C2^3xM4(2) - GroupNames

G = C₂³×M₄(2) order 128 = 2⁷

Direct product of C₂³ and M₄(2)

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

Aliases: C₂³×M₄(2), C₈⋊₄C₂⁴, C₂⁵.₉C₄, C₄.₁₉C₂⁵, (C₂³×C₈)⋊₁₆C₂, (C₂×C₈)⋊₁₇C₂³, (C₂⁴×C₄).₁₅C₂, C₂.₁₃(C₂⁴×C₄), C₄.₅₅(C₂³×C₄), (C₂³×C₄).₄₄C₄, C₄ ○(C₂²×M₄(2)), M₄(2)○(C₂²×C₄), (C₂×C₄).₆₉₄C₂⁴, C₂⁴.₁₃₂(C₂×C₄), (C₂²×C₈)⋊₇₁C₂², C₂².₄₉(C₂³×C₄), C₂³.₂₃₅(C₂²×C₄), (C₂³×C₄).₇₀₉C₂², (C₂²×C₄).₁₆₅₂C₂³, (C₂×C₄)○₂(C₂×M₄(2)), (C₂²×C₄)○(C₂×M₄(2)), (C₂×C₄)○(C₂²×M₄(2)), (C₂²×C₄).₄₉₉(C₂×C₄), (C₂×C₄).₅₇₆(C₂²×C₄), (C₂²×C₄)○(C₂²×M₄(2)), SmallGroup(128,2302)

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

Derived series

C₁ — C₂ — C₂³×M₄(2)

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂⁴×C₄ — C₂³×M₄(2)

Lower central

C₁ — C₂ — C₂³×M₄(2)

Upper central

C₁ — C₂³×C₄ — C₂³×M₄(2)

Jennings

C₁ — C₂ — C₂ — C₄ — C₂³×M₄(2)

Generators and relations for C₂³×M₄(2)
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d⁵ >

Subgroups: 988 in 860 conjugacy classes, 732 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂³, C₂³, C₂×C₈, M₄(2), C₂²×C₄, C₂⁴, C₂⁴, C₂⁴, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂⁵, C₂³×C₈, C₂²×M₄(2), C₂⁴×C₄, C₂³×M₄(2)
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, M₄(2), C₂²×C₄, C₂⁴, C₂×M₄(2), C₂³×C₄, C₂⁵, C₂²×M₄(2), C₂⁴×C₄, C₂³×M₄(2)

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

Generators in S₆₄

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

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

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

(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)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 37)(35 39)(42 46)(44 48)(49 53)(51 55)(58 62)(60 64)

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

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

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

80 conjugacy classes

class	1	2A	···	2O	2P	···	2W	4A	···	4P	4Q	···	4X	8A	···	8AF
order	1	2	···	2	2	···	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	2	···	2	1	···	1	2	···	2	2	···	2

80 irreducible representations

dim	1	1	1	1	1	1	2
type	+	+	+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	M₄(2)
kernel	C₂³×M₄(2)	C₂³×C₈	C₂²×M₄(2)	C₂⁴×C₄	C₂³×C₄	C₂⁵	C₂³
# reps	1	2	28	1	30	2	16

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

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

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

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

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

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

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[13,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,16,0],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,16] >;

C₂³×M₄(2) in GAP, Magma, Sage, TeX

C_2^3\times M_4(2)

% in TeX

G:=Group("C2^3xM4(2)");

// GroupNames label

G:=SmallGroup(128,2302);

// by ID

G=gap.SmallGroup(128,2302);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,-2,224,925,124]);

// Polycyclic

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

// generators/relations

G = C23×M4(2) order 128 = 27

Direct product of C23 and M4(2)

G = C₂³×M₄(2) order 128 = 2⁷

Direct product of C₂³ and M₄(2)