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G = C2×C42.28C22  order 128 = 27

Direct product of C2 and C42.28C22

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

Aliases: C2×C42.28C22, C42.236D4, C42.362C23, C4⋊Q8⋊62C22, C4⋊C4.85C23, C8⋊C4⋊62C22, (C2×C4).330C24, (C2×C8).451C23, (C2×D4).99C23, (C22×C4).456D4, C23.873(C2×D4), (C2×Q8).87C23, Q8⋊C4⋊97C22, C4.21(C4.4D4), (C22×C8).457C22, (C2×C42).843C22, C22.590(C22×D4), D4⋊C4.200C22, C22.123(C8⋊C22), (C22×C4).1552C23, C4.4D4.134C22, C22.83(C4.4D4), (C22×D4).366C22, (C22×Q8).299C22, C22.112(C8.C22), (C2×C4⋊Q8)⋊35C2, (C2×C8⋊C4)⋊37C2, C4.39(C2×C4○D4), (C2×C4).510(C2×D4), C2.37(C2×C8⋊C22), (C2×Q8⋊C4)⋊57C2, C2.41(C2×C4.4D4), C2.37(C2×C8.C22), (C2×D4⋊C4).38C2, (C2×C4).709(C4○D4), (C2×C4⋊C4).621C22, (C2×C4.4D4).39C2, SmallGroup(128,1864)

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

Derived series
C1 — C2×C4 — C2×C42.28C22
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C8 — C2×C8⋊C4 — C2×C42.28C22
Lower central
C1 — C2 — C2×C4 — C2×C42.28C22
Upper central
C1 — C23 — C2×C42 — C2×C42.28C22
Jennings
C1 — C2 — C2 — C2×C4 — C2×C42.28C22

Generators and relations for C2×C42.28C22
 G = < a,b,c,d,e | a2=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1c2, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 452 in 218 conjugacy classes, 100 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C24, C8⋊C4, D4⋊C4, Q8⋊C4, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4.4D4, C4.4D4, C4⋊Q8, C4⋊Q8, C22×C8, C22×D4, C22×Q8, C22×Q8, C2×C8⋊C4, C2×D4⋊C4, C2×Q8⋊C4, C42.28C22, C2×C4.4D4, C2×C4⋊Q8, C2×C42.28C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C4.4D4, C8⋊C22, C8.C22, C22×D4, C2×C4○D4, C42.28C22, C2×C4.4D4, C2×C8⋊C22, C2×C8.C22, C2×C42.28C22

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

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G4H4I···4N8A···8H
order12···222444444444···48···8
size11···188222244448···84···4

32 irreducible representations

dim111111122244
type++++++++++-
imageC1C2C2C2C2C2C2D4D4C4○D4C8⋊C22C8.C22
kernelC2×C42.28C22C2×C8⋊C4C2×D4⋊C4C2×Q8⋊C4C42.28C22C2×C4.4D4C2×C4⋊Q8C42C22×C4C2×C4C22C22
# reps112281122822

Matrix representation of C2×C42.28C22 ►in GL8(𝔽17)

10000000
01000000
001600000
000160000
00001000
00000100
00000010
00000001
,
18000000
416000000
00100000
00010000
0000212010
00001221010
0000512100
000014253
,
160000000
016000000
001600000
000160000
00000100
000016000
0000111615
000016011
,
160000000
131000000
00100000
000160000
000016000
00000100
0000161612
000001016
,
40000000
04000000
00010000
001600000
000022103
0000512100
000015507
0000214153

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,4,0,0,0,0,0,0,8,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,12,5,14,0,0,0,0,12,2,12,2,0,0,0,0,0,10,10,5,0,0,0,0,10,10,0,3],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,1,16,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[16,13,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,16,0,0,0,0,0,0,1,16,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,16],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,5,15,2,0,0,0,0,2,12,5,14,0,0,0,0,10,10,0,15,0,0,0,0,3,0,7,3] >;
 

C2×C42.28C22 in GAP, Magma, Sage, TeX 

C_2\times C_4^2._{28}C_2^2
 
% in TeX
 
G:=Group("C2xC4^2.28C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,1864);
 
// by ID
 
G=gap.SmallGroup(128,1864);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,232,758,723,100,2804,172,4037,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^4=d^2=1,e^2=c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d=b^-1*c^2,e*b*e^-1=b*c^2,d*c*d=c^-1,c*e=e*c,e*d*e^-1=b^2*c^-1*d>;
 
// generators/relations
 

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