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G = C22×C8.C22  order 128 = 27

Direct product of C22 and C8.C22

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

Aliases: C22×C8.C22, C8.1C24, C4.6C25, Q16⋊3C23, D4.3C24, Q8.3C24, SD16⋊2C23, C24.186D4, M4(2)⋊5C23, (C2×Q8)⋊21C23, (Q8×C23)⋊14C2, C4.32(C22×D4), C2.41(D4×C23), (C2×C8).297C23, (C2×C4).612C24, (C2×Q16)⋊58C22, (C22×Q16)⋊22C2, (C22×SD16)⋊9C2, C4○D4.33C23, C23.711(C2×D4), (C22×C4).537D4, (C2×SD16)⋊60C22, (C2×D4).491C23, (C22×M4(2))⋊7C2, (C22×Q8)⋊69C22, C22.53(C22×D4), (C2×M4(2))⋊57C22, (C22×C8).297C22, (C23×C4).623C22, (C22×C4).1223C23, (C22×D4).604C22, (C2×C4).668(C2×D4), (C22×C4○D4).29C2, (C2×C4○D4).336C22, SmallGroup(128,2311)

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

Derived series
C1 — C4 — C22×C8.C22
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — Q8×C23 — C22×C8.C22
Lower central
C1 — C2 — C4 — C22×C8.C22
Upper central
C1 — C23 — C23×C4 — C22×C8.C22
Jennings
C1 — C2 — C2 — C4 — C22×C8.C22

Generators and relations for C22×C8.C22
 G = < a,b,c,d,e | a2=b2=c8=d2=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c3, ece=c5, ede=c4d >

Subgroups: 1068 in 732 conjugacy classes, 436 normal (14 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C23, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C4○D4, C24, C24, C22×C8, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C22×Q8, C22×Q8, C2×C4○D4, C2×C4○D4, C22×M4(2), C22×SD16, C22×Q16, C2×C8.C22, Q8×C23, C22×C4○D4, C22×C8.C22
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C8.C22, C22×D4, C25, C2×C8.C22, D4×C23, C22×C8.C22

Smallest permutation representation of C22×C8.C22
►On 64 points
Generators in S64
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 64)(18 57)(19 58)(20 59)(21 60)(22 61)(23 62)(24 63)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 49)(34 50)(35 51)(36 52)(37 53)(38 54)(39 55)(40 56)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4T8A···8H
order12···2222222224···44···48···8
size11···1222244442···24···44···4

44 irreducible representations

dim1111111224
type+++++++++-
imageC1C2C2C2C2C2C2D4D4C8.C22
kernelC22×C8.C22C22×M4(2)C22×SD16C22×Q16C2×C8.C22Q8×C23C22×C4○D4C22×C4C24C22
# reps11222411714

Matrix representation of C22×C8.C22 ►in GL8(𝔽17)

10000000
01000000
001600000
000160000
000016000
000001600
000000160
000000016
,
160000000
016000000
001600000
000160000
00001000
00000100
00000010
00000001
,
12000000
1616000000
001150000
001160000
000000413
0000013013
000013004
000013404
,
160000000
11000000
001600000
001610000
00001000
000001600
000000160
000001161
,
160000000
016000000
001600000
000160000
00000010
0000116115
00001000
00000001

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

C22×C8.C22 in GAP, Magma, Sage, TeX 

C_2^2\times C_8.C_2^2
 
% in TeX
 
G:=Group("C2^2xC8.C2^2");
 
// GroupNames label
 
G:=SmallGroup(128,2311);
 
// by ID
 
G=gap.SmallGroup(128,2311);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,2,-2,-2,477,456,1430,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^8=d^2=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,d*c*d=c^3,e*c*e=c^5,e*d*e=c^4*d>;
 
// generators/relations
 

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