C2^3.9M4(2) - GroupNames

G = C₂³.₉M₄(2) order 128 = 2⁷

5^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²

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

Aliases: C₂³.₉M₄(2), (C₂×C₈).₁₉₈D₄, C₂⁴.₆₁(C₂×C₄), C₂.₁₅(C₈⋊₆D₄), C₂.₁₉(C₈⋊₉D₄), C₂².₁₆₁(C₄×D₄), (C₂×C₄).₂₀M₄(2), (C₂×C₄²).₆C₂², C₄.₁₉₂(C₄⋊D₄), C₄.₈₅(C₄.₄D₄), C₂².₅₄(C₈○D₄), (C₂²×C₈).₄₇C₂², (C₂³×C₄).₂₃C₂², C₄.₄₉(C₄²⋊₂C₂), C₂.C₄².₂₃C₄, C₂.₉(C₄².₆C₄), C₂³.₃₁₅(C₂²×C₄), C₂².₆₈(C₂×M₄(2)), (C₂²×C₄).₁₆₃₄C₂³, C₂².₇C₄²⋊₄₁C₂, C₂².₈₃(C₄²⋊C₂), C₄.₁₄₁(C₂².D₄), C₂.₉(C₂⁴.C₂²), C₂.₁₁(C₄².₇C₂²), (C₂×C₄⋊C₈)⋊₄₀C₂, (C₂×C₈⋊C₄)⋊₂₃C₂, (C₂×C₄).₁₅₃₆(C₂×D₄), (C₄×C₂²⋊C₄).₁₆C₂, (C₂×C₂²⋊C₈).₄₄C₂, (C₂×C₂²⋊C₄).₄₀C₄, (C₂×C₄).₉₄₀(C₄○D₄), (C₂²×C₄).₁₂₄(C₂×C₄), SmallGroup(128,656)

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

Derived series

C₁ — C₂³ — C₂³.₉M₄(2)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₂×C₈⋊C₄ — C₂³.₉M₄(2)

Lower central

C₁ — C₂³ — C₂³.₉M₄(2)

Upper central

C₁ — C₂²×C₄ — C₂³.₉M₄(2)

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₉M₄(2)

Generators and relations for C₂³.₉M₄(2)
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=b, ab=ba, eae^-1=ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=cd⁵ >

Subgroups: 252 in 140 conjugacy classes, 60 normal (52 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂²×C₈, C₂³×C₄, C₂².₇C₄², C₄×C₂²⋊C₄, C₂×C₈⋊C₄, C₂×C₂²⋊C₈, C₂×C₄⋊C₈, C₂³.₉M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₄²⋊C₂, C₄×D₄, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂×M₄(2), C₈○D₄, C₂⁴.C₂², C₄².₆C₄, C₄².₇C₂², C₈⋊₉D₄, C₈⋊₆D₄, C₂³.₉M₄(2)

Smallest permutation representation of C₂³.₉M₄(2)
►On 64 points

Generators in S₆₄

(2 16)(4 10)(6 12)(8 14)(17 33)(18 47)(19 35)(20 41)(21 37)(22 43)(23 39)(24 45)(26 61)(28 63)(30 57)(32 59)(34 52)(36 54)(38 56)(40 50)(42 55)(44 49)(46 51)(48 53)

(1 31)(2 32)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 60)(10 61)(11 62)(12 63)(13 64)(14 57)(15 58)(16 59)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(24 40)(41 54)(42 55)(43 56)(44 49)(45 50)(46 51)(47 52)(48 53)

(1 58)(2 59)(3 60)(4 61)(5 62)(6 63)(7 64)(8 57)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 46)(18 47)(19 48)(20 41)(21 42)(22 43)(23 44)(24 45)(33 51)(34 52)(35 53)(36 54)(37 55)(38 56)(39 49)(40 50)

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 52 31 47)(2 39 32 23)(3 54 25 41)(4 33 26 17)(5 56 27 43)(6 35 28 19)(7 50 29 45)(8 37 30 21)(9 20 60 36)(10 46 61 51)(11 22 62 38)(12 48 63 53)(13 24 64 40)(14 42 57 55)(15 18 58 34)(16 44 59 49)

G:=sub<Sym(64)| (2,16)(4,10)(6,12)(8,14)(17,33)(18,47)(19,35)(20,41)(21,37)(22,43)(23,39)(24,45)(26,61)(28,63)(30,57)(32,59)(34,52)(36,54)(38,56)(40,50)(42,55)(44,49)(46,51)(48,53), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,52,31,47)(2,39,32,23)(3,54,25,41)(4,33,26,17)(5,56,27,43)(6,35,28,19)(7,50,29,45)(8,37,30,21)(9,20,60,36)(10,46,61,51)(11,22,62,38)(12,48,63,53)(13,24,64,40)(14,42,57,55)(15,18,58,34)(16,44,59,49)>;

G:=Group( (2,16)(4,10)(6,12)(8,14)(17,33)(18,47)(19,35)(20,41)(21,37)(22,43)(23,39)(24,45)(26,61)(28,63)(30,57)(32,59)(34,52)(36,54)(38,56)(40,50)(42,55)(44,49)(46,51)(48,53), (1,31)(2,32)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,60)(10,61)(11,62)(12,63)(13,64)(14,57)(15,58)(16,59)(17,33)(18,34)(19,35)(20,36)(21,37)(22,38)(23,39)(24,40)(41,54)(42,55)(43,56)(44,49)(45,50)(46,51)(47,52)(48,53), (1,58)(2,59)(3,60)(4,61)(5,62)(6,63)(7,64)(8,57)(9,25)(10,26)(11,27)(12,28)(13,29)(14,30)(15,31)(16,32)(17,46)(18,47)(19,48)(20,41)(21,42)(22,43)(23,44)(24,45)(33,51)(34,52)(35,53)(36,54)(37,55)(38,56)(39,49)(40,50), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,52,31,47)(2,39,32,23)(3,54,25,41)(4,33,26,17)(5,56,27,43)(6,35,28,19)(7,50,29,45)(8,37,30,21)(9,20,60,36)(10,46,61,51)(11,22,62,38)(12,48,63,53)(13,24,64,40)(14,42,57,55)(15,18,58,34)(16,44,59,49) );

G=PermutationGroup([[(2,16),(4,10),(6,12),(8,14),(17,33),(18,47),(19,35),(20,41),(21,37),(22,43),(23,39),(24,45),(26,61),(28,63),(30,57),(32,59),(34,52),(36,54),(38,56),(40,50),(42,55),(44,49),(46,51),(48,53)], [(1,31),(2,32),(3,25),(4,26),(5,27),(6,28),(7,29),(8,30),(9,60),(10,61),(11,62),(12,63),(13,64),(14,57),(15,58),(16,59),(17,33),(18,34),(19,35),(20,36),(21,37),(22,38),(23,39),(24,40),(41,54),(42,55),(43,56),(44,49),(45,50),(46,51),(47,52),(48,53)], [(1,58),(2,59),(3,60),(4,61),(5,62),(6,63),(7,64),(8,57),(9,25),(10,26),(11,27),(12,28),(13,29),(14,30),(15,31),(16,32),(17,46),(18,47),(19,48),(20,41),(21,42),(22,43),(23,44),(24,45),(33,51),(34,52),(35,53),(36,54),(37,55),(38,56),(39,49),(40,50)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,52,31,47),(2,39,32,23),(3,54,25,41),(4,33,26,17),(5,56,27,43),(6,35,28,19),(7,50,29,45),(8,37,30,21),(9,20,60,36),(10,46,61,51),(11,22,62,38),(12,48,63,53),(13,24,64,40),(14,42,57,55),(15,18,58,34),(16,44,59,49)]])

44 conjugacy classes

class	1	2A	···	2G	2H	2I	4A	···	4H	4I	···	4R	8A	···	8P
order	1	2	···	2	2	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	4	4	1	···	1	4	···	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2
type	+	+	+	+	+	+			+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	D₄	M₄(2)	C₄○D₄	M₄(2)	C₈○D₄
kernel	C₂³.₉M₄(2)	C₂².₇C₄²	C₄×C₂²⋊C₄	C₂×C₈⋊C₄	C₂×C₂²⋊C₈	C₂×C₄⋊C₈	C₂.C₄²	C₂×C₂²⋊C₄	C₂×C₈	C₂×C₄	C₂×C₄	C₂³	C₂²
# reps	1	2	1	1	2	1	4	4	4	4	8	4	8

Matrix representation of C₂³.₉M₄(2) ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	13	16	0	0
0	0	0	0	1	0
0	0	0	0	15	16

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	1	0
0	0	0	0	0	1

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	0	0	16	0
0	0	0	0	0	16

0	4	0	0	0	0
16	0	0	0	0	0
0	0	1	9	0	0
0	0	0	16	0	0
0	0	0	0	9	9
0	0	0	0	16	8

4	0	0	0	0	0
0	13	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	13
0	0	0	0	8	4

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

C₂³.₉M₄(2) in GAP, Magma, Sage, TeX

C_2^3._9M_4(2)

% in TeX

G:=Group("C2^3.9M4(2)");

// GroupNames label

G:=SmallGroup(128,656);

// by ID

G=gap.SmallGroup(128,656);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,723,58,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,a*b=b*a,e*a*e^-1=a*c=c*a,d*a*d^-1=a*b*c,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;

// generators/relations

G = C23.9M4(2) order 128 = 27

5th non-split extension by C23 of M4(2) acting via M4(2)/C4=C22

G = C₂³.₉M₄(2) order 128 = 2⁷

5^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₄=C₂²