C28.48D4 - GroupNames

G = C₂₈.₄₈D₄ order 224 = 2⁵·7

5^th non-split extension by C₂₈ of D₄ acting via D₄/C₂²=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂₈.₄₈D₄, C₂²:₂Dic₁₄, C₂³.₂₀D₁₄, (C₂xC₁₄):₃Q₈, C₄:Dic₇:₈C₂, C₇:₄(C₂²:Q₈), C₁₄.₈(C₂xQ₈), Dic₇:C₄:₂C₂, C₁₄.₃₉(C₂xD₄), (C₂xC₄).₈₃D₁₄, (C₂²xC₄).₅D₇, (C₂xDic₁₄):₆C₂, C₄.₂₃(C₇:D₄), (C₂²xC₂₈).₆C₂, C₂.₉(C₂xDic₁₄), C₂³.D₇.₄C₂, C₂.₁₇(C₄oD₂₈), C₁₄.₁₅(C₄oD₄), (C₂xC₁₄).₄₂C₂³, (C₂xC₂₈).₉₁C₂², C₂².₅₄(C₂²xD₇), (C₂²xC₁₄).₃₄C₂², (C₂xDic₇).₁₄C₂², C₂.₅(C₂xC₇:D₄), SmallGroup(224,119)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₁₄ — C₂₈.₄₈D₄

Chief series

C₁ — C₇ — C₁₄ — C₂xC₁₄ — C₂xDic₇ — C₂xDic₁₄ — C₂₈.₄₈D₄

Lower central

C₇ — C₂xC₁₄ — C₂₈.₄₈D₄

Upper central

C₁ — C₂² — C₂²xC₄

Generators and relations for C₂₈.₄₈D₄
G = < a,b,c | a²⁸=b⁴=1, c²=a¹⁴, bab^-1=cac^-1=a^-1, cbc^-1=a¹⁴b^-1 >

Subgroups: 238 in 74 conjugacy classes, 37 normal (21 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₇, C₂xC₄, C₂xC₄, Q₈, C₂³, C₁₄, C₁₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xQ₈, Dic₇, C₂₈, C₂₈, C₂xC₁₄, C₂xC₁₄, C₂xC₁₄, C₂²:Q₈, Dic₁₄, C₂xDic₇, C₂xC₂₈, C₂xC₂₈, C₂²xC₁₄, Dic₇:C₄, C₄:Dic₇, C₂³.D₇, C₂xDic₁₄, C₂²xC₂₈, C₂₈.₄₈D₄
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, D₇, C₂xD₄, C₂xQ₈, C₄oD₄, D₁₄, C₂²:Q₈, Dic₁₄, C₇:D₄, C₂²xD₇, C₂xDic₁₄, C₄oD₂₈, C₂xC₇:D₄, C₂₈.₄₈D₄

Smallest permutation representation of C₂₈.₄₈D₄
►On 112 points

Generators in S₁₁₂

(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)

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

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

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

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

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

62 conjugacy classes

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	7A	7B	7C	14A	···	14U	28A	···	28X
order	1	2	2	2	2	2	4	4	4	4	4	4	4	4	7	7	7	14	···	14	28	···	28
size	1	1	1	1	2	2	2	2	2	2	28	28	28	28	2	2	2	2	···	2	2	···	2

62 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+	+	+	+	+	+	-	+		+	+		-
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	Q₈	D₇	C₄oD₄	D₁₄	D₁₄	C₇:D₄	Dic₁₄	C₄oD₂₈
kernel	C₂₈.₄₈D₄	Dic₇:C₄	C₄:Dic₇	C₂³.D₇	C₂xDic₁₄	C₂²xC₂₈	C₂₈	C₂xC₁₄	C₂²xC₄	C₁₄	C₂xC₄	C₂³	C₄	C₂²	C₂
# reps	1	2	1	2	1	1	2	2	3	2	6	3	12	12	12

Matrix representation of C₂₈.₄₈D₄ ►in GL₄(F₂₉) generated by

21	0	0	0
23	18	0	0
0	0	16	0
0	0	0	20

22	11	0	0
22	7	0	0
0	0	0	1
0	0	28	0

22	11	0	0
6	7	0	0
0	0	0	1
0	0	1	0

G:=sub<GL(4,GF(29))| [21,23,0,0,0,18,0,0,0,0,16,0,0,0,0,20],[22,22,0,0,11,7,0,0,0,0,0,28,0,0,1,0],[22,6,0,0,11,7,0,0,0,0,0,1,0,0,1,0] >;

C₂₈.₄₈D₄ in GAP, Magma, Sage, TeX

C_{28}._{48}D_4

% in TeX

G:=Group("C28.48D4");

// GroupNames label

G:=SmallGroup(224,119);

// by ID

G=gap.SmallGroup(224,119);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,217,103,218,6917]);

// Polycyclic

G:=Group<a,b,c|a^28=b^4=1,c^2=a^14,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^14*b^-1>;

// generators/relations

G = C28.48D4 order 224 = 25·7

5th non-split extension by C28 of D4 acting via D4/C22=C2

G = C₂₈.₄₈D₄ order 224 = 2⁵·7

5^th non-split extension by C₂₈ of D₄ acting via D₄/C₂²=C₂