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G = C22×C4.4D4order 128 = 27

Direct product of C22 and C4.4D4

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

Aliases: C22×C4.4D4, C4221C23, C25.70C22, C23.13C24, C22.25C25, C24.475C23, (Q8×C23)⋊8C2, C2.9(D4×C23), (C2×Q8)⋊15C23, C4.25(C22×D4), C22⋊C418C23, (C2×C4).163C24, (C22×C42)⋊21C2, (C2×C42)⋊89C22, (D4×C23).20C2, (C22×C4).625D4, C23.890(C2×D4), (C2×D4).443C23, (C22×Q8)⋊57C22, C23.380(C4○D4), (C23×C4).662C22, C22.158(C22×D4), (C22×C4).1172C23, (C22×D4).582C22, (C2×C4).878(C2×D4), C2.9(C22×C4○D4), (C22×C22⋊C4)⋊29C2, (C2×C22⋊C4)⋊84C22, C22.149(C2×C4○D4), SmallGroup(128,2168)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22×C4.4D4
C1C2C22C23C24C23×C4C22×C42 — C22×C4.4D4
C1C22 — C22×C4.4D4
C1C24 — C22×C4.4D4
C1C22 — C22×C4.4D4

Generators and relations for C22×C4.4D4
 G = < a,b,c,d,e | a2=b2=c4=d4=1, e2=c2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=c2d-1 >

Subgroups: 1580 in 940 conjugacy classes, 476 normal (9 characteristic)
C1, C2, C2 [×14], C2 [×8], C4 [×8], C4 [×16], C22, C22 [×34], C22 [×72], C2×C4 [×44], C2×C4 [×48], D4 [×32], Q8 [×32], C23 [×23], C23 [×104], C42 [×16], C22⋊C4 [×64], C22×C4 [×38], C22×C4 [×16], C2×D4 [×16], C2×D4 [×48], C2×Q8 [×16], C2×Q8 [×48], C24, C24 [×12], C24 [×24], C2×C42 [×12], C2×C22⋊C4 [×48], C4.4D4 [×64], C23×C4, C23×C4 [×4], C22×D4 [×12], C22×D4 [×8], C22×Q8 [×12], C22×Q8 [×8], C25 [×2], C22×C42, C22×C22⋊C4 [×4], C2×C4.4D4 [×24], D4×C23, Q8×C23, C22×C4.4D4
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C4○D4 [×8], C24 [×31], C4.4D4 [×16], C22×D4 [×14], C2×C4○D4 [×12], C25, C2×C4.4D4 [×12], D4×C23, C22×C4○D4 [×2], C22×C4.4D4

Smallest permutation representation of C22×C4.4D4
On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 55)(6 56)(7 53)(8 54)(9 47)(10 48)(11 45)(12 46)(13 63)(14 64)(15 61)(16 62)(17 57)(18 58)(19 59)(20 60)(21 33)(22 34)(23 35)(24 36)(25 41)(26 42)(27 43)(28 44)(29 37)(30 38)(31 39)(32 40)
(1 15)(2 16)(3 13)(4 14)(5 45)(6 46)(7 47)(8 48)(9 53)(10 54)(11 55)(12 56)(17 35)(18 36)(19 33)(20 34)(21 59)(22 60)(23 57)(24 58)(25 39)(26 40)(27 37)(28 38)(29 43)(30 44)(31 41)(32 42)(49 61)(50 62)(51 63)(52 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 54 17 43)(2 55 18 44)(3 56 19 41)(4 53 20 42)(5 58 28 50)(6 59 25 51)(7 60 26 52)(8 57 27 49)(9 34 32 14)(10 35 29 15)(11 36 30 16)(12 33 31 13)(21 39 63 46)(22 40 64 47)(23 37 61 48)(24 38 62 45)
(1 31 3 29)(2 30 4 32)(5 22 7 24)(6 21 8 23)(9 18 11 20)(10 17 12 19)(13 43 15 41)(14 42 16 44)(25 63 27 61)(26 62 28 64)(33 54 35 56)(34 53 36 55)(37 49 39 51)(38 52 40 50)(45 60 47 58)(46 59 48 57)

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

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

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

56 conjugacy classes

class 1 2A···2O2P···2W4A···4X4Y···4AF
order12···22···24···44···4
size11···14···42···24···4

56 irreducible representations

dim11111122
type+++++++
imageC1C2C2C2C2C2D4C4○D4
kernelC22×C4.4D4C22×C42C22×C22⋊C4C2×C4.4D4D4×C23Q8×C23C22×C4C23
# reps1142411816

Matrix representation of C22×C4.4D4 in GL7(𝔽5)

4000000
0100000
0010000
0001000
0000100
0000010
0000001
,
1000000
0400000
0040000
0001000
0000100
0000040
0000004
,
4000000
0200000
0430000
0003400
0000200
0000020
0000013
,
4000000
0200000
0020000
0003000
0000300
0000010
0000034
,
1000000
0320000
0020000
0003000
0003200
0000011
0000034

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

C22×C4.4D4 in GAP, Magma, Sage, TeX

C_2^2\times C_4._4D_4
% in TeX

G:=Group("C2^2xC4.4D4");
// GroupNames label

G:=SmallGroup(128,2168);
// by ID

G=gap.SmallGroup(128,2168);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,456,1430,184]);
// Polycyclic

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

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