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G = C42.495C23  order 128 = 27

356th non-split extension by C42 of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.495C23, C4.892- (1+4), (C4×D8)⋊33C2, C8⋊5(C4○D4), C8⋊3Q8⋊9C2, C8⋊6D4⋊23C2, C8⋊2D4⋊33C2, C4⋊C4.384D4, D4⋊6D4⋊16C2, D4⋊2Q8⋊23C2, D4⋊Q8⋊41C2, (C2×D4).334D4, C22⋊C4.67D4, C4⋊C4.259C23, C4⋊C8.127C22, C4.49(C8⋊C22), (C2×C8).111C23, (C2×C4).546C24, (C4×C8).196C22, C23.351(C2×D4), C4⋊Q8.176C22, C2.99(D4⋊6D4), C2.94(D4○SD16), (C4×D4).186C22, (C2×D4).262C23, (C2×D8).166C22, M4(2)⋊C4⋊41C2, C2.D8.226C22, C4.Q8.112C22, D4⋊C4.85C22, C23.19D4⋊49C2, C23.46D4⋊24C2, C4⋊D4.111C22, C22⋊C8.105C22, (C22×C4).346C23, C22.806(C22×D4), C42⋊C2.217C22, C22.49C24⋊10C2, (C2×M4(2)).139C22, C4.128(C2×C4○D4), (C2×C4).630(C2×D4), C2.85(C2×C8⋊C22), (C2×C4⋊C4).695C22, SmallGroup(128,2086)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.495C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — D4⋊6D4 — C42.495C23
Lower central
C1 — C2 — C2×C4 — C42.495C23
Upper central
C1 — C22 — C4×D4 — C42.495C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.495C23

Subgroups: 392 in 193 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×12], Q8 [×3], C23 [×2], C23 [×2], C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], D8 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C4○D4 [×4], C4×C8, C22⋊C8 [×2], D4⋊C4 [×2], D4⋊C4 [×4], C4⋊C8, C4.Q8 [×6], C2.D8, C2.D8 [×2], C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2, C4×D4 [×3], C4⋊D4 [×4], C4⋊D4, C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C2×D8, C2×C4○D4, M4(2)⋊C4 [×2], C8⋊6D4, C4×D8, C8⋊2D4 [×2], D4⋊Q8, D4⋊2Q8, C23.46D4 [×2], C23.19D4 [×2], C8⋊3Q8, D4⋊6D4, C22.49C24, C42.495C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, C2×C8⋊C22, D4○SD16, C42.495C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, e2=b2, ab=ba, cac-1=a-1, dad-1=ab2, eae-1=a-1b2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, ede-1=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

400000
0130000
000100
0016000
00116115
0010116
,
100000
010000
000100
0016000
00116115
0010116
,
040000
400000
00169710
009807
001998
008181
,
400000
040000
0001600
0016000
00161162
000001
,
0130000
400000
00116115
0000160
000100
0010116

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

Character table of C42.495C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-11-111-1-11-1-111-1-111-1-1-1111-1    linear of order 2
ρ3111111-1111111111-1-11-11-11-1-1-1-1-1-1    linear of order 2
ρ41111-1111-111-1-11-1-1-1-1-111-1-111-1-1-11    linear of order 2
ρ5111111-1-11111-1111-1-1-1-1-1-1-1111111    linear of order 2
ρ61111-111-1-111-111-1-1-1-111-1-11-1-1111-1    linear of order 2
ρ71111111-11111-111111-11-11-1-1-1-1-1-1-1    linear of order 2
ρ81111-11-1-1-111-111-1-1111-1-11111-1-1-11    linear of order 2
ρ91111-1-1-1111111-11-1-1-111-11-1-1-1-1-111    linear of order 2
ρ1011111-111-111-1-1-1-11-1-1-1-1-11111-1-11-1    linear of order 2
ρ111111-1-11111111-11-1111-1-1-1-11111-1-1    linear of order 2
ρ1211111-1-11-111-1-1-1-1111-11-1-11-1-111-11    linear of order 2
ρ131111-1-11-11111-1-11-111-1-11-11-1-1-1-111    linear of order 2
ρ1411111-1-1-1-111-11-1-1111111-1-111-1-11-1    linear of order 2
ρ151111-1-1-1-11111-1-11-1-1-1-111111111-1-1    linear of order 2
ρ1611111-11-1-111-11-1-11-1-11-111-1-1-111-11    linear of order 2
ρ1722222-200-2-2-2-2022-20000000000000    orthogonal lifted from D4
ρ18222222002-2-220-2-2-20000000000000    orthogonal lifted from D4
ρ192222-2200-2-2-2-20-2220000000000000    orthogonal lifted from D4
ρ202222-2-2002-2-2202-220000000000000    orthogonal lifted from D4
ρ212-22-200000-2202i0002i2i2i0000-220000    complex lifted from C4○D4
ρ222-22-200000-2202i0002i2i2i00002-20000    complex lifted from C4○D4
ρ232-22-200000-2202i0002i2i2i00002-20000    complex lifted from C4○D4
ρ242-22-200000-2202i0002i2i2i0000-220000    complex lifted from C4○D4
ρ254-4-440000400-400000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-440000-400400000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-4000004-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2844-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16
ρ2944-4-40000000000000000000002√-22√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{495}C_2^3
 
% in TeX
 
G:=Group("C4^2.495C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2086);
 
// by ID
 
G=gap.SmallGroup(128,2086);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,100,2019,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,e*a*e^-1=a^-1*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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