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G = C2×C22.C42  order 128 = 27

Direct product of C2 and C22.C42

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C2×C22.C42, C23.31C42, (C23×C4).9C4, (C2×C4).22C42, (C22×C4).39Q8, C23.64(C4⋊C4), (C2×M4(2))⋊15C4, M4(2)⋊20(C2×C4), C24.108(C2×C4), (C22×C4).756D4, C22.6(C2×C42), (C22×C4).646C23, C23.176(C22×C4), (C23×C4).221C22, C23.227(C22⋊C4), C4.22(C2.C42), C22.30(C4.D4), (C22×M4(2)).15C2, C22.20(C4.10D4), (C2×M4(2)).299C22, C22.32(C2.C42), C4.29(C2×C4⋊C4), (C2×C4⋊C4).45C4, C4.84(C2×C22⋊C4), C22.14(C2×C4⋊C4), (C2×C4).112(C2×Q8), C2.3(C2×C4.D4), (C2×C4).126(C4⋊C4), (C2×C4).1296(C2×D4), (C22×C4⋊C4).11C2, (C22×C4).47(C2×C4), C2.3(C2×C4.10D4), (C2×C4⋊C4).742C22, (C2×C4).174(C22×C4), (C2×C4).114(C22⋊C4), C22.109(C2×C22⋊C4), C2.18(C2×C2.C42), SmallGroup(128,473)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

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

Generators and relations for C2×C22.C42
 G = < a,b,c,d,e | a2=b2=c2=e4=1, d4=c, ab=ba, ac=ca, ad=da, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=bcd >

Subgroups: 356 in 216 conjugacy classes, 116 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, C23, C23, C23, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C24, C2×C4⋊C4, C2×C4⋊C4, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22.C42, C22×C4⋊C4, C22×M4(2), C2×C22.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C4.D4, C4.10D4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C22.C42, C2×C2.C42, C2×C4.D4, C2×C4.10D4, C2×C22.C42

Smallest permutation representation of C2×C22.C42
►On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 18)(10 19)(11 20)(12 21)(13 22)(14 23)(15 24)(16 17)(25 39)(26 40)(27 33)(28 34)(29 35)(30 36)(31 37)(32 38)(41 61)(42 62)(43 63)(44 64)(45 57)(46 58)(47 59)(48 60)

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4P8A···8P
order12···222224···44···48···8
size11···122222···24···44···4

44 irreducible representations

dim11111112244
type+++++-+-
imageC1C2C2C2C4C4C4D4Q8C4.D4C4.10D4
kernelC2×C22.C42C22.C42C22×C4⋊C4C22×M4(2)C2×C4⋊C4C2×M4(2)C23×C4C22×C4C22×C4C22C22
# reps141241646222

Matrix representation of C2×C22.C42 ►in GL8(𝔽17)

160000000
016000000
00100000
00010000
00001000
00000100
00000010
00000001
,
160000000
016000000
001600000
000160000
000016000
000001600
00000010
00000001
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
1116000000
16000000
00040000
00400000
000000160
000000016
000016200
000016100
,
016000000
160000000
000160000
00100000
000015900
000011200
00000028
000000615

G:=sub<GL(8,GF(17))| [16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[11,1,0,0,0,0,0,0,16,6,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0],[0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,15,11,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,2,6,0,0,0,0,0,0,8,15] >;
 

C2×C22.C42 in GAP, Magma, Sage, TeX 

C_2\times C_2^2.C_4^2
 
% in TeX
 
G:=Group("C2xC2^2.C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,473);
 
// by ID
 
G=gap.SmallGroup(128,473);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,2019,1411,172]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^4=1,d^4=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=b*c*d>;
 
// generators/relations
 

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