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G = C22×C422C2order 128 = 27

Direct product of C22 and C422C2

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

Aliases: C22×C422C2, C4218C23, C25.71C22, C22.27C25, C23.270C24, C24.658C23, C4⋊C417C23, (C2×C4).31C24, (C2×C42)⋊84C22, (C22×C42)⋊10C2, C22⋊C4.70C23, C23.382(C4○D4), (C23×C4).663C22, (C22×C4).1174C23, (C22×C4⋊C4)⋊40C2, (C2×C4⋊C4)⋊126C22, C2.11(C22×C4○D4), C22.151(C2×C4○D4), (C22×C22⋊C4).28C2, (C2×C22⋊C4).525C22, SmallGroup(128,2170)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C22×C422C2
C1C2C22C23C24C23×C4C22×C42 — C22×C422C2
C1C22 — C22×C422C2
C1C24 — C22×C422C2
C1C22 — C22×C422C2

Generators and relations for C22×C422C2
 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=cd2, ede=c2d-1 >

Subgroups: 972 in 660 conjugacy classes, 436 normal (6 characteristic)
C1, C2 [×15], C2 [×4], C4 [×24], C22, C22 [×34], C22 [×36], C2×C4 [×24], C2×C4 [×72], C23 [×19], C23 [×52], C42 [×16], C22⋊C4 [×48], C4⋊C4 [×48], C22×C4 [×36], C22×C4 [×24], C24, C24 [×6], C24 [×12], C2×C42 [×12], C2×C22⋊C4 [×36], C2×C4⋊C4 [×36], C422C2 [×64], C23×C4 [×6], C25, C22×C42, C22×C22⋊C4 [×3], C22×C4⋊C4 [×3], C2×C422C2 [×24], C22×C422C2
Quotients: C1, C2 [×31], C22 [×155], C23 [×155], C4○D4 [×12], C24 [×31], C422C2 [×16], C2×C4○D4 [×18], C25, C2×C422C2 [×12], C22×C4○D4 [×3], C22×C422C2

Smallest permutation representation of C22×C422C2
On 64 points
Generators in S64
(1 29)(2 30)(3 31)(4 32)(5 39)(6 40)(7 37)(8 38)(9 34)(10 35)(11 36)(12 33)(13 47)(14 48)(15 45)(16 46)(17 51)(18 52)(19 49)(20 50)(21 63)(22 64)(23 61)(24 62)(25 55)(26 56)(27 53)(28 54)(41 59)(42 60)(43 57)(44 58)
(1 19)(2 20)(3 17)(4 18)(5 53)(6 54)(7 55)(8 56)(9 45)(10 46)(11 47)(12 48)(13 36)(14 33)(15 34)(16 35)(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 61)(42 62)(43 63)(44 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 25 9 59)(2 26 10 60)(3 27 11 57)(4 28 12 58)(5 13 63 51)(6 14 64 52)(7 15 61 49)(8 16 62 50)(17 39 47 21)(18 40 48 22)(19 37 45 23)(20 38 46 24)(29 55 34 41)(30 56 35 42)(31 53 36 43)(32 54 33 44)
(1 51)(2 14)(3 49)(4 16)(5 57)(6 28)(7 59)(8 26)(9 13)(10 52)(11 15)(12 50)(17 29)(18 35)(19 31)(20 33)(21 53)(22 44)(23 55)(24 42)(25 61)(27 63)(30 48)(32 46)(34 47)(36 45)(37 41)(38 56)(39 43)(40 54)(58 64)(60 62)

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

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

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

56 conjugacy classes

class 1 2A···2O2P2Q2R2S4A···4X4Y···4AJ
order12···222224···44···4
size11···144442···24···4

56 irreducible representations

dim111112
type+++++
imageC1C2C2C2C2C4○D4
kernelC22×C422C2C22×C42C22×C22⋊C4C22×C4⋊C4C2×C422C2C23
# reps11332424

Matrix representation of C22×C422C2 in GL8(𝔽5)

40000000
04000000
00100000
00010000
00001000
00000100
00000010
00000001
,
40000000
04000000
00100000
00010000
00004000
00000400
00000040
00000004
,
42000000
41000000
00040000
00100000
00000100
00004000
00000010
00000014
,
20000000
02000000
00200000
00020000
00002000
00000200
00000030
00000032
,
40000000
41000000
00400000
00010000
00001000
00000400
00000042
00000001

G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,4],[2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,2],[4,4,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1] >;

C22×C422C2 in GAP, Magma, Sage, TeX

C_2^2\times C_4^2\rtimes_2C_2
% in TeX

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

G:=SmallGroup(128,2170);
// by ID

G=gap.SmallGroup(128,2170);
# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,680,1430,184]);
// 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*d^2,e*d*e=c^2*d^-1>;
// generators/relations

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