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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

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 [×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 ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I ··· 4T 8A ··· 8H order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 ··· 4 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 4 type + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 D4 D4 C8.C22 kernel C22×C8.C22 C22×M4(2) C22×SD16 C22×Q16 C2×C8.C22 Q8×C23 C22×C4○D4 C22×C4 C24 C22 # reps 1 1 2 2 24 1 1 7 1 4

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

 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 2 0 0 0 0 0 0 16 16 0 0 0 0 0 0 0 0 1 15 0 0 0 0 0 0 1 16 0 0 0 0 0 0 0 0 0 0 4 13 0 0 0 0 0 13 0 13 0 0 0 0 13 0 0 4 0 0 0 0 13 4 0 4
,
 16 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0 0 16 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 1 16 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 0 1 0 0 0 0 0 1 16 1 15 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1

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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