C4^2.16C2^3 - GroupNames

G = C₄².₁₆C₂³ order 128 = 2⁷

16^th non-split extension by C₄² of C₂³ acting faithfully

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

Aliases: C₄².₁₆C₂³, C₄⋊C₈⋊₇C₂², (C₂×Q₈)⋊₂₂D₄, Q₈.₇(C₂×D₄), C₄⋊C₄.₃₃₇D₄, C₄⋊SD₁₆⋊₃C₂, (C₄×Q₈)⋊₃C₂², (C₂×C₈).₁₈C₂³, C₄.₇₄(C₂²×D₄), C₄.₃₅(C₄⋊D₄), C₄⋊C₄.₃₈₄C₂³, (C₂×C₄).₂₄₇C₂⁴, Q₈.D₄⋊₁₄C₂, C₂²⋊C₄.₁₃₈D₄, (C₂×Q₁₆)⋊₁₇C₂², (C₂×D₄).₅₃C₂³, C₂³.₄₄₄(C₂×D₄), D₄⋊C₄⋊₇₆C₂², C₂.₁₃(D₄○SD₁₆), Q₈⋊C₄⋊₇₉C₂², (C₂²×SD₁₆)⋊₂₄C₂, (C₂×SD₁₆)⋊₁₀C₂², C₄⋊₁D₄.₅₈C₂², C₂³.₃₇D₄⋊₉C₂, (C₂×Q₈).₃₆₀C₂³, C₂³.₂₄D₄⋊₂₄C₂, C₂².₈₂(C₄⋊D₄), C₄².₆C₂²⋊₅C₂, (C₂²×C₄).₉₇₇C₂³, (C₂²×C₈).₃₄₁C₂², C₄.₄D₄.₂₄C₂², C₂².₅₀₇(C₂²×D₄), C₂³.₃₂C₂³⋊₇C₂, C₂².₂₉C₂⁴.₁₀C₂, (C₂²×D₄).₃₄₂C₂², (C₂×M₄(2)).₅₄C₂², (C₂²×Q₈).₂₇₄C₂², C₄²⋊C₂.₁₀₂C₂², C₄.₁₅₇(C₂×C₄○D₄), (C₂×C₄).₄₆₇(C₂×D₄), C₂.₆₅(C₂×C₄⋊D₄), (C₂×C₈.C₂²)⋊₁₆C₂, (C₂×C₄).₂₇₈(C₄○D₄), (C₂×C₄○D₄).₁₁₉C₂², SmallGroup(128,1775)

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

Derived series

C₁ — C₂×C₄ — C₄².₁₆C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂³.₃₂C₂³ — C₄².₁₆C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₁₆C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₁₆C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₁₆C₂³

Generators and relations for C₄².₁₆C₂³
G = < a,b,c,d,e | a⁴=b⁴=c²=1, d²=b², e²=a²b², ab=ba, cac=a^-1b², ad=da, eae^-1=ab², cbc=dbd^-1=b^-1, be=eb, dcd^-1=bc, ece^-1=a²b²c, de=ed >

Subgroups: 508 in 245 conjugacy classes, 100 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, D₄⋊C₄, Q₈⋊C₄, C₄⋊C₈, C₄²⋊C₂, C₄²⋊C₂, C₄×Q₈, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₄.₄D₄, C₄⋊₁D₄, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³.₂₄D₄, C₂³.₃₇D₄, C₄².₆C₂², C₄⋊SD₁₆, Q₈.D₄, C₂³.₃₂C₂³, C₂².₂₉C₂⁴, C₂²×SD₁₆, C₂×C₈.C₂², C₄².₁₆C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄⋊D₄, D₄○SD₁₆, C₄².₁₆C₂³

Smallest permutation representation of C₄².₁₆C₂³
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₁₆C₂³

C₂³.₂₄D₄

C₂³.₃₇D₄

C₄².₆C₂²

C₄⋊SD₁₆

Q₈.D₄

C₂³.₃₂C₂³

C₂².₂₉C₂⁴

C₂²×SD₁₆

C₂×C₈.C₂²

C₂²⋊C₄

C₄⋊C₄

C₂×Q₈

C₂×C₄

C₂

# reps

Matrix representation of C₄².₁₆C₂³ ►in GL₆(𝔽₁₇)

2	15	0	0	0	0
11	15	0	0	0	0
0	0	16	0	15	0
0	0	0	0	1	1
0	0	1	0	1	0
0	0	16	16	16	0

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

1	0	0	0	0	0
2	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	1	0	0	16

16	0	0	0	0	0
0	16	0	0	0	0
0	0	0	10	0	0
0	0	5	0	0	0
0	0	0	12	12	12
0	0	12	5	12	5

2	15	0	0	0	0
11	15	0	0	0	0
0	0	16	0	15	0
0	0	0	0	1	1
0	0	0	0	1	0
0	0	0	1	16	0

G:=sub<GL(6,GF(17))| [2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,1,16,0,0,0,0,0,16,0,0,15,1,1,16,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,1,0,16,0,0,15,1,1,16,0,0,0,0,0,16,0,0,0,0,1,0],[1,2,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,1,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,5,0,12,0,0,10,0,12,5,0,0,0,0,12,12,0,0,0,0,12,5],[2,11,0,0,0,0,15,15,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,15,1,1,16,0,0,0,1,0,0] >;

C₄².₁₆C₂³ in GAP, Magma, Sage, TeX

C_4^2._{16}C_2^3

% in TeX

G:=Group("C4^2.16C2^3");

// GroupNames label

G:=SmallGroup(128,1775);

// by ID

G=gap.SmallGroup(128,1775);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,352,1018,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.16C23 order 128 = 27

16th non-split extension by C42 of C23 acting faithfully

G = C₄².₁₆C₂³ order 128 = 2⁷

16^th non-split extension by C₄² of C₂³ acting faithfully