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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

Aliases: C2×(C22×C8)⋊C2, (C23×C8)⋊4C2, C24.72(C2×C4), C22⋊C872C22, (C2×C4).628C24, (C2×C8).388C23, (C22×C8)⋊61C22, (C22×D4).33C4, (C22×C4).773D4, C4.177(C22×D4), (C22×Q8).26C4, C22.37(C8○D4), (C2×M4(2))⋊63C22, (C22×M4(2))⋊14C2, (C23×C4).505C22, C22.158(C23×C4), C23.289(C22×C4), C23.127(C22⋊C4), (C22×C4).1492C23, C2.6(C2×C8○D4), (C2×C22⋊C8)⋊40C2, (C2×C4○D4).21C4, C4.66(C2×C22⋊C4), C4((C22×C8)⋊C2), (C2×D4).218(C2×C4), (C2×C4).1397(C2×D4), (C2×Q8).197(C2×C4), (C22×C4).318(C2×C4), (C2×C4).239(C22×C4), (C22×C4○D4).13C2, C22.17(C2×C22⋊C4), C2.21(C22×C22⋊C4), (C2×C4).277(C22⋊C4), (C2×C4○D4).267C22, (C2×C4)((C22×C8)⋊C2), SmallGroup(128,1610)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C2×(C22×C8)⋊C2
C1C2C4C2×C4C22×C4C23×C4C22×C4○D4 — C2×(C22×C8)⋊C2
C1C22 — C2×(C22×C8)⋊C2
C1C22×C4 — C2×(C22×C8)⋊C2
C1C2C2C2×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 [×6], C2 [×8], C4 [×2], C4 [×6], C4 [×4], C22, C22 [×10], C22 [×32], C8 [×8], C2×C4 [×2], C2×C4 [×30], C2×C4 [×20], D4 [×24], Q8 [×8], C23, C23 [×10], C23 [×16], C2×C8 [×8], C2×C8 [×16], M4(2) [×8], C22×C4 [×2], C22×C4 [×22], C22×C4 [×8], C2×D4 [×12], C2×D4 [×12], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×32], C24, C24 [×2], C22⋊C8 [×16], C22×C8 [×8], C22×C8 [×4], C2×M4(2) [×4], C2×M4(2) [×4], C23×C4, C23×C4 [×2], C22×D4, C22×D4 [×2], C22×Q8, C2×C4○D4 [×8], C2×C4○D4 [×8], C2×C22⋊C8 [×4], (C22×C8)⋊C2 [×8], C23×C8, C22×M4(2), C22×C4○D4, C2×(C22×C8)⋊C2
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], C23 [×15], C22⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C24, C2×C22⋊C4 [×12], C8○D4 [×4], C23×C4, C22×D4 [×2], (C22×C8)⋊C2 [×4], C22×C22⋊C4, C2×C8○D4 [×2], 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 23)(10 24)(11 17)(12 18)(13 19)(14 20)(15 21)(16 22)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 57)(32 58)(33 44)(34 45)(35 46)(36 47)(37 48)(38 41)(39 42)(40 43)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 33)(7 34)(8 35)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(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 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 49)(16 50)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 41)(32 42)(33 60)(34 61)(35 62)(36 63)(37 64)(38 57)(39 58)(40 59)
(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 34)(2 62)(3 36)(4 64)(5 38)(6 58)(7 40)(8 60)(9 27)(10 46)(11 29)(12 48)(13 31)(14 42)(15 25)(16 44)(17 63)(18 37)(19 57)(20 39)(21 59)(22 33)(23 61)(24 35)(26 50)(28 52)(30 54)(32 56)(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)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,36)(2,37)(3,38)(4,39)(5,40)(6,33)(7,34)(8,35)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(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)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59), (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), (1,34)(2,62)(3,36)(4,64)(5,38)(6,58)(7,40)(8,60)(9,27)(10,46)(11,29)(12,48)(13,31)(14,42)(15,25)(16,44)(17,63)(18,37)(19,57)(20,39)(21,59)(22,33)(23,61)(24,35)(26,50)(28,52)(30,54)(32,56)(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)(9,23)(10,24)(11,17)(12,18)(13,19)(14,20)(15,21)(16,22)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,57)(32,58)(33,44)(34,45)(35,46)(36,47)(37,48)(38,41)(39,42)(40,43), (1,36)(2,37)(3,38)(4,39)(5,40)(6,33)(7,34)(8,35)(9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)(17,57)(18,58)(19,59)(20,60)(21,61)(22,62)(23,63)(24,64)(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)(9,51)(10,52)(11,53)(12,54)(13,55)(14,56)(15,49)(16,50)(25,43)(26,44)(27,45)(28,46)(29,47)(30,48)(31,41)(32,42)(33,60)(34,61)(35,62)(36,63)(37,64)(38,57)(39,58)(40,59), (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), (1,34)(2,62)(3,36)(4,64)(5,38)(6,58)(7,40)(8,60)(9,27)(10,46)(11,29)(12,48)(13,31)(14,42)(15,25)(16,44)(17,63)(18,37)(19,57)(20,39)(21,59)(22,33)(23,61)(24,35)(26,50)(28,52)(30,54)(32,56)(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),(9,23),(10,24),(11,17),(12,18),(13,19),(14,20),(15,21),(16,22),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,57),(32,58),(33,44),(34,45),(35,46),(36,47),(37,48),(38,41),(39,42),(40,43)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,33),(7,34),(8,35),(9,29),(10,30),(11,31),(12,32),(13,25),(14,26),(15,27),(16,28),(17,57),(18,58),(19,59),(20,60),(21,61),(22,62),(23,63),(24,64),(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),(9,51),(10,52),(11,53),(12,54),(13,55),(14,56),(15,49),(16,50),(25,43),(26,44),(27,45),(28,46),(29,47),(30,48),(31,41),(32,42),(33,60),(34,61),(35,62),(36,63),(37,64),(38,57),(39,58),(40,59)], [(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)], [(1,34),(2,62),(3,36),(4,64),(5,38),(6,58),(7,40),(8,60),(9,27),(10,46),(11,29),(12,48),(13,31),(14,42),(15,25),(16,44),(17,63),(18,37),(19,57),(20,39),(21,59),(22,33),(23,61),(24,35),(26,50),(28,52),(30,54),(32,56),(41,55),(43,49),(45,51),(47,53)])

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I4J4K4L4M4N4O4P8A···8P8Q···8X
order12···2222222224···4444444448···88···8
size11···1222244441···1222244442···24···4

56 irreducible representations

dim11111111122
type+++++++
imageC1C2C2C2C2C2C4C4C4D4C8○D4
kernelC2×(C22×C8)⋊C2C2×C22⋊C8(C22×C8)⋊C2C23×C8C22×M4(2)C22×C4○D4C22×D4C22×Q8C2×C4○D4C22×C4C22
# reps148111628816

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

160000
01000
00100
000160
000016
,
10000
016000
02100
000016
000160
,
10000
01000
00100
000160
000016
,
10000
02000
00200
00002
00020
,
160000
04400
091300
000013
00040

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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