(55 101)

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

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

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

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

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

46 conjugacy classes

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

46 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+	+	+			+			-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₇	D₁₄	D₂₈	C₄×D₇	C₇⋊D₄	C₂³⋊C₄	C₂³.D₄	C₂³.₁D₁₄	(C₂×C₂₈).D₄
kernel	(C₂×C₂₈).D₄	C₂₈.D₄	C₇×C₂³⋊C₄	C₂³.₁₈D₁₄	C₂³.D₇	C₂²×Dic₇	C₂×C₂₈	C₂²×C₁₄	C₂³⋊C₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₄	C₇	C₂	C₁
# reps	1	1	1	1	2	2	1	1	3	3	6	6	6	1	2	6	3

Matrix representation of (C₂×C₂₈).D₄ ►in GL₆(𝔽₁₁₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	112	0	0	0
0	0	0	112	0	0
0	0	0	0	1	0
0	0	96	27	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	15	0	0	0
0	0	85	98	0	0
0	0	0	0	98	0
0	0	39	66	15	15

0	104	0	0	0	0
88	79	0	0	0	0
0	0	0	0	1	0
0	0	46	20	81	81
0	0	98	105	0	0
0	0	57	73	45	93

112	80	0	0	0	0
0	1	0	0	0	0
0	0	29	47	98	83
0	0	101	39	28	28
0	0	1	106	0	0
0	0	13	46	3	45

G:=sub<GL(6,GF(113))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,112,0,0,96,0,0,0,112,0,27,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,15,85,0,39,0,0,0,98,0,66,0,0,0,0,98,15,0,0,0,0,0,15],[0,88,0,0,0,0,104,79,0,0,0,0,0,0,0,46,98,57,0,0,0,20,105,73,0,0,1,81,0,45,0,0,0,81,0,93],[112,0,0,0,0,0,80,1,0,0,0,0,0,0,29,101,1,13,0,0,47,39,106,46,0,0,98,28,0,3,0,0,83,28,0,45] >;

(C₂×C₂₈).D₄ in GAP, Magma, Sage, TeX

(C_2\times C_{28}).D_4

% in TeX

G:=Group("(C2xC28).D4");

// GroupNames label

G:=SmallGroup(448,29);

// by ID

G=gap.SmallGroup(448,29);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,224,141,36,422,346,297,851,18822]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^28=1,d^2=b^-1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a*b^-1,b*d=d*b,d*c*d^-1=b*c^-1>;

// generators/relations

G = (C2×C28).D4 order 448 = 26·7

2nd non-split extension by C2×C28 of D4 acting faithfully

G = (C₂×C₂₈).D₄ order 448 = 2⁶·7

2^nd non-split extension by C₂×C₂₈ of D₄ acting faithfully