C2^4.61D4 - GroupNames

G = C₂⁴.₆₁D₄ order 128 = 2⁷

16^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₆₁D₄, C₂³.₇Q₁₆, C₂³.₁₆SD₁₆, C₂²⋊Q₈⋊₈C₄, (C₂²×Q₈)⋊₅C₄, C₄.₁₄(C₂³⋊C₄), C₂³.₅₁₀(C₂×D₄), (C₂²×C₄).₂₂₁D₄, C₂².₁₀(C₂×Q₁₆), C₂³.₃₁D₄⋊₂C₂, C₂².₁₁(C₂×SD₁₆), C₂³.₇Q₈.₄C₂, C₂²⋊C₈.₁₆₈C₂², (C₂²×C₄).₆₄₂C₂³, (C₂³×C₄).₂₁₄C₂², C₂²⋊Q₈.₁₄₇C₂², C₂³.₁₇₆(C₂²⋊C₄), C₂².₂₁(Q₈⋊C₄), C₂.C₄².₉C₂², C₂.₂₅(C₄²⋊C₂²), (C₂×C₄⋊C₄)⋊₁₂C₄, C₄⋊C₄.₂₀(C₂×C₄), C₂.₂₃(C₂×C₂³⋊C₄), (C₂×Q₈).₁₅(C₂×C₄), (C₂×C₂²⋊Q₈).₅C₂, (C₂×C₄).₁₁₆₆(C₂×D₄), (C₂×C₂²⋊C₈).₁₁C₂, C₂.₁₀(C₂×Q₈⋊C₄), (C₂×C₄).₁₃₂(C₂²×C₄), (C₂²×C₄).₂₀₅(C₂×C₄), (C₂×C₄).₂₄₂(C₂²⋊C₄), C₂².₁₉₆(C₂×C₂²⋊C₄), SmallGroup(128,252)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₆₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊Q₈ — C₂⁴.₆₁D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₂⁴.₆₁D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₆₁D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₆₁D₄

Generators and relations for C₂⁴.₆₁D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, ad=da, ae=ea, faf^-1=acd, bc=cb, bd=db, ebe^-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bcde³ >

Subgroups: 324 in 144 conjugacy classes, 54 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂³×C₄, C₂²×Q₈, C₂³.₃₁D₄, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×C₂²⋊Q₈, C₂⁴.₆₁D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, SD₁₆, Q₁₆, C₂²×C₄, C₂×D₄, C₂³⋊C₄, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×SD₁₆, C₂×Q₁₆, C₂×C₂³⋊C₄, C₂×Q₈⋊C₄, C₄²⋊C₂², C₂⁴.₆₁D₄

Smallest permutation representation of C₂⁴.₆₁D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+				+	+		-	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	SD₁₆	Q₁₆	C₂³⋊C₄	C₄²⋊C₂²
kernel	C₂⁴.₆₁D₄	C₂³.₃₁D₄	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×C₂²⋊Q₈	C₂×C₄⋊C₄	C₂²⋊Q₈	C₂²×Q₈	C₂²×C₄	C₂⁴	C₂³	C₂³	C₄	C₂
# reps	1	4	1	1	1	2	4	2	3	1	4	4	2	2

Matrix representation of C₂⁴.₆₁D₄ ►in GL₆(𝔽₁₇)

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

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	15	0	16	0
0	0	0	2	0	16

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

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

3	14	0	0	0	0
3	3	0	0	0	0
0	0	0	8	0	9
0	0	9	0	9	0
0	0	0	1	0	8
0	0	1	0	9	0

0	4	0	0	0	0
4	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	3	0	13
0	0	14	0	13	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,0,0,0,0,1,0,2,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,0,9,0,1,0,0,8,0,1,0,0,0,0,9,0,9,0,0,9,0,8,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,14,0,0,1,0,3,0,0,0,0,0,0,13,0,0,0,0,13,0] >;

C₂⁴.₆₁D₄ in GAP, Magma, Sage, TeX

C_2^4._{61}D_4

% in TeX

G:=Group("C2^4.61D4");

// GroupNames label

G:=SmallGroup(128,252);

// by ID

G=gap.SmallGroup(128,252);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,387,352,1123,1018,248,1971]);

// Polycyclic

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

// generators/relations

G = C24.61D4 order 128 = 27

16th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₆₁D₄ order 128 = 2⁷

16^th non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²