C2^3.32M4(2) - GroupNames

G = C₂³.₃₂M₄(2) order 128 = 2⁷

9^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₂×C₄=C₂

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

Aliases: C₂³.₃₂M₄(2), (C₂²×C₄)⋊₇C₈, C₂³.₃₃(C₂×C₈), (C₂×C₄²).₃₇C₄, (C₂³×C₄).₃₀C₄, C₂⁴.₁₁₆(C₂×C₄), (C₂²×C₄).₆₇₇D₄, (C₂×C₄).₆₁M₄(2), (C₂²×C₄²).₇C₂, (C₂²×C₈).₁₄C₂², C₂².₃₅(C₂²×C₈), C₂².₂₉(C₂²⋊C₈), C₂.₅(C₄².₆C₄), C₂.₅(C₂⁴.₄C₄), (C₂³×C₄).₆₃₄C₂², (C₂×C₄²).₉₉₆C₂², C₂³.₂₆₄(C₂²×C₄), C₂².₇C₄²⋊₄C₂, C₂².₄₆(C₂×M₄(2)), C₂³.₁₉₇(C₂²⋊C₄), C₂.₆(C₄².₁₂C₄), (C₂²×C₄).₁₆₁₉C₂³, C₂.₃(C₂³.₃₄D₄), C₂².₅₅(C₄²⋊C₂), C₄.₁₂₈(C₂².D₄), (C₂×C₄).₆₀(C₂×C₈), C₂.₁₈(C₂×C₂²⋊C₈), (C₂×C₄).₁₅₁₆(C₂×D₄), (C₂×C₂²⋊C₈).₁₅C₂, (C₂×C₄).₉₂₇(C₄○D₄), (C₂²×C₄).₄₄₂(C₂×C₄), (C₂×C₄).₃₃₃(C₂²⋊C₄), C₂².₁₂₅(C₂×C₂²⋊C₄), SmallGroup(128,549)

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

Derived series

C₁ — C₂² — C₂³.₃₂M₄(2)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₂²×C₄² — C₂³.₃₂M₄(2)

Lower central

C₁ — C₂² — C₂³.₃₂M₄(2)

Upper central

C₁ — C₂²×C₄ — C₂³.₃₂M₄(2)

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³.₃₂M₄(2)

Generators and relations for C₂³.₃₂M₄(2)
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=b, dad^-1=ab=ba, ac=ca, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede^-1=cd⁵ >

Subgroups: 308 in 190 conjugacy classes, 80 normal (16 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂×C₈, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, C₂×C₄², C₂×C₄², C₂²×C₈, C₂³×C₄, C₂².₇C₄², C₂×C₂²⋊C₈, C₂²×C₄², C₂³.₃₂M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₂²⋊C₈, C₂×C₂²⋊C₄, C₄²⋊C₂, C₂².D₄, C₂²×C₈, C₂×M₄(2), C₂³.₃₄D₄, C₂×C₂²⋊C₈, C₂⁴.₄C₄, C₄².₁₂C₄, C₄².₆C₄, C₂³.₃₂M₄(2)

Smallest permutation representation of C₂³.₃₂M₄(2)
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4AB	8A	···	8P
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	2	2	2	2	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+				+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₈	D₄	M₄(2)	C₄○D₄	M₄(2)
kernel	C₂³.₃₂M₄(2)	C₂².₇C₄²	C₂×C₂²⋊C₈	C₂²×C₄²	C₂×C₄²	C₂³×C₄	C₂²×C₄	C₂²×C₄	C₂×C₄	C₂×C₄	C₂³
# reps	1	4	2	1	4	4	16	4	8	8	4

Matrix representation of C₂³.₃₂M₄(2) ►in GL₅(𝔽₁₇)

16	0	0	0	0
0	1	0	0	0
0	9	16	0	0
0	0	0	16	0
0	0	0	0	16

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

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

9	0	0	0	0
0	2	9	0	0
0	0	15	0	0
0	0	0	2	0
0	0	0	9	15

16	0	0	0	0
0	13	0	0	0
0	0	13	0	0
0	0	0	1	9
0	0	0	0	16

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,9,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[9,0,0,0,0,0,2,0,0,0,0,9,15,0,0,0,0,0,2,9,0,0,0,0,15],[16,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,1,0,0,0,0,9,16] >;

C₂³.₃₂M₄(2) in GAP, Magma, Sage, TeX

C_2^3._{32}M_4(2)

% in TeX

G:=Group("C2^3.32M4(2)");

// GroupNames label

G:=SmallGroup(128,549);

// by ID

G=gap.SmallGroup(128,549);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=b,d*a*d^-1=a*b=b*a,a*c=c*a,a*e=e*a,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 = C23.32M4(2) order 128 = 27

9th non-split extension by C23 of M4(2) acting via M4(2)/C2×C4=C2

G = C₂³.₃₂M₄(2) order 128 = 2⁷

9^th non-split extension by C₂³ of M₄(2) acting via M₄(2)/C₂×C₄=C₂