C2^2xC4:D4 - GroupNames

G = C₂²×C₄⋊D₄ order 128 = 2⁷

Direct product of C₂² and C₄⋊D₄

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

Aliases: C₂²×C₄⋊D₄, C₂⁴⋊₁₄D₄, C₂⁵.₆₈C₂², C₂³.₁₁C₂⁴, C₂².₂₁C₂⁵, C₂⁴.₆₀₆C₂³, (C₂⁴×C₄)⋊₈C₂, C₄⋊₃(C₂²×D₄), C₂³⋊₈(C₂×D₄), C₄⋊C₄⋊₁₃C₂³, (C₂²×C₄)⋊₅₅D₄, C₂.₅(D₄×C₂³), (D₄×C₂³)⋊₁₀C₂, (C₂×D₄)⋊₁₄C₂³, C₂²⋊₁(C₂²×D₄), C₂²⋊C₄⋊₁₅C₂³, (C₂×C₄).₅₉₆C₂⁴, (C₂³×C₄)⋊₅₉C₂², (C₂²×C₄)⋊₂₄C₂³, (C₂²×D₄)⋊₅₇C₂², C₂³.₃₇₇(C₄○D₄), (C₂×C₄)⋊₂₁(C₂×D₄), (C₂²×C₄⋊C₄)⋊₃₇C₂, C₂.₅(C₂²×C₄○D₄), (C₂×C₄⋊C₄)⋊₁₂₂C₂², (C₂²×C₂²⋊C₄)⋊₂₇C₂, (C₂×C₂²⋊C₄)⋊₈₁C₂², C₂².₁₄₆(C₂×C₄○D₄), SmallGroup(128,2164)

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

Derived series

C₁ — C₂² — C₂²×C₄⋊D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁴×C₄ — C₂²×C₄⋊D₄

Lower central

C₁ — C₂² — C₂²×C₄⋊D₄

Upper central

C₁ — C₂⁴ — C₂²×C₄⋊D₄

Jennings

C₁ — C₂² — C₂²×C₄⋊D₄

Generators and relations for C₂²×C₄⋊D₄
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, dcd^-1=ece=c^-1, ede=d^-1 >

Subgroups: 2188 in 1264 conjugacy classes, 516 normal (13 characteristic)
C₁, C₂, C₂, C₂, C₄, 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, C₄⋊D₄, C₂³×C₄, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂⁵, C₂²×C₂²⋊C₄, C₂²×C₄⋊C₄, C₂×C₄⋊D₄, C₂⁴×C₄, D₄×C₂³, D₄×C₂³, C₂²×C₄⋊D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂⁴, C₄⋊D₄, C₂²×D₄, C₂×C₄○D₄, C₂⁵, C₂×C₄⋊D₄, D₄×C₂³, C₂²×C₄○D₄, C₂²×C₄⋊D₄

Smallest permutation representation of C₂²×C₄⋊D₄
►On 64 points

Generators in S₆₄

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

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

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

(2 4)(5 61)(6 64)(7 63)(8 62)(9 44)(10 43)(11 42)(12 41)(14 16)(18 20)(21 37)(22 40)(23 39)(24 38)(25 57)(26 60)(27 59)(28 58)(30 32)(33 35)(46 48)(50 52)(54 56)

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

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

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

56 conjugacy classes

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

56 irreducible representations

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

Matrix representation of C₂²×C₄⋊D₄ ►in GL₇(𝔽₅)

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

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

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

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

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

G:=sub<GL(7,GF(5))| [4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,4,1,0,0,0,0,0,3,1],[1,0,0,0,0,0,0,0,4,2,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1,4,0,0,0,0,0,2,4,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,4] >;

C₂²×C₄⋊D₄ in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes D_4

% in TeX

G:=Group("C2^2xC4:D4");

// GroupNames label

G:=SmallGroup(128,2164);

// by ID

G=gap.SmallGroup(128,2164);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430]);

// 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,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = C22×C4⋊D4 order 128 = 27

Direct product of C22 and C4⋊D4

G = C₂²×C₄⋊D₄ order 128 = 2⁷

Direct product of C₂² and C₄⋊D₄