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G = C22×C8.C22order 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, Q163C23, D4.3C24, Q8.3C24, SD162C23, 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

C1C4 — C22×C8.C22
C1C2C4C2×C4C22×C4C23×C4Q8×C23 — C22×C8.C22
C1C2C4 — C22×C8.C22
C1C23C23×C4 — C22×C8.C22
C1C2C2C4 — 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 [×6], C2 [×8], C4, C4 [×7], C4 [×12], C22 [×11], C22 [×28], C8 [×8], C2×C4 [×28], C2×C4 [×50], D4 [×4], D4 [×22], Q8 [×12], Q8 [×34], C23, C23 [×6], C23 [×14], C2×C8 [×12], M4(2) [×16], SD16 [×32], Q16 [×32], C22×C4 [×2], C22×C4 [×12], C22×C4 [×27], C2×D4 [×6], C2×D4 [×15], C2×Q8 [×34], C2×Q8 [×45], C4○D4 [×16], C4○D4 [×24], C24, C24, C22×C8 [×2], C2×M4(2) [×12], C2×SD16 [×24], C2×Q16 [×24], C8.C22 [×64], C23×C4, C23×C4 [×2], C22×D4, C22×D4, C22×Q8, C22×Q8 [×14], C22×Q8 [×7], C2×C4○D4 [×12], C2×C4○D4 [×6], C22×M4(2), C22×SD16 [×2], C22×Q16 [×2], C2×C8.C22 [×24], Q8×C23, C22×C4○D4, C22×C8.C22
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C8.C22 [×4], C22×D4 [×14], C25, C2×C8.C22 [×6], 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 50)(18 51)(19 52)(20 53)(21 54)(22 55)(23 56)(24 49)(25 45)(26 46)(27 47)(28 48)(29 41)(30 42)(31 43)(32 44)(33 64)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)(40 63)
(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 54)(10 55)(11 56)(12 49)(13 50)(14 51)(15 52)(16 53)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 41)
(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 24)(2 19)(3 22)(4 17)(5 20)(6 23)(7 18)(8 21)(9 56)(10 51)(11 54)(12 49)(13 52)(14 55)(15 50)(16 53)(25 59)(26 62)(27 57)(28 60)(29 63)(30 58)(31 61)(32 64)(33 44)(34 47)(35 42)(36 45)(37 48)(38 43)(39 46)(40 41)
(1 57)(2 62)(3 59)(4 64)(5 61)(6 58)(7 63)(8 60)(9 35)(10 40)(11 37)(12 34)(13 39)(14 36)(15 33)(16 38)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)(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)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,24)(2,19)(3,22)(4,17)(5,20)(6,23)(7,18)(8,21)(9,56)(10,51)(11,54)(12,49)(13,52)(14,55)(15,50)(16,53)(25,59)(26,62)(27,57)(28,60)(29,63)(30,58)(31,61)(32,64)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41), (1,57)(2,62)(3,59)(4,64)(5,61)(6,58)(7,63)(8,60)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(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)(17,50)(18,51)(19,52)(20,53)(21,54)(22,55)(23,56)(24,49)(25,45)(26,46)(27,47)(28,48)(29,41)(30,42)(31,43)(32,44)(33,64)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62)(40,63), (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,54)(10,55)(11,56)(12,49)(13,50)(14,51)(15,52)(16,53)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,42)(34,43)(35,44)(36,45)(37,46)(38,47)(39,48)(40,41), (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), (1,24)(2,19)(3,22)(4,17)(5,20)(6,23)(7,18)(8,21)(9,56)(10,51)(11,54)(12,49)(13,52)(14,55)(15,50)(16,53)(25,59)(26,62)(27,57)(28,60)(29,63)(30,58)(31,61)(32,64)(33,44)(34,47)(35,42)(36,45)(37,48)(38,43)(39,46)(40,41), (1,57)(2,62)(3,59)(4,64)(5,61)(6,58)(7,63)(8,60)(9,35)(10,40)(11,37)(12,34)(13,39)(14,36)(15,33)(16,38)(17,28)(18,25)(19,30)(20,27)(21,32)(22,29)(23,26)(24,31)(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),(17,50),(18,51),(19,52),(20,53),(21,54),(22,55),(23,56),(24,49),(25,45),(26,46),(27,47),(28,48),(29,41),(30,42),(31,43),(32,44),(33,64),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62),(40,63)], [(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,54),(10,55),(11,56),(12,49),(13,50),(14,51),(15,52),(16,53),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,42),(34,43),(35,44),(36,45),(37,46),(38,47),(39,48),(40,41)], [(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)], [(1,24),(2,19),(3,22),(4,17),(5,20),(6,23),(7,18),(8,21),(9,56),(10,51),(11,54),(12,49),(13,52),(14,55),(15,50),(16,53),(25,59),(26,62),(27,57),(28,60),(29,63),(30,58),(31,61),(32,64),(33,44),(34,47),(35,42),(36,45),(37,48),(38,43),(39,46),(40,41)], [(1,57),(2,62),(3,59),(4,64),(5,61),(6,58),(7,63),(8,60),(9,35),(10,40),(11,37),(12,34),(13,39),(14,36),(15,33),(16,38),(17,28),(18,25),(19,30),(20,27),(21,32),(22,29),(23,26),(24,31),(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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