(C2xD4).301D4 - GroupNames

G = (C₂xD₄).₃₀₁D₄ order 128 = 2⁷

54^th non-split extension by C₂xD₄ of D₄ acting via D₄/C₂²=C₂

p-group, metabelian, nilpotent (class 3), monomial

Aliases: (C₂xD₄).₃₀₁D₄, C₂.₁₅(D₄oD₈), (C₂xQ₈).₂₃₆D₄, C₄.Q₈:₁₁C₂², C₂.D₈:₂₂C₂², C₄:C₄.₃₉₄C₂³, C₂²:C₈:₁₂C₂², (C₂xC₄).₂₉₄C₂⁴, (C₂xC₈).₁₄₄C₂³, (C₂xD₄).₈₂C₂³, C₂³.₂₄₅(C₂xD₄), C₂.₂₃(D₄oSD₁₆), C₄:D₄.₂₃C₂², C₂².D₈:₁₃C₂, C₂³.₄₆D₄:₃C₂, C₂³.₂₅D₄:₇C₂, M₄(2):C₄:₂₆C₂, C₄²:C₂:₁₅C₂², C₂³.₃₇D₄:₁₁C₂, C₂³.₁₉D₄:₁₄C₂, (C₂²xC₈).₁₈₄C₂², C₂².₅₅₄(C₂²xD₄), D₄:C₄.₁₈₀C₂², C₂².₂₉C₂⁴.₁₂C₂, C₂³.₃₃C₂³:₉C₂, (C₂²xC₄).₁₀₁₀C₂³, C₄.₈₂(C₂².D₄), (C₂²xD₄).₃₆₀C₂², (C₂xM₄(2)).₇₆C₂², C₂².₁₈(C₂².D₄), (C₂xC₄:C₄):₄₉C₂², C₄.₁₀₄(C₂xC₄oD₄), (C₂xC₄).₄₈₉(C₂xD₄), (C₂xD₄:C₄):₃₀C₂, (C₂²xC₈):C₂:₉C₂, (C₂xC₄).₂₉₆(C₄oD₄), (C₂xC₄oD₄).₁₃₉C₂², C₂.₅₉(C₂xC₂².D₄), SmallGroup(128,1828)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂xC₄ — (C₂xD₄).₃₀₁D₄

Chief series

C₁ — C₂ — C₄ — C₂xC₄ — C₂²xC₄ — C₂xC₄:C₄ — C₂³.₃₃C₂³ — (C₂xD₄).₃₀₁D₄

Lower central

C₁ — C₂ — C₂xC₄ — (C₂xD₄).₃₀₁D₄

Upper central

C₁ — C₂² — C₂xC₄oD₄ — (C₂xD₄).₃₀₁D₄

Jennings

C₁ — C₂ — C₂ — C₂xC₄ — (C₂xD₄).₃₀₁D₄

Generators and relations for (C₂xD₄).₃₀₁D₄
G = < a,b,c,d,e | a²=b⁴=c²=e²=1, d⁴=b², ab=ba, ac=ca, ad=da, ae=ea, cbc=b^-1, dbd^-1=ebe=ab^-1, dcd^-1=ece=ab²c, ede=ad³ >

Subgroups: 452 in 209 conjugacy classes, 92 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²:C₄, C₄:C₄, C₄:C₄, C₄:C₄, C₂xC₈, C₂xC₈, C₂xC₈, M₄(2), C₂²xC₄, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₂²:C₈, D₄:C₄, C₄.Q₈, C₄.Q₈, C₂.D₈, C₂.D₈, C₂xC₄:C₄, C₂xC₄:C₄, C₄²:C₂, C₄²:C₂, C₄xD₄, C₄xQ₈, C₂²wrC₂, C₄:D₄, C₄.₄D₄, C₄:₁D₄, C₂²xC₈, C₂xM₄(2), C₂²xD₄, C₂xC₄oD₄, (C₂²xC₈):C₂, C₂xD₄:C₄, C₂³.₃₇D₄, C₂³.₂₅D₄, M₄(2):C₄, C₂².D₈, C₂³.₄₆D₄, C₂³.₁₉D₄, C₂³.₃₃C₂³, C₂².₂₉C₂⁴, (C₂xD₄).₃₀₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂xD₄, C₄oD₄, C₂⁴, C₂².D₄, C₂²xD₄, C₂xC₄oD₄, C₂xC₂².D₄, D₄oD₈, D₄oSD₁₆, (C₂xD₄).₃₀₁D₄

Smallest permutation representation of (C₂xD₄).₃₀₁D₄
►On 32 points

Generators in S₃₂

(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 17)(8 18)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)

(1 28 5 32)(2 15 6 11)(3 30 7 26)(4 9 8 13)(10 23 14 19)(12 17 16 21)(18 31 22 27)(20 25 24 29)

(1 32)(2 11)(3 26)(4 13)(5 28)(6 15)(7 30)(8 9)(10 23)(12 17)(14 19)(16 21)(18 27)(20 29)(22 31)(24 25)

(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)

(2 22)(3 7)(4 20)(6 18)(8 24)(9 11)(10 32)(12 30)(13 15)(14 28)(16 26)(17 21)(25 31)(27 29)

G:=sub<Sym(32)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,28,5,32)(2,15,6,11)(3,30,7,26)(4,9,8,13)(10,23,14,19)(12,17,16,21)(18,31,22,27)(20,25,24,29), (1,32)(2,11)(3,26)(4,13)(5,28)(6,15)(7,30)(8,9)(10,23)(12,17)(14,19)(16,21)(18,27)(20,29)(22,31)(24,25), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,32)(12,30)(13,15)(14,28)(16,26)(17,21)(25,31)(27,29)>;

G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,17)(8,18)(9,27)(10,28)(11,29)(12,30)(13,31)(14,32)(15,25)(16,26), (1,28,5,32)(2,15,6,11)(3,30,7,26)(4,9,8,13)(10,23,14,19)(12,17,16,21)(18,31,22,27)(20,25,24,29), (1,32)(2,11)(3,26)(4,13)(5,28)(6,15)(7,30)(8,9)(10,23)(12,17)(14,19)(16,21)(18,27)(20,29)(22,31)(24,25), (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), (2,22)(3,7)(4,20)(6,18)(8,24)(9,11)(10,32)(12,30)(13,15)(14,28)(16,26)(17,21)(25,31)(27,29) );

G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,17),(8,18),(9,27),(10,28),(11,29),(12,30),(13,31),(14,32),(15,25),(16,26)], [(1,28,5,32),(2,15,6,11),(3,30,7,26),(4,9,8,13),(10,23,14,19),(12,17,16,21),(18,31,22,27),(20,25,24,29)], [(1,32),(2,11),(3,26),(4,13),(5,28),(6,15),(7,30),(8,9),(10,23),(12,17),(14,19),(16,21),(18,27),(20,29),(22,31),(24,25)], [(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)], [(2,22),(3,7),(4,20),(6,18),(8,24),(9,11),(10,32),(12,30),(13,15),(14,28),(16,26),(17,21),(25,31),(27,29)]])

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	···	4N	4O	4P	8A	8B	8C	8D	8E	8F
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	···	4	4	4	8	8	8	8	8	8
size	1	1	1	1	2	2	4	4	8	8	2	2	2	2	4	···	4	8	8	4	4	4	4	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄oD₄

D₄oD₈

D₄oSD₁₆

kernel

(C₂xD₄).₃₀₁D₄

(C₂²xC₈):C₂

C₂xD₄:C₄

C₂³.₃₇D₄

C₂³.₂₅D₄

M₄(2):C₄

C₂².D₈

C₂³.₄₆D₄

C₂³.₁₉D₄

C₂³.₃₃C₂³

C₂².₂₉C₂⁴

C₂xD₄

C₂xQ₈

C₂xC₄

C₂

# reps

Matrix representation of (C₂xD₄).₃₀₁D₄ ►in GL₆(F₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

0	16	0	0	0	0
16	0	0	0	0	0
0	0	1	0	15	0
0	0	0	0	1	1
0	0	1	0	16	0
0	0	16	16	1	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	1	0	15	0
0	0	0	0	1	1
0	0	0	0	16	0
0	0	0	1	1	0

13	0	0	0	0	0
0	4	0	0	0	0
0	0	6	6	0	0
0	0	14	0	0	0
0	0	0	3	3	14
0	0	14	14	3	3

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	16	16	0	0
0	0	0	0	1	0
0	0	16	0	0	16

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,1,16,0,0,0,0,0,16,0,0,15,1,16,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,15,1,16,1,0,0,0,1,0,0],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,6,14,0,14,0,0,6,0,3,14,0,0,0,0,3,3,0,0,0,0,14,3],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16] >;

(C₂xD₄).₃₀₁D₄ in GAP, Magma, Sage, TeX

(C_2\times D_4)._{301}D_4

% in TeX

G:=Group("(C2xD4).301D4");

// GroupNames label

G:=SmallGroup(128,1828);

// by ID

G=gap.SmallGroup(128,1828);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,1018,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^4=c^2=e^2=1,d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,d*b*d^-1=e*b*e=a*b^-1,d*c*d^-1=e*c*e=a*b^2*c,e*d*e=a*d^3>;

// generators/relations

G = (C2xD4).301D4 order 128 = 27

54th non-split extension by C2xD4 of D4 acting via D4/C22=C2

G = (C₂xD₄).₃₀₁D₄ order 128 = 2⁷

54^th non-split extension by C₂xD₄ of D₄ acting via D₄/C₂²=C₂