(C2^2xC4).276D4 - GroupNames

G = (C₂²×C₄).₂₇₆D₄ order 128 = 2⁷

161^st non-split extension by C₂²×C₄ of D₄ acting via D₄/C₂=C₂²

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

Aliases: (C₂×C₄²).₃C₄, C₂⁴.₅₄(C₂×C₄), (C₂²×C₄).₂₇₆D₄, C₂².C₄²⋊₁₆C₂, C₂³.₇₅(C₂²⋊C₄), C₂⁴.₄C₄.₁₉C₂, (C₂²×C₄).₆₆₈C₂³, (C₂³×C₄).₂₄₂C₂², C₂³.₁₉₁(C₂²×C₄), C₂².₃₁(C₄²⋊C₂), C₄.₁₀₂(C₂².D₄), C₂.₁₀(C₂³.₃₄D₄), (C₂×M₄(2)).₁₆₁C₂², C₂.₂₅(M₄(2).₈C₂²), (C₂×C₄).₁₃₂₃(C₂×D₄), (C₂×C₂²⋊C₄).₂₇C₄, (C₂²×C₄).₅₅(C₂×C₄), (C₂×C₄).₃₁₁(C₄○D₄), (C₂×C₄⋊C₄).₇₅₆C₂², (C₂×C₄).₃₃₄(C₂²⋊C₄), (C₂×C₄²⋊C₂).₁₆C₂, C₂².₂₅₆(C₂×C₂²⋊C₄), SmallGroup(128,554)

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

Derived series

C₁ — C₂³ — (C₂²×C₄).₂₇₆D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂×C₄²⋊C₂ — (C₂²×C₄).₂₇₆D₄

Lower central

C₁ — C₂ — C₂³ — (C₂²×C₄).₂₇₆D₄

Upper central

C₁ — C₂² — C₂³×C₄ — (C₂²×C₄).₂₇₆D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂²×C₄).₂₇₆D₄

Generators and relations for (C₂²×C₄).₂₇₆D₄
G = < a,b,c,d,e | a²=b²=c⁴=1, d⁴=c², e²=bc^-1, ebe^-1=ab=ba, dcd^-1=ece^-1=ac=ca, ad=da, ae=ea, bc=cb, dbd^-1=abc², ede^-1=abcd³ >

Subgroups: 260 in 136 conjugacy classes, 52 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂³×C₄, C₂².C₄², C₂⁴.₄C₄, C₂×C₄²⋊C₂, (C₂²×C₄).₂₇₆D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄²⋊C₂, C₂².D₄, C₂³.₃₄D₄, M₄(2).₈C₂², (C₂²×C₄).₂₇₆D₄

Smallest permutation representation of (C₂²×C₄).₂₇₆D₄
►On 32 points

Generators in S₃₂

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

(2 31)(4 25)(6 27)(8 29)(9 13)(10 24)(11 15)(12 18)(14 20)(16 22)(17 21)(19 23)

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

(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)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	2	2	4
type	+	+	+	+			+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	C₄○D₄	M₄(2).₈C₂²
kernel	(C₂²×C₄).₂₇₆D₄	C₂².C₄²	C₂⁴.₄C₄	C₂×C₄²⋊C₂	C₂×C₄²	C₂×C₂²⋊C₄	C₂²×C₄	C₂×C₄	C₂
# reps	1	4	2	1	4	4	4	8	4

Matrix representation of (C₂²×C₄).₂₇₆D₄ ►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

16	0	0	0	0	0
13	1	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	1

16	0	0	0	0	0
13	1	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	4	0
0	0	0	0	0	4

4	15	0	0	0	0
0	13	0	0	0	0
0	0	0	0	0	1
0	0	0	0	16	0
0	0	0	13	0	0
0	0	13	0	0	0

16	9	0	0	0	0
0	1	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	4	0	0	0
0	0	0	13	0	0

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],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,13,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,15,13,0,0,0,0,0,0,0,0,0,13,0,0,0,0,13,0,0,0,0,16,0,0,0,0,1,0,0,0],[16,0,0,0,0,0,9,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,1,0,0,0,0,0,0,1,0,0] >;

(C₂²×C₄).₂₇₆D₄ in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{276}D_4

% in TeX

G:=Group("(C2^2xC4).276D4");

// GroupNames label

G:=SmallGroup(128,554);

// by ID

G=gap.SmallGroup(128,554);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,456,422,58,2804,1027,124]);

// Polycyclic

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

// generators/relations

G = (C22×C4).276D4 order 128 = 27

161st non-split extension by C22×C4 of D4 acting via D4/C2=C22

G = (C₂²×C₄).₂₇₆D₄ order 128 = 2⁷

161^st non-split extension by C₂²×C₄ of D₄ acting via D₄/C₂=C₂²