C4^2.693C2^3

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₄².₆₉₃C₂³, C₄.₁₆₉2₊⁽¹⁺⁴⁾, C₈⋊₆D₄⋊₄₀C₂, C₄⋊C₈⋊₉₁C₂², (C₄×C₈)⋊₆₀C₂², (C₄×D₄).₃₅C₄, (C₂×C₄)⋊₄M₄(2), C₂⁴.₈₆(C₂×C₄), C₂²⋊C₈⋊₄₆C₂², (C₂×C₄).₆₇₂C₂⁴, (C₂×C₈).₄₃₃C₂³, C₄².₂₂₃(C₂×C₄), (C₂²×D₄).₄₄C₄, C₄.₁₅(C₂×M₄(2)), C₂⁴.₄C₄⋊₃₆C₂, (C₄×D₄).₂₉₉C₂², C₄².₁₂C₄⋊₅₃C₂, C₂.₂₈(Q₈○M₄(2)), (C₂×M₄(2))⋊₄₆C₂², (C₂²×C₄).₉₄₀C₂³, (C₂³×C₄).₅₃₁C₂², C₂³.₁₀₆(C₂²×C₄), C₂².₁₉₆(C₂³×C₄), (C₂×C₄²).₇₈₂C₂², C₂².₂₉(C₂×M₄(2)), C₂.₂₀(C₂²×M₄(2)), C₂.₄₆(C₂².₁₁C₂⁴), (C₂×C₄×D₄).₇₈C₂, (C₂×C₄⋊C₄).₇₈C₄, C₄⋊C₄.₂₃₀(C₂×C₄), (C₂×D₄).₂₃₆(C₂×C₄), C₂²⋊C₄.₇₈(C₂×C₄), (C₂×C₂²⋊C₄).₅₂C₄, (C₂²×C₄).₃₅₃(C₂×C₄), (C₂×C₄).₂₉₈(C₂²×C₄), SmallGroup(128,1707)

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

Derived series

C₁ — C₂² — C₄².₆₉₃C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₄².₆₉₃C₂³

Lower central

C₁ — C₂² — C₄².₆₉₃C₂³

Upper central

C₁ — C₂×C₄ — C₄².₆₉₃C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₆₉₃C₂³

Subgroups: 388 in 226 conjugacy classes, 134 normal (16 characteristic)
C₁, C₂^[×3], C₂^[×6], C₄^[×6], C₄^[×7], C₂², C₂²^[×2], C₂²^[×22], C₈^[×8], C₂×C₄^[×2], C₂×C₄^[×12], C₂×C₄^[×17], D₄^[×8], C₂³, C₂³^[×4], C₂³^[×8], C₄²^[×4], C₂²⋊C₄^[×8], C₄⋊C₄^[×4], C₂×C₈^[×8], M₄(2)^[×8], C₂²×C₄^[×3], C₂²×C₄^[×10], C₂²×C₄^[×4], C₂×D₄^[×4], C₂×D₄^[×4], C₂⁴^[×2], C₄×C₈^[×4], C₂²⋊C₈^[×12], C₄⋊C₈^[×4], C₂×C₄², C₂×C₂²⋊C₄^[×2], C₂×C₄⋊C₄, C₄×D₄^[×8], C₂×M₄(2)^[×8], C₂³×C₄^[×2], C₂²×D₄, C₂⁴.₄C₄^[×4], C₄².₁₂C₄^[×2], C₈⋊₆D₄^[×8], C₂×C₄×D₄, C₄².₆₉₃C₂³

Quotients:
C₁, C₂^[×15], C₄^[×8], C₂²^[×35], C₂×C₄^[×28], C₂³^[×15], M₄(2)^[×4], C₂²×C₄^[×14], C₂⁴, C₂×M₄(2)^[×6], C₂³×C₄, 2₊⁽¹⁺⁴⁾^[×2], C₂².₁₁C₂⁴, C₂²×M₄(2), Q₈○M₄(2), C₄².₆₉₃C₂³

Generators and relations
G = < a,b,c,d,e | a⁴=b⁴=d²=1, c²=b, e²=a², ab=ba, ac=ca, dad=a^-1, ae=ea, bc=cb, bd=db, be=eb, dcd=b²c, ece^-1=a²c, ede^-1=a²d >

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

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

13	0	0	0	0	0
0	13	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	M₄(2)	2₊⁽¹⁺⁴⁾	Q₈○M₄(2)
kernel	C₄².₆₉₃C₂³	C₂⁴.₄C₄	C₄².₁₂C₄	C₈⋊₆D₄	C₂×C₄×D₄	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₄×D₄	C₂²×D₄	C₂×C₄	C₄	C₂
# reps	1	4	2	8	1	4	2	8	2	8	2	2

In GAP, Magma, Sage, TeX

C_4^2._{693}C_2^3

% in TeX

G:=Group("C4^2.693C2^3");

// GroupNames label

G:=SmallGroup(128,1707);

// by ID

G=gap.SmallGroup(128,1707);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,891,2467,124]);

// Polycyclic

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

// generators/relations

G = C₄².₆₉₃C₂³ order 128 = 2⁷

108^th non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂

G = C42.693C23 order 128 = 27

108th non-split extension by C42 of C23 acting via C23/C22=C2

G = C₄².₆₉₃C₂³ order 128 = 2⁷

108^th non-split extension by C₄² of C₂³ acting via C₂³/C₂²=C₂