C2^2xD4:C4 - GroupNames

G = C₂²×D₄⋊C₄ order 128 = 2⁷

Direct product of C₂² and D₄⋊C₄

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂²×D₄⋊C₄, C₂³.₆₁D₈, C₂⁴.₁₈₉D₄, C₂³.₄₆SD₁₆, (C₂³×C₈)⋊₅C₂, C₄⋊C₄⋊₁₁C₂³, (C₂×C₈)⋊₁₃C₂³, D₄⋊₄(C₂²×C₄), C₄.₁(C₂³×C₄), C₂.₁(C₂²×D₈), (C₂²×D₄)⋊₂₆C₄, C₂².₆₇(C₂×D₈), (C₂×C₄).₁₇₁C₂⁴, (C₂²×C₈)⋊₆₂C₂², (D₄×C₂³).₁₅C₂, C₄.₁₃₆(C₂²×D₄), (C₂²×C₄).₆₀₀D₄, C₂³.₈₃₆(C₂×D₄), C₂.₁(C₂²×SD₁₆), (C₂×D₄).₃₅₇C₂³, C₂².₇₈(C₂×SD₁₆), (C₂³×C₄).₆₈₈C₂², C₂².₁₂₁(C₂²×D₄), C₂³.₂₃₂(C₂²⋊C₄), (C₂²×C₄).₁₄₉₅C₂³, (C₂²×D₄).₅₅₀C₂², (C₂×D₄)⋊₄₆(C₂×C₄), (C₂²×C₄⋊C₄)⋊₃₁C₂, C₄.₇₁(C₂×C₂²⋊C₄), (C₂×C₄⋊C₄)⋊₁₁₂C₂², (C₂×C₄).₁₄₀₂(C₂×D₄), (C₂×C₄).₄₅₆(C₂²×C₄), (C₂²×C₄).₄₁₃(C₂×C₄), C₂.₃₃(C₂²×C₂²⋊C₄), (C₂×C₄).₂₈₂(C₂²⋊C₄), C₂².₁₃₆(C₂×C₂²⋊C₄), SmallGroup(128,1622)

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

Derived series

C₁ — C₄ — C₂²×D₄⋊C₄

Chief series

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

Lower central

C₁ — C₂ — C₄ — C₂²×D₄⋊C₄

Upper central

C₁ — C₂⁴ — C₂³×C₄ — C₂²×D₄⋊C₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂²×D₄⋊C₄

Generators and relations for C₂²×D₄⋊C₄
G = < a,b,c,d,e | a²=b²=c⁴=d²=e⁴=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=ece^-1=c^-1, ede^-1=cd >

Subgroups: 1148 in 536 conjugacy classes, 220 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, C₂³, C₂³, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, D₄⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁵, C₂×D₄⋊C₄, C₂²×C₄⋊C₄, C₂³×C₈, D₄×C₂³, C₂²×D₄⋊C₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂⁴, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×D₈, C₂×SD₁₆, C₂³×C₄, C₂²×D₄, C₂×D₄⋊C₄, C₂²×C₂²⋊C₄, C₂²×D₈, C₂²×SD₁₆, C₂²×D₄⋊C₄

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

Generators in S₆₄

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2W	4A	···	4H	4I	···	4P	8A	···	8P
order	1	2	···	2	2	···	2	4	···	4	4	···	4	8	···	8
size	1	1	···	1	4	···	4	2	···	2	4	···	4	2	···	2

56 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+	+	+		+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₄	D₄	D₄	D₈	SD₁₆
kernel	C₂²×D₄⋊C₄	C₂×D₄⋊C₄	C₂²×C₄⋊C₄	C₂³×C₈	D₄×C₂³	C₂²×D₄	C₂²×C₄	C₂⁴	C₂³	C₂³
# reps	1	12	1	1	1	16	7	1	8	8

Matrix representation of C₂²×D₄⋊C₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	1	0	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	16

1	0	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	16	0
0	0	0	0	0	16

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

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

16	0	0	0	0	0
0	16	0	0	0	0
0	0	1	2	0	0
0	0	16	16	0	0
0	0	0	0	12	5
0	0	0	0	5	5

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,12,5,0,0,0,0,5,5] >;

C₂²×D₄⋊C₄ in GAP, Magma, Sage, TeX

C_2^2\times D_4\rtimes C_4

% in TeX

G:=Group("C2^2xD4:C4");

// GroupNames label

G:=SmallGroup(128,1622);

// by ID

G=gap.SmallGroup(128,1622);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,2804,1411,172]);

// Polycyclic

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

// generators/relations

G = C22×D4⋊C4 order 128 = 27

Direct product of C22 and D4⋊C4

G = C₂²×D₄⋊C₄ order 128 = 2⁷

Direct product of C₂² and D₄⋊C₄