C2^3:2M4(2) - GroupNames

G = C₂³⋊₂M₄(2) order 128 = 2⁷

1^st semidirect product of C₂³ and M₄(2) acting via M₄(2)/C₄=C₂²

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

Aliases: C₂³⋊₂M₄(2), (C₂×C₈)⋊₂₂D₄, (C₂×C₄)⋊₃M₄(2), C₂⁴.₅₇(C₂×C₄), C₄.₁₂₂C₂²≀C₂, C₂.₁₇(C₈⋊₉D₄), C₂.₁₃(C₈⋊₆D₄), (C₂²×D₄).₃₀C₄, C₂².₁₄₉(C₄×D₄), (C₂²×C₄).₂₈₄D₄, C₄.₁₈₆(C₄⋊D₄), C₂².₅₂(C₈○D₄), (C₂²×M₄(2))⋊₉C₂, (C₂²×C₈).₃₀C₂², C₂³.₇₈(C₂²⋊C₄), (C₂×C₄²).₂₇₄C₂², C₂³.₃₁₃(C₂²×C₄), (C₂³×C₄).₂₅₂C₂², C₂².₆₆(C₂×M₄(2)), C₂.₁₈(C₂⁴.₄C₄), (C₂²×C₄).₁₆₃₀C₂³, C₂².₇C₄²⋊₄₀C₂, C₂.₉(C₂³.₂₃D₄), C₄.₁₃₅(C₂².D₄), (C₂×C₄×D₄).₂₀C₂, (C₂×C₄⋊C₄).₅₅C₄, (C₂×C₂²⋊C₈)⋊₃₇C₂, (C₂×C₄).₁₅₂₉(C₂×D₄), (C₂×C₂²⋊C₄).₃₉C₄, (C₂×C₄).₉₃₆(C₄○D₄), (C₂²×C₄).₁₁₉(C₂×C₄), (C₂×C₄).₁₂₉(C₂²⋊C₄), C₂².₂₆₁(C₂×C₂²⋊C₄), C₂.₂₇((C₂²×C₈)⋊C₂), SmallGroup(128,602)

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

Derived series

C₁ — C₂³ — C₂³⋊₂M₄(2)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×M₄(2) — C₂³⋊₂M₄(2)

Lower central

C₁ — C₂³ — C₂³⋊₂M₄(2)

Upper central

C₁ — C₂²×C₄ — C₂³⋊₂M₄(2)

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — C₂³⋊₂M₄(2)

Generators and relations for C₂³⋊₂M₄(2)
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, eae=ac=ca, dad^-1=abc, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d⁵ >

Subgroups: 380 in 204 conjugacy classes, 68 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂²×D₄, C₂².₇C₄², C₂×C₂²⋊C₈, C₂×C₂²⋊C₈, C₂×C₄×D₄, C₂²×M₄(2), C₂³⋊₂M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, M₄(2), C₂²×C₄, C₂×D₄, C₄○D₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₂×M₄(2), C₈○D₄, C₂³.₂₃D₄, C₂⁴.₄C₄, (C₂²×C₈)⋊C₂, C₈⋊₉D₄, C₈⋊₆D₄, C₂³⋊₂M₄(2)

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

Generators in S₆₄

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

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

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

(2 6)(4 8)(9 17)(10 22)(11 19)(12 24)(13 21)(14 18)(15 23)(16 20)(25 57)(26 62)(27 59)(28 64)(29 61)(30 58)(31 63)(32 60)(34 38)(36 40)(41 45)(43 47)(50 54)(52 56)

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

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

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

44 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	···	4H	4I	···	4P	8A	···	8P
order	1	2	···	2	2	2	2	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	4	4	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	2
type	+	+	+	+	+				+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	M₄(2)	C₄○D₄	M₄(2)	C₈○D₄
kernel	C₂³⋊₂M₄(2)	C₂².₇C₄²	C₂×C₂²⋊C₈	C₂×C₄×D₄	C₂²×M₄(2)	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₂²×D₄	C₂×C₈	C₂²×C₄	C₂×C₄	C₂×C₄	C₂³	C₂²
# reps	1	2	3	1	1	4	2	2	4	4	4	4	4	8

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

1	4	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	2
0	0	0	0	0	16

16	0	0	0	0	0
0	16	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	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

16	6	0	0	0	0
9	1	0	0	0	0
0	0	0	16	0	0
0	0	13	0	0	0
0	0	0	0	16	0
0	0	0	0	1	1

1	4	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	0
0	0	0	0	16	16

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

C₂³⋊₂M₄(2) in GAP, Magma, Sage, TeX

C_2^3\rtimes_2M_4(2)

% in TeX

G:=Group("C2^3:2M4(2)");

// GroupNames label

G:=SmallGroup(128,602);

// by ID

G=gap.SmallGroup(128,602);

# by ID

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

// Polycyclic

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

// generators/relations

G = C23⋊2M4(2) order 128 = 27

1st semidirect product of C23 and M4(2) acting via M4(2)/C4=C22

G = C₂³⋊₂M₄(2) order 128 = 2⁷

1^st semidirect product of C₂³ and M₄(2) acting via M₄(2)/C₄=C₂²