C4^2.15C2^3 - GroupNames

G = C₄².₁₅C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₁₅C₂³, C₄⋊C₈⋊₆C₂², D₄.₈(C₂×D₄), C₄⋊Q₈⋊₆C₂², C₄⋊C₄.₃₃₆D₄, (C₂×D₄).₁₅₀D₄, D₄.D₄⋊₃C₂, (C₂×C₈).₁₇C₂³, C₄.₇₃(C₂²×D₄), D₄.₂D₄⋊₁₅C₂, C₄.₃₄(C₄⋊D₄), C₄⋊C₄.₃₈₃C₂³, (C₂×C₄).₂₄₆C₂⁴, C₂²⋊C₄.₁₃₇D₄, (C₄×D₄).₆₇C₂², (C₂×D₈).₅₅C₂², C₄.₄D₄⋊₆C₂², C₂³.₄₄₃(C₂×D₄), (C₂×Q₈).₄₀C₂³, D₄⋊C₄⋊₇₅C₂², C₂.₁₂(D₄○SD₁₆), Q₈⋊C₄⋊₇₈C₂², (C₂²×SD₁₆)⋊₂₃C₂, (C₂×D₄).₃₈₈C₂³, C₂³.₃₈D₄⋊₈C₂, (C₂×SD₁₆).₅C₂², C₂².₁₁C₂⁴⋊₁₀C₂, C₂³.₂₄D₄⋊₂₃C₂, C₂².₈₁(C₄⋊D₄), (C₂²×C₄).₉₇₆C₂³, (C₂²×C₈).₃₄₀C₂², C₄².₆C₂²⋊₄C₂, C₂².₅₀₆(C₂²×D₄), C₂³.₃₈C₂³⋊₈C₂, (C₂²×D₄).₃₄₁C₂², (C₂×M₄(2)).₅₃C₂², (C₂²×Q₈).₂₇₃C₂², C₄²⋊C₂.₁₀₁C₂², C₄.₁₅₆(C₂×C₄○D₄), (C₂×C₄).₄₆₆(C₂×D₄), C₂.₆₄(C₂×C₄⋊D₄), (C₂×C₈⋊C₂²).₁₁C₂, (C₂×C₄).₂₇₇(C₄○D₄), (C₂×C₄○D₄).₁₁₈C₂², SmallGroup(128,1774)

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

Derived series

C₁ — C₂×C₄ — C₄².₁₅C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂².₁₁C₂⁴ — C₄².₁₅C₂³

Lower central

C₁ — C₂ — C₂×C₄ — C₄².₁₅C₂³

Upper central

C₁ — C₂² — C₄²⋊C₂ — C₄².₁₅C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₁₅C₂³

Generators and relations for C₄².₁₅C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=b², e²=a²b², ab=ba, cac^-1=a^-1b², ad=da, eae^-1=ab², cbc^-1=dbd=b^-1, be=eb, dcd=bc, ece^-1=a²b²c, de=ed >

Subgroups: 508 in 247 conjugacy classes, 100 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), D₈, SD₁₆, 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, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊Q₈, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂³.₂₄D₄, C₂³.₃₈D₄, C₄².₆C₂², D₄.D₄, D₄.₂D₄, C₂².₁₁C₂⁴, C₂³.₃₈C₂³, C₂²×SD₁₆, C₂×C₈⋊C₂², C₄².₁₅C₂³
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄⋊D₄, D₄○SD₁₆, C₄².₁₅C₂³

Smallest permutation representation of C₄².₁₅C₂³
►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 21 18 26)(2 22 19 27)(3 23 20 28)(4 24 17 25)(5 30 11 16)(6 31 12 13)(7 32 9 14)(8 29 10 15)

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

(5 30)(6 31)(7 32)(8 29)(9 14)(10 15)(11 16)(12 13)(21 26)(22 27)(23 28)(24 25)

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₄○SD₁₆

kernel

C₄².₁₅C₂³

C₂³.₂₄D₄

C₂³.₃₈D₄

C₄².₆C₂²

D₄.D₄

D₄.₂D₄

C₂².₁₁C₂⁴

C₂³.₃₈C₂³

C₂²×SD₁₆

C₂×C₈⋊C₂²

C₂²⋊C₄

C₄⋊C₄

C₂×D₄

C₂×C₄

C₂

# reps

Matrix representation of C₄².₁₅C₂³ ►in GL₆(𝔽₁₇)

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	16	0	0
0	0	2	1	0	0
0	0	0	0	16	16
0	0	0	0	2	1

0	2	0	0	0	0
9	0	0	0	0	0
0	0	0	5	0	0
0	0	10	0	0	0
0	0	0	0	0	12
0	0	0	0	7	0

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

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

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,16,2,0,0,0,0,16,1],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,10,0,0,0,0,5,0,0,0,0,0,0,0,0,7,0,0,0,0,12,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,1,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₄².₁₅C₂³ in GAP, Magma, Sage, TeX

C_4^2._{15}C_2^3

% in TeX

G:=Group("C4^2.15C2^3");

// GroupNames label

G:=SmallGroup(128,1774);

// by ID

G=gap.SmallGroup(128,1774);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C42.15C23 order 128 = 27

15th non-split extension by C42 of C23 acting faithfully

G = C₄².₁₅C₂³ order 128 = 2⁷

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