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G = C22×C42⋊2C2  order 128 = 27

Direct product of C22 and C42⋊2C2

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

Aliases: C22×C42⋊2C2, C42⋊18C23, C25.71C22, C22.27C25, C23.270C24, C24.658C23, C4⋊C4⋊17C23, (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

Derived series
C1 — C22 — C22×C42⋊2C2
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C42 — C22×C42⋊2C2
Lower central
C1 — C22 — C22×C42⋊2C2
Upper central
C1 — C24 — C22×C42⋊2C2
Jennings
C1 — C22 — C22×C42⋊2C2

Generators and relations for C22×C42⋊2C2
 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, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C24, C24, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C42⋊2C2, C23×C4, C25, C22×C42, C22×C22⋊C4, C22×C4⋊C4, C2×C42⋊2C2, C22×C42⋊2C2
Quotients: C1, C2, C22, C23, C4○D4, C24, C42⋊2C2, C2×C4○D4, C25, C2×C42⋊2C2, C22×C4○D4, C22×C42⋊2C2

Smallest permutation representation of C22×C42⋊2C2
►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×C42⋊2C2C22×C42C22×C22⋊C4C22×C4⋊C4C2×C42⋊2C2C23
# reps11332424

Matrix representation of C22×C42⋊2C2 ►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×C42⋊2C2 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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