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G = C22×C22.D4order 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⋊C415C23, C2.7(D4×C23), (C2×C4).28C24, C22⋊C416C23, (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

C1C22 — C22×C22.D4
C1C2C22C23C24C23×C4C24×C4 — C22×C22.D4
C1C22 — C22×C22.D4
C1C24 — C22×C22.D4
C1C22 — 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 [×14], C2 [×12], C4 [×20], C22, C22 [×42], C22 [×92], C2×C4 [×20], C2×C4 [×92], D4 [×32], C23 [×47], C23 [×108], C22⋊C4 [×48], C4⋊C4 [×32], C22×C4 [×46], C22×C4 [×68], C2×D4 [×16], C2×D4 [×48], C24, C24 [×20], C24 [×20], C2×C22⋊C4 [×36], C2×C4⋊C4 [×24], C22.D4 [×64], C23×C4, C23×C4 [×16], C23×C4 [×8], C22×D4 [×12], C22×D4 [×8], C25 [×2], C22×C22⋊C4, C22×C22⋊C4 [×2], C22×C4⋊C4 [×2], C2×C22.D4 [×24], C24×C4, D4×C23, C22×C22.D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×8], C24 [×31], C22.D4 [×16], C22×D4 [×14], C2×C4○D4 [×12], C25, C2×C22.D4 [×12], D4×C23, C22×C4○D4 [×2], 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 51)(38 52)(39 49)(40 50)(41 54)(42 55)(43 56)(44 53)(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 56)(38 53)(39 54)(40 55)(41 49)(42 50)(43 51)(44 52)(45 60)(46 57)(47 58)(48 59)
(1 44)(2 64)(3 42)(4 62)(5 52)(6 19)(7 50)(8 17)(9 55)(10 45)(11 53)(12 47)(13 49)(14 20)(15 51)(16 18)(21 58)(22 40)(23 60)(24 38)(25 63)(26 41)(27 61)(28 43)(29 59)(30 37)(31 57)(32 39)(33 54)(34 48)(35 56)(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 51)(18 52)(19 49)(20 50)(21 32)(22 29)(23 30)(24 31)(37 60)(38 57)(39 58)(40 59)(41 64)(42 61)(43 62)(44 63)(45 56)(46 53)(47 54)(48 55)
(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 55)(19 47)(20 53)(21 26)(23 28)(25 29)(27 31)(37 43)(38 61)(39 41)(40 63)(42 57)(44 59)(46 50)(48 52)(49 54)(51 56)(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)(37,51)(38,52)(39,49)(40,50)(41,54)(42,55)(43,56)(44,53)(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)(37,56)(38,53)(39,54)(40,55)(41,49)(42,50)(43,51)(44,52)(45,60)(46,57)(47,58)(48,59), (1,44)(2,64)(3,42)(4,62)(5,52)(6,19)(7,50)(8,17)(9,55)(10,45)(11,53)(12,47)(13,49)(14,20)(15,51)(16,18)(21,58)(22,40)(23,60)(24,38)(25,63)(26,41)(27,61)(28,43)(29,59)(30,37)(31,57)(32,39)(33,54)(34,48)(35,56)(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)(17,51)(18,52)(19,49)(20,50)(21,32)(22,29)(23,30)(24,31)(37,60)(38,57)(39,58)(40,59)(41,64)(42,61)(43,62)(44,63)(45,56)(46,53)(47,54)(48,55), (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)(18,55)(19,47)(20,53)(21,26)(23,28)(25,29)(27,31)(37,43)(38,61)(39,41)(40,63)(42,57)(44,59)(46,50)(48,52)(49,54)(51,56)(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)(37,51)(38,52)(39,49)(40,50)(41,54)(42,55)(43,56)(44,53)(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)(37,56)(38,53)(39,54)(40,55)(41,49)(42,50)(43,51)(44,52)(45,60)(46,57)(47,58)(48,59), (1,44)(2,64)(3,42)(4,62)(5,52)(6,19)(7,50)(8,17)(9,55)(10,45)(11,53)(12,47)(13,49)(14,20)(15,51)(16,18)(21,58)(22,40)(23,60)(24,38)(25,63)(26,41)(27,61)(28,43)(29,59)(30,37)(31,57)(32,39)(33,54)(34,48)(35,56)(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)(17,51)(18,52)(19,49)(20,50)(21,32)(22,29)(23,30)(24,31)(37,60)(38,57)(39,58)(40,59)(41,64)(42,61)(43,62)(44,63)(45,56)(46,53)(47,54)(48,55), (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)(18,55)(19,47)(20,53)(21,26)(23,28)(25,29)(27,31)(37,43)(38,61)(39,41)(40,63)(42,57)(44,59)(46,50)(48,52)(49,54)(51,56)(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),(37,51),(38,52),(39,49),(40,50),(41,54),(42,55),(43,56),(44,53),(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),(37,56),(38,53),(39,54),(40,55),(41,49),(42,50),(43,51),(44,52),(45,60),(46,57),(47,58),(48,59)], [(1,44),(2,64),(3,42),(4,62),(5,52),(6,19),(7,50),(8,17),(9,55),(10,45),(11,53),(12,47),(13,49),(14,20),(15,51),(16,18),(21,58),(22,40),(23,60),(24,38),(25,63),(26,41),(27,61),(28,43),(29,59),(30,37),(31,57),(32,39),(33,54),(34,48),(35,56),(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),(17,51),(18,52),(19,49),(20,50),(21,32),(22,29),(23,30),(24,31),(37,60),(38,57),(39,58),(40,59),(41,64),(42,61),(43,62),(44,63),(45,56),(46,53),(47,54),(48,55)], [(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),(18,55),(19,47),(20,53),(21,26),(23,28),(25,29),(27,31),(37,43),(38,61),(39,41),(40,63),(42,57),(44,59),(46,50),(48,52),(49,54),(51,56),(58,64),(60,62)])

56 conjugacy classes

class 1 2A···2O2P···2W2X2Y2Z2AA4A···4P4Q···4AB
order12···22···222224···44···4
size11···12···244442···24···4

56 irreducible representations

dim11111122
type+++++++
imageC1C2C2C2C2C2D4C4○D4
kernelC22×C22.D4C22×C22⋊C4C22×C4⋊C4C2×C22.D4C24×C4D4×C23C24C23
# reps1322411816

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

400000
040000
004000
000400
000040
000004
,
100000
040000
004000
000400
000010
000001
,
100000
040000
000400
004000
000003
000020
,
100000
010000
004000
000400
000040
000004
,
100000
040000
002000
000300
000004
000040
,
100000
040000
004000
000100
000040
000001

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

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