(C2^2xC4).275D4 - GroupNames

G = (C₂²×C₄).₂₇₅D₄ order 128 = 2⁷

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

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

Aliases: (C₂³×C₄).₁₁C₄, C₂⁴.₁₁₈(C₂×C₄), (C₂²×C₄).₂₇₅D₄, C₂².C₄²⋊₁₅C₂, C₂⁴.₄C₄.₁₈C₂, (C₂²×C₄).₆₆₇C₂³, C₂³.₁₉₀(C₂²×C₄), (C₂³×C₄).₂₄₁C₂², C₂³.₁₉₈(C₂²⋊C₄), C₂².₂₀(C₄.D₄), C₂.₉(C₂³.₃₄D₄), C₂².₃₀(C₄²⋊C₂), C₄.₁₀₁(C₂².D₄), C₂².₁₂(C₄.₁₀D₄), (C₂×M₄(2)).₁₆₀C₂², (C₂×C₄⋊C₄).₅₃C₄, (C₂×C₄).₁₃₂₂(C₂×D₄), (C₂²×C₄⋊C₄).₁₂C₂, (C₂²×C₄).₅₄(C₂×C₄), C₂.₂₆(C₂×C₄.D₄), (C₂×C₄).₃₁₀(C₄○D₄), (C₂×C₄⋊C₄).₇₅₅C₂², C₂.₂₄(C₂×C₄.₁₀D₄), (C₂×C₄).₁₂₅(C₂²⋊C₄), C₂².₂₅₅(C₂×C₂²⋊C₄), SmallGroup(128,553)

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

Derived series

C₁ — C₂³ — (C₂²×C₄).₂₇₅D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — (C₂²×C₄).₂₇₅D₄

Lower central

C₁ — C₂ — C₂³ — (C₂²×C₄).₂₇₅D₄

Upper central

C₁ — C₂² — C₂³×C₄ — (C₂²×C₄).₂₇₅D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — (C₂²×C₄).₂₇₅D₄

Generators and relations for (C₂²×C₄).₂₇₅D₄
G = < a,b,c,d,e | a²=b²=c⁴=1, d⁴=c², e²=c, dad^-1=eae^-1=ab=ba, ac=ca, bc=cb, bd=db, be=eb, dcd^-1=c^-1, ce=ec, ede^-1=bc^-1d³ >

Subgroups: 308 in 158 conjugacy classes, 56 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂².C₄², C₂⁴.₄C₄, C₂²×C₄⋊C₄, (C₂²×C₄).₂₇₅D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₄.D₄, C₄.₁₀D₄, C₂×C₂²⋊C₄, C₄²⋊C₂, C₂².D₄, C₂³.₃₄D₄, C₂×C₄.D₄, C₂×C₄.₁₀D₄, (C₂²×C₄).₂₇₅D₄

Smallest permutation representation of (C₂²×C₄).₂₇₅D₄
►On 32 points

Generators in S₃₂

(2 27)(4 29)(6 31)(8 25)(10 24)(12 18)(14 20)(16 22)

(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)

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

(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)| (2,27)(4,29)(6,31)(8,25)(10,24)(12,18)(14,20)(16,22), (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), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (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( (2,27)(4,29)(6,31)(8,25)(10,24)(12,18)(14,20)(16,22), (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), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,23,21,19)(18,20,22,24)(25,27,29,31)(26,32,30,28), (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([[(2,27),(4,29),(6,31),(8,25),(10,24),(12,18),(14,20),(16,22)], [(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)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,23,21,19),(18,20,22,24),(25,27,29,31),(26,32,30,28)], [(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	···	2I	4A	4B	4C	4D	4E	···	4N	8A	···	8H
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	2	2	2	2	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	2	2	4	4
type	+	+	+	+			+		+	-
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	C₄○D₄	C₄.D₄	C₄.₁₀D₄
kernel	(C₂²×C₄).₂₇₅D₄	C₂².C₄²	C₂⁴.₄C₄	C₂²×C₄⋊C₄	C₂×C₄⋊C₄	C₂³×C₄	C₂²×C₄	C₂×C₄	C₂²	C₂²
# reps	1	4	2	1	4	4	4	8	2	2

Matrix representation of (C₂²×C₄).₂₇₅D₄ ►in GL₆(𝔽₁₇)

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

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

0	15	0	0	0	0
9	0	0	0	0	0
0	0	10	0	5	9
0	0	6	0	15	7
0	0	6	11	11	13
0	0	10	13	11	13

0	8	0	0	0	0
15	0	0	0	0	0
0	0	1	0	15	0
0	0	0	0	16	1
0	0	0	1	16	0
0	0	1	0	16	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,1,0,0,15,16,16,16,0,0,0,0,0,16,0,0,0,0,1,0],[0,9,0,0,0,0,15,0,0,0,0,0,0,0,10,6,6,10,0,0,0,0,11,13,0,0,5,15,11,11,0,0,9,7,13,13],[0,15,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,1,0,0,0,15,16,16,16,0,0,0,1,0,0] >;

(C₂²×C₄).₂₇₅D₄ in GAP, Magma, Sage, TeX

(C_2^2\times C_4)._{275}D_4

% in TeX

G:=Group("(C2^2xC4).275D4");

// GroupNames label

G:=SmallGroup(128,553);

// by ID

G=gap.SmallGroup(128,553);

# by ID

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

// Polycyclic

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

// generators/relations

G = (C22×C4).275D4 order 128 = 27

160th non-split extension by C22×C4 of D4 acting via D4/C2=C22

G = (C₂²×C₄).₂₇₅D₄ order 128 = 2⁷

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