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G = C22×C41D4order 128 = 27

Direct product of C22 and C41D4

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

Aliases: C22×C41D4, C4222C23, C25.72C22, C23.14C24, C22.29C25, C24.476C23, C41(C22×D4), (C22×C4)⋊49D4, (C2×D4)⋊16C23, (D4×C23)⋊11C2, C2.10(D4×C23), (C2×C4).597C24, (C2×C42)⋊91C22, (C22×C42)⋊23C2, C23.891(C2×D4), (C22×D4)⋊59C22, (C23×C4).705C22, C22.159(C22×D4), (C22×C4).1583C23, (C2×C4)⋊15(C2×D4), SmallGroup(128,2172)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22×C41D4
C1C2C22C23C24C23×C4C22×C42 — C22×C41D4
C1C22 — C22×C41D4
C1C24 — C22×C41D4
C1C22 — C22×C41D4

Generators and relations for C22×C41D4
 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=c-1, ede=d-1 >

Subgroups: 2796 in 1500 conjugacy classes, 556 normal (5 characteristic)
C1, C2 [×15], C2 [×16], C4 [×24], C22, C22 [×34], C22 [×144], C2×C4 [×84], D4 [×192], C23 [×31], C23 [×208], C42 [×16], C22×C4 [×42], C2×D4 [×96], C2×D4 [×288], C24, C24 [×24], C24 [×48], C2×C42 [×12], C41D4 [×64], C23×C4 [×3], C22×D4 [×72], C22×D4 [×48], C25 [×4], C22×C42, C2×C41D4 [×24], D4×C23 [×6], C22×C41D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×24], C23 [×155], C2×D4 [×84], C24 [×31], C41D4 [×16], C22×D4 [×42], C25, C2×C41D4 [×12], D4×C23 [×3], C22×C41D4

Smallest permutation representation of C22×C41D4
On 64 points
Generators in S64
(1 24)(2 21)(3 22)(4 23)(5 56)(6 53)(7 54)(8 55)(9 48)(10 45)(11 46)(12 47)(13 64)(14 61)(15 62)(16 63)(17 35)(18 36)(19 33)(20 34)(25 38)(26 39)(27 40)(28 37)(29 43)(30 44)(31 41)(32 42)(49 60)(50 57)(51 58)(52 59)
(1 18)(2 19)(3 20)(4 17)(5 46)(6 47)(7 48)(8 45)(9 54)(10 55)(11 56)(12 53)(13 58)(14 59)(15 60)(16 57)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(29 40)(30 37)(31 38)(32 39)(49 62)(50 63)(51 64)(52 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 31 59 55)(2 32 60 56)(3 29 57 53)(4 30 58 54)(5 21 42 49)(6 22 43 50)(7 23 44 51)(8 24 41 52)(9 17 37 13)(10 18 38 14)(11 19 39 15)(12 20 40 16)(25 61 45 36)(26 62 46 33)(27 63 47 34)(28 64 48 35)
(1 50)(2 49)(3 52)(4 51)(5 56)(6 55)(7 54)(8 53)(9 48)(10 47)(11 46)(12 45)(13 35)(14 34)(15 33)(16 36)(17 64)(18 63)(19 62)(20 61)(21 60)(22 59)(23 58)(24 57)(25 40)(26 39)(27 38)(28 37)(29 41)(30 44)(31 43)(32 42)

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2AE4A···4X
order12···22···24···4
size11···14···42···2

56 irreducible representations

dim11112
type+++++
imageC1C2C2C2D4
kernelC22×C41D4C22×C42C2×C41D4D4×C23C22×C4
# reps1124624

Matrix representation of C22×C41D4 in GL6(ℤ)

-100000
010000
001000
000100
0000-10
00000-1
,
-100000
0-10000
00-1000
000-100
000010
000001
,
100000
0-10000
00-1000
000-100
00001-1
00002-1
,
-100000
0-10000
000-100
001000
000010
000001
,
100000
0-10000
001000
000-100
000010
00002-1

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,2,0,0,0,0,-1,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,2,0,0,0,0,0,-1] >;

C22×C41D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes_1D_4
% in TeX

G:=Group("C2^2xC4:1D4");
// GroupNames label

G:=SmallGroup(128,2172);
// by ID

G=gap.SmallGroup(128,2172);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,232,1430,352]);
// 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^-1,e*d*e=d^-1>;
// generators/relations

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