C2^2xC2^2.D4 - GroupNames

G = C₂²×C₂².D₄ order 128 = 2⁷

Direct product of C₂² and C₂².D₄

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

Aliases: C₂²×C₂².D₄, C₂⁴.₁₈₄D₄, C₂⁵.₆₉C₂², C₂².₂₃C₂⁵, C₂³.₂₆₈C₂⁴, C₂⁴.₆₅₆C₂³, (C₂⁴×C₄)⋊₆C₂, C₄⋊C₄⋊₁₅C₂³, C₂.₇(D₄×C₂³), (C₂×C₄).₂₈C₂⁴, C₂²⋊C₄⋊₁₆C₂³, (C₂²×C₄)⋊₂₃C₂³, (C₂³×C₄)⋊₅₈C₂², (D₄×C₂³).₁₉C₂, C₂³.₇₀₆(C₂×D₄), (C₂×D₄).₄₄₂C₂³, C₂².₄₅(C₂²×D₄), C₂³.₃₇₉(C₄○D₄), (C₂²×D₄).₅₈₁C₂², (C₂²×C₄⋊C₄)⋊₃₉C₂, C₂.₇(C₂²×C₄○D₄), (C₂×C₄⋊C₄)⋊₁₂₄C₂², (C₂²×C₂²⋊C₄)⋊₂₈C₂, (C₂×C₂²⋊C₄)⋊₈₂C₂², C₂².₁₄₈(C₂×C₄○D₄), SmallGroup(128,2166)

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

Derived series

C₁ — C₂² — C₂²×C₂².D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂⁴×C₄ — C₂²×C₂².D₄

Lower central

C₁ — C₂² — C₂²×C₂².D₄

Upper central

C₁ — C₂⁴ — C₂²×C₂².D₄

Jennings

C₁ — C₂² — C₂²×C₂².D₄

Generators and relations for C₂²×C₂².D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece^-1=fcf=cd=dc, de=ed, df=fd, fef=de^-1 >

Subgroups: 1580 in 984 conjugacy classes, 476 normal (11 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂⁴, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂³×C₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂²×C₂²⋊C₄, C₂²×C₂²⋊C₄, C₂²×C₄⋊C₄, C₂×C₂².D₄, C₂⁴×C₄, D₄×C₂³, C₂²×C₂².D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₂².D₄, C₂²×D₄, C₂×C₄○D₄, C₂⁵, C₂×C₂².D₄, D₄×C₂³, C₂²×C₄○D₄, C₂²×C₂².D₄

Smallest permutation representation of C₂²×C₂².D₄
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2W	2X	2Y	2Z	2AA	4A	···	4P	4Q	···	4AB
order	1	2	···	2	2	···	2	2	2	2	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	4	4	4	4	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	2	2
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	C₄○D₄
kernel	C₂²×C₂².D₄	C₂²×C₂²⋊C₄	C₂²×C₄⋊C₄	C₂×C₂².D₄	C₂⁴×C₄	D₄×C₂³	C₂⁴	C₂³
# reps	1	3	2	24	1	1	8	16

Matrix representation of C₂²×C₂².D₄ ►in GL₆(𝔽₅)

4	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
0	0	0	0	0	4

1	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	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	4	0	0	0	0
0	0	0	4	0	0
0	0	4	0	0	0
0	0	0	0	0	3
0	0	0	0	2	0

1	0	0	0	0	0
0	1	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

1	0	0	0	0	0
0	4	0	0	0	0
0	0	2	0	0	0
0	0	0	3	0	0
0	0	0	0	0	4
0	0	0	0	4	0

1	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	1	0	0
0	0	0	0	4	0
0	0	0	0	0	1

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,3,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,4,0],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;

C₂²×C₂².D₄ in GAP, Magma, Sage, TeX

C_2^2\times C_2^2.D_4

% in TeX

G:=Group("C2^2xC2^2.D4");

// GroupNames label

G:=SmallGroup(128,2166);

// by ID

G=gap.SmallGroup(128,2166);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,184]);

// Polycyclic

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

// generators/relations

G = C22×C22.D4 order 128 = 27

Direct product of C22 and C22.D4

G = C₂²×C₂².D₄ order 128 = 2⁷

Direct product of C₂² and C₂².D₄