C2^2xC4^2:C2 - GroupNames

G = C₂²×C₄²⋊C₂ order 128 = 2⁷

Direct product of C₂² and C₄²⋊C₂

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

Aliases: C₂²×C₄²⋊C₂, C₄²⋊₁₇C₂³, C₂².₇C₂⁵, C₂⁵.₉₃C₂², C₂³.₁₀₂C₂⁴, C₂⁴.₆₅₄C₂³, C₄⋊C₄⋊₂₂C₂³, (C₂³×C₄)⋊₁₉C₄, C₂.₃(C₂⁴×C₄), (C₂⁴×C₄).₁₃C₂, C₄.₆₅(C₂³×C₄), (C₂²×C₄²)⋊₉C₂, C₂⁴.₁₃₁(C₂×C₄), (C₂×C₄).₁₅₄C₂⁴, (C₂×C₄²)⋊₈₃C₂², C₂².₄₇(C₂³×C₄), C₂³.₃₇₄(C₄○D₄), C₂²⋊C₄.₁₂₃C₂³, (C₂³×C₄).₇₀₄C₂², C₂³.₂₃₃(C₂²×C₄), (C₂²×C₄).₁₂₉₁C₂³, C₄ ○(C₂²×C₄⋊C₄), C₄⋊C₄ ○₂(C₂²×C₄), C₄ ○(C₂×C₄²⋊C₂), C₄ ○(C₂²×C₂²⋊C₄), (C₂²×C₄⋊C₄)⋊₄₈C₂, (C₂×C₄)⋊₁₃(C₂²×C₄), (C₂²×C₄)⋊₅₃(C₂×C₄), C₂²⋊C₄ ○₂(C₂²×C₄), C₂.₁(C₂²×C₄○D₄), (C₂×C₄⋊C₄)⋊₁₄₃C₂², (C₂×C₄)○₂(C₄²⋊C₂), C₂².₁₄₂(C₂×C₄○D₄), (C₂²×C₄)○(C₄²⋊C₂), (C₂²×C₂²⋊C₄).₃₁C₂, (C₂×C₂²⋊C₄).₅₅₉C₂², (C₂×C₄)○₄(C₂×C₄⋊C₄), (C₂×C₄)○(C₂²×C₄⋊C₄), (C₂×C₄)○₃(C₂×C₂²⋊C₄), (C₂²×C₄)○₂(C₂×C₄⋊C₄), (C₂×C₄)○(C₂×C₄²⋊C₂), (C₂²×C₄)○(C₂²×C₄⋊C₄), (C₂²×C₄)○(C₂×C₄²⋊C₂), SmallGroup(128,2153)

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

Derived series

C₁ — C₂ — C₂²×C₄²⋊C₂

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂⁴×C₄ — C₂²×C₄²⋊C₂

Lower central

C₁ — C₂ — C₂²×C₄²⋊C₂

Upper central

C₁ — C₂³×C₄ — C₂²×C₄²⋊C₂

Jennings

C₁ — C₂² — C₂²×C₄²⋊C₂

Generators and relations for C₂²×C₄²⋊C₂
G = < a,b,c,d,e | a²=b²=c⁴=d⁴=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=cd², de=ed >

Subgroups: 1148 in 940 conjugacy classes, 732 normal (9 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂³×C₄, C₂³×C₄, C₂⁵, C₂²×C₄², C₂²×C₂²⋊C₄, C₂²×C₄⋊C₄, C₂×C₄²⋊C₂, C₂⁴×C₄, C₂²×C₄²⋊C₂
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, C₂⁴, C₄²⋊C₂, C₂³×C₄, C₂×C₄○D₄, C₂⁵, C₂×C₄²⋊C₂, C₂⁴×C₄, C₂²×C₄○D₄, C₂²×C₄²⋊C₂

Smallest permutation representation of C₂²×C₄²⋊C₂
►On 64 points

Generators in S₆₄

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

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

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

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

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

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

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

80 conjugacy classes

class	1	2A	···	2O	2P	···	2W	4A	···	4P	4Q	···	4BD
order	1	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	···	1	2	···	2	1	···	1	2	···	2

80 irreducible representations

dim	1	1	1	1	1	1	1	2
type	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄○D₄
kernel	C₂²×C₄²⋊C₂	C₂²×C₄²	C₂²×C₂²⋊C₄	C₂²×C₄⋊C₄	C₂×C₄²⋊C₂	C₂⁴×C₄	C₂³×C₄	C₂³
# reps	1	2	2	2	24	1	32	16

Matrix representation of C₂²×C₄²⋊C₂ ►in GL₅(𝔽₅)

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

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

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

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

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

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

C₂²×C₄²⋊C₂ in GAP, Magma, Sage, TeX

C_2^2\times C_4^2\rtimes C_2

% in TeX

G:=Group("C2^2xC4^2:C2");

// GroupNames label

G:=SmallGroup(128,2153);

// by ID

G=gap.SmallGroup(128,2153);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,184]);

// Polycyclic

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

// generators/relations

G = C22×C42⋊C2 order 128 = 27

Direct product of C22 and C42⋊C2

G = C₂²×C₄²⋊C₂ order 128 = 2⁷

Direct product of C₂² and C₄²⋊C₂