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## G = C2×(C22×C8)⋊C2order 128 = 27

### Direct product of C2 and (C22×C8)⋊C2

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×(C22×C8)⋊C2
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C23×C4 — C22×C4○D4 — C2×(C22×C8)⋊C2
 Lower central C1 — C22 — C2×(C22×C8)⋊C2
 Upper central C1 — C22×C4 — C2×(C22×C8)⋊C2
 Jennings C1 — C2 — C2 — C2×C4 — C2×(C22×C8)⋊C2

Generators and relations for C2×(C22×C8)⋊C2
G = < a,b,c,d,e | a2=b2=c2=d8=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe=bd4, ede=cd=dc, ce=ec >

Subgroups: 636 in 396 conjugacy classes, 180 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C22⋊C8, C22×C8, C22×C8, C2×M4(2), C2×M4(2), C23×C4, C23×C4, C22×D4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C2×C22⋊C8, (C22×C8)⋊C2, C23×C8, C22×M4(2), C22×C4○D4, C2×(C22×C8)⋊C2
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C24, C2×C22⋊C4, C8○D4, C23×C4, C22×D4, (C22×C8)⋊C2, C22×C22⋊C4, C2×C8○D4, C2×(C22×C8)⋊C2

Smallest permutation representation of C2×(C22×C8)⋊C2
On 64 points
Generators in S64
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 42)(10 43)(11 44)(12 45)(13 46)(14 47)(15 48)(16 41)(17 31)(18 32)(19 25)(20 26)(21 27)(22 28)(23 29)(24 30)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)
(1 14)(2 15)(3 16)(4 9)(5 10)(6 11)(7 12)(8 13)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 37)(26 38)(27 39)(28 40)(29 33)(30 34)(31 35)(32 36)(41 53)(42 54)(43 55)(44 56)(45 49)(46 50)(47 51)(48 52)
(1 23)(2 24)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 58)(10 59)(11 60)(12 61)(13 62)(14 63)(15 64)(16 57)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 47)(34 48)(35 41)(36 42)(37 43)(38 44)(39 45)(40 46)
(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 12)(2 62)(3 14)(4 64)(5 16)(6 58)(7 10)(8 60)(9 20)(11 22)(13 24)(15 18)(17 63)(19 57)(21 59)(23 61)(25 35)(26 42)(27 37)(28 44)(29 39)(30 46)(31 33)(32 48)(34 54)(36 56)(38 50)(40 52)(41 55)(43 49)(45 51)(47 53)

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

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

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

56 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A ··· 4H 4I 4J 4K 4L 4M 4N 4O 4P 8A ··· 8P 8Q ··· 8X order 1 2 ··· 2 2 2 2 2 2 2 2 2 4 ··· 4 4 4 4 4 4 4 4 4 8 ··· 8 8 ··· 8 size 1 1 ··· 1 2 2 2 2 4 4 4 4 1 ··· 1 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 type + + + + + + + image C1 C2 C2 C2 C2 C2 C4 C4 C4 D4 C8○D4 kernel C2×(C22×C8)⋊C2 C2×C22⋊C8 (C22×C8)⋊C2 C23×C8 C22×M4(2) C22×C4○D4 C22×D4 C22×Q8 C2×C4○D4 C22×C4 C22 # reps 1 4 8 1 1 1 6 2 8 8 16

Matrix representation of C2×(C22×C8)⋊C2 in GL5(𝔽17)

 16 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 16 0 0 0 0 2 1 0 0 0 0 0 0 16 0 0 0 16 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 16 0 0 0 0 0 16
,
 1 0 0 0 0 0 2 0 0 0 0 0 2 0 0 0 0 0 0 2 0 0 0 2 0
,
 16 0 0 0 0 0 4 4 0 0 0 9 13 0 0 0 0 0 0 13 0 0 0 4 0

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,2,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,16,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,2,0],[16,0,0,0,0,0,4,9,0,0,0,4,13,0,0,0,0,0,0,4,0,0,0,13,0] >;

C2×(C22×C8)⋊C2 in GAP, Magma, Sage, TeX

C_2\times (C_2^2\times C_8)\rtimes C_2
% in TeX

G:=Group("C2x(C2^2xC8):C2");
// GroupNames label

G:=SmallGroup(128,1610);
// by ID

G=gap.SmallGroup(128,1610);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=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,e*b*e=b*d^4,e*d*e=c*d=d*c,c*e=e*c>;
// generators/relations

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