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G = C42.266C23order 128 = 27

127th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.266C23, (C2×C8)⋊27D4, C4.41(C4×D4), C4⋊Q8.28C4, C89D431C2, C8.128(C2×D4), C22.3(C4×D4), C4⋊D4.18C4, C41D4.17C4, C4⋊C8.232C22, C42.209(C2×C4), (C2×C8).404C23, (C2×C4).651C24, C4.4D4.15C4, (C4×D4).54C22, C4.197(C22×D4), C4⋊M4(2)⋊34C2, C23.35(C22×C4), C8⋊C4.155C22, C22⋊C8.140C22, C2.15(Q8○M4(2)), (C2×C42).759C22, C22.178(C23×C4), (C22×C4).918C23, (C22×C8).434C22, (C2×M4(2)).348C22, C22.26C24.25C2, C2.49(C2×C4×D4), (C2×C8○D4)⋊22C2, (C2×C8⋊C4)⋊34C2, C4⋊C4.115(C2×C4), C4.302(C2×C4○D4), (C2×C4).844(C2×D4), (C2×D4).137(C2×C4), C22⋊C4.16(C2×C4), (C2×Q8).155(C2×C4), (C22×C8)⋊C229C2, (C2×C4).684(C4○D4), (C2×C4).262(C22×C4), (C22×C4).341(C2×C4), (C2×C4○D4).286C22, SmallGroup(128,1664)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.266C23
C1C2C4C2×C4C22×C4C22×C8C2×C8○D4 — C42.266C23
C1C22 — C42.266C23
C1C2×C4 — C42.266C23
C1C2C2C2×C4 — C42.266C23

Generators and relations for C42.266C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2b, ab=ba, cac-1=a-1, dad-1=ab2, ae=ea, bc=cb, bd=db, be=eb, dcd-1=a2c, ece=b2c, de=ed >

Subgroups: 364 in 242 conjugacy classes, 140 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×14], C8 [×4], C8 [×6], C2×C4 [×2], C2×C4 [×12], C2×C4 [×10], D4 [×16], Q8 [×4], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×12], C2×C8 [×10], M4(2) [×12], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×8], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×2], C22×C8 [×4], C2×M4(2) [×6], C8○D4 [×8], C2×C4○D4 [×2], C2×C8⋊C4, (C22×C8)⋊C2 [×2], C4⋊M4(2), C89D4 [×8], C22.26C24, C2×C8○D4 [×2], C42.266C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, Q8○M4(2) [×2], C42.266C23

Smallest permutation representation of C42.266C23
On 64 points
Generators in S64
(1 45 32 24)(2 42 25 21)(3 47 26 18)(4 44 27 23)(5 41 28 20)(6 46 29 17)(7 43 30 22)(8 48 31 19)(9 37 51 58)(10 34 52 63)(11 39 53 60)(12 36 54 57)(13 33 55 62)(14 38 56 59)(15 35 49 64)(16 40 50 61)
(1 26 5 30)(2 27 6 31)(3 28 7 32)(4 29 8 25)(9 53 13 49)(10 54 14 50)(11 55 15 51)(12 56 16 52)(17 48 21 44)(18 41 22 45)(19 42 23 46)(20 43 24 47)(33 64 37 60)(34 57 38 61)(35 58 39 62)(36 59 40 63)
(1 15 5 11)(2 50 6 54)(3 9 7 13)(4 52 8 56)(10 31 14 27)(12 25 16 29)(17 57 21 61)(18 37 22 33)(19 59 23 63)(20 39 24 35)(26 51 30 55)(28 53 32 49)(34 48 38 44)(36 42 40 46)(41 60 45 64)(43 62 47 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 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 25)(17 36)(18 37)(19 38)(20 39)(21 40)(22 33)(23 34)(24 35)(41 60)(42 61)(43 62)(44 63)(45 64)(46 57)(47 58)(48 59)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4N8A···8H8I···8T
order12222222224444444···48···88···8
size11112244441111224···42···24···4

44 irreducible representations

dim11111111111224
type++++++++
imageC1C2C2C2C2C2C2C4C4C4C4D4C4○D4Q8○M4(2)
kernelC42.266C23C2×C8⋊C4(C22×C8)⋊C2C4⋊M4(2)C89D4C22.26C24C2×C8○D4C4⋊D4C4.4D4C41D4C4⋊Q8C2×C8C2×C4C2
# reps11218128422444

Matrix representation of C42.266C23 in GL6(𝔽17)

040000
400000
0086614
000830
000190
0016069
,
1600000
0160000
0013000
0001300
0000130
0000013
,
100000
0160000
0001111
0016061
0000016
000010
,
0160000
1600000
001612127
00016100
000210
00150121
,
1600000
0160000
0001011
0010110
0000016
0000160

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,4,0,0,0,0,0,0,0,8,0,0,16,0,0,6,8,1,0,0,0,6,3,9,6,0,0,14,0,0,9],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,1,6,0,1,0,0,11,1,16,0],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,15,0,0,12,16,2,0,0,0,12,10,1,12,0,0,7,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,11,0,16,0,0,11,0,16,0] >;

C42.266C23 in GAP, Magma, Sage, TeX

C_4^2._{266}C_2^3
% in TeX

G:=Group("C4^2.266C2^3");
// GroupNames label

G:=SmallGroup(128,1664);
// by ID

G=gap.SmallGroup(128,1664);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,184,2019,1018,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=b^2,d^2=a^2*b,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=a^2*c,e*c*e=b^2*c,d*e=e*d>;
// generators/relations

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