C2^4.100D4 - GroupNames

G = C₂⁴.₁₀₀D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₀₀D₄, C₄.₅(C₂²×Q₈), (C₂×M₄(2))⋊₈C₄, C₄.₄₉(C₂³×C₄), C₈.₁₃(C₂²×C₄), C₂.D₈⋊₆₃C₂², C₄.Q₈⋊₄₃C₂², (C₂²×C₄).₆₅Q₈, C₄⋊C₄.₃₅₀C₂³, C₂³.₃₂(C₄⋊C₄), M₄(2)⋊₁₅(C₂×C₄), (C₂×C₈).₂₄₄C₂³, (C₂×C₄).₁₈₇C₂⁴, C₂³.₃₇₉(C₂×D₄), (C₂²×C₄).₇₈₅D₄, C₄ ○(M₄(2)⋊C₄), M₄(2)⋊C₄⋊₄₅C₂, C₂.₂(D₈⋊C₂²), C₂³.₂₅D₄⋊₂₀C₂, (C₂²×C₈).₂₄₃C₂², (C₂³×C₄).₅₁₉C₂², (C₂²×C₄).₉₀₆C₂³, C₂².₁₃₄(C₂²×D₄), (C₂²×M₄(2)).₄C₂, C₄²⋊C₂.₂₈₇C₂², (C₂×M₄(2)).₂₅₇C₂², (C₂×C₈)⋊₈(C₂×C₄), C₄.₆₆(C₂×C₄⋊C₄), (C₂×C₄).₆₂(C₄⋊C₄), C₂².₃₇(C₂×C₄⋊C₄), C₂.₂₆(C₂²×C₄⋊C₄), (C₂×C₄).₁₄₂(C₂×Q₈), (C₂×C₄).₁₅₆₈(C₂×D₄), (C₂×C₄⋊C₄).₉₀₅C₂², (C₂×C₄).₂₅₀(C₂²×C₄), (C₂²×C₄).₃₃₁(C₂×C₄), (C₂×C₄²⋊C₂).₅₅C₂, SmallGroup(128,1643)

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

Derived series

C₁ — C₄ — C₂⁴.₁₀₀D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×M₄(2) — C₂⁴.₁₀₀D₄

Lower central

C₁ — C₂ — C₄ — C₂⁴.₁₀₀D₄

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂⁴.₁₀₀D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₀₀D₄

Generators and relations for C₂⁴.₁₀₀D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=dc=cd, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, bc=cb, fbf^-1=bd=db, be=eb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 364 in 246 conjugacy classes, 172 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₄.Q₈, C₂.D₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂³.₂₅D₄, M₄(2)⋊C₄, C₂×C₄²⋊C₂, C₂²×M₄(2), C₂⁴.₁₀₀D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂×C₄⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂²×C₄⋊C₄, D₈⋊C₂², C₂⁴.₁₀₀D₄

Smallest permutation representation of C₂⁴.₁₀₀D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	4A	4B	4C	4D	4E	···	4J	4K	···	4Z	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	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+		+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	D₄	D₈⋊C₂²
kernel	C₂⁴.₁₀₀D₄	C₂³.₂₅D₄	M₄(2)⋊C₄	C₂×C₄²⋊C₂	C₂²×M₄(2)	C₂×M₄(2)	C₂²×C₄	C₂²×C₄	C₂⁴	C₂
# reps	1	4	8	2	1	16	3	4	1	4

Matrix representation of C₂⁴.₁₀₀D₄ ►in GL₆(𝔽₁₇)

16	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

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	1	0	16	0
0	0	16	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

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	16	0
0	0	0	0	0	16

13	0	0	0	0	0
9	4	0	0	0	0
0	0	16	15	0	0
0	0	11	1	0	0
0	0	6	16	0	13
0	0	1	1	16	0

4	13	0	0	0	0
0	13	0	0	0	0
0	0	4	0	0	8
0	0	7	0	16	13
0	0	11	1	0	4
0	0	0	0	0	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,16,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[13,9,0,0,0,0,0,4,0,0,0,0,0,0,16,11,6,1,0,0,15,1,16,1,0,0,0,0,0,16,0,0,0,0,13,0],[4,0,0,0,0,0,13,13,0,0,0,0,0,0,4,7,11,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,8,13,4,13] >;

C₂⁴.₁₀₀D₄ in GAP, Magma, Sage, TeX

C_2^4._{100}D_4

% in TeX

G:=Group("C2^4.100D4");

// GroupNames label

G:=SmallGroup(128,1643);

// by ID

G=gap.SmallGroup(128,1643);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,120,723,248,2804,172]);

// Polycyclic

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

// generators/relations

G = C24.100D4 order 128 = 27

55th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₀₀D₄ order 128 = 2⁷

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