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G = C42.492C23  order 128 = 27

353rd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.492C23, C4.862- (1+4), C8⋊6D4⋊21C2, C8⋊D4⋊54C2, C4⋊C4.169D4, Q8.Q8⋊44C2, D4.Q8⋊44C2, (C4×SD16)⋊22C2, (C2×D4).183D4, C8.15(C4○D4), C8.5Q8⋊21C2, C4⋊C4.256C23, C4⋊C8.124C22, (C2×C4).543C24, (C2×C8).366C23, (C4×C8).193C22, C22⋊C4.179D4, C23.348(C2×D4), C2.96(D4⋊6D4), C2.93(D4○SD16), (C2×D4).260C23, (C4×D4).183C22, (C4×Q8).182C22, (C2×Q8).245C23, M4(2)⋊C4⋊38C2, C4.Q8.111C22, C2.D8.133C22, C23.46D4⋊23C2, C23.20D4⋊48C2, C4⋊D4.109C22, C23.47D4⋊23C2, C23.19D4⋊47C2, C22⋊C8.102C22, (C22×C4).343C23, Q8⋊C4.82C22, C22.803(C22×D4), C22⋊Q8.108C22, C42.C2.56C22, D4⋊C4.188C22, C2.98(D8⋊C22), (C2×SD16).120C22, C42⋊C2.214C22, C22.46C24⋊11C2, (C2×M4(2)).136C22, C22.47C24.5C2, C4.125(C2×C4○D4), (C2×C4).627(C2×D4), (C2×C4⋊C4).692C22, SmallGroup(128,2083)

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

Derived series
C1 — C2×C4 — C42.492C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.46C24 — C42.492C23
Lower central
C1 — C2 — C2×C4 — C42.492C23
Upper central
C1 — C22 — C4×D4 — C42.492C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.492C23

Subgroups: 320 in 174 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×15], D4 [×6], Q8 [×2], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C4⋊C4 [×9], C2×C8 [×4], M4(2) [×4], SD16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8, C4×C8, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8 [×5], C2.D8 [×4], C2×C4⋊C4 [×2], C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22.D4 [×2], C42.C2 [×2], C42.C2, C42⋊2C2 [×2], C2×M4(2) [×2], C2×SD16, M4(2)⋊C4 [×2], C8⋊6D4, C4×SD16, C8⋊D4 [×2], D4.Q8, Q8.Q8, C23.46D4, C23.19D4, C23.47D4, C23.20D4, C8.5Q8, C22.46C24, C22.47C24, C42.492C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2- (1+4), D4⋊6D4, D8⋊C22, D4○SD16, C42.492C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=a2, ab=ba, cac-1=a-1b2, dad-1=ab2, eae=a-1, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=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 23 57 9)(2 24 58 10)(3 21 59 11)(4 22 60 12)(5 47 40 28)(6 48 37 25)(7 45 38 26)(8 46 39 27)(13 33 19 53)(14 34 20 54)(15 35 17 55)(16 36 18 56)(29 61 44 49)(30 62 41 50)(31 63 42 51)(32 64 43 52)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

700100000
1055120000
5512120000
1200100000
0000011016
00006010
00000106
0000160110
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
10101000000
01212120000
12121250000
501000000
0000016011
0000160110
00000601
00006010
,
1601500000
00110000
10100000
16161600000
000012500
00005500
000000125
00000055
,
77700000
1055120000
55550000
120700000
00000106
0000160110
0000011016
00006010

G:=sub<GL(8,GF(17))| [7,10,5,12,0,0,0,0,0,5,5,0,0,0,0,0,0,5,12,0,0,0,0,0,10,12,12,10,0,0,0,0,0,0,0,0,0,6,0,16,0,0,0,0,11,0,1,0,0,0,0,0,0,1,0,11,0,0,0,0,16,0,6,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[10,0,12,5,0,0,0,0,10,12,12,0,0,0,0,0,10,12,12,10,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,16,0,6,0,0,0,0,16,0,6,0,0,0,0,0,0,11,0,1,0,0,0,0,11,0,1,0],[16,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,15,1,1,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,5,5],[7,10,5,12,0,0,0,0,7,5,5,0,0,0,0,0,7,5,5,7,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,0,0,16,0,6,0,0,0,0,1,0,11,0,0,0,0,0,0,11,0,1,0,0,0,0,6,0,16,0] >;
 

Character table of C42.492C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11114482222444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-11-1-111-111-1-111111-1-1-11-1-111-1    linear of order 2
ρ31111-1-1-11111-1-11-1-1-1-111-1111111-1-1    linear of order 2
ρ411111-11-111-1-1-1-11-1-1-1111-1-11-1-11-11    linear of order 2
ρ511111-1-1-111-11-1-11-1-11-111-11-111-11-1    linear of order 2
ρ61111-1-1111111-11-1-1-11-11-11-1-1-1-1-111    linear of order 2
ρ71111-111-111-1-11-1-111-1-11-1-11-111-1-11    linear of order 2
ρ8111111-11111-111111-1-1111-1-1-1-1-1-1-1    linear of order 2
ρ9111111-11111-1-111-11-1-1-1-1-1-1111111    linear of order 2
ρ101111-111-111-1-1-1-1-1-11-1-1-11111-1-111-1    linear of order 2
ρ111111-1-111111111-11-11-1-11-1-11111-1-1    linear of order 2
ρ1211111-1-1-111-111-111-11-1-1-1111-1-11-11    linear of order 2
ρ1311111-11-111-1-11-111-1-11-1-11-1-111-11-1    linear of order 2
ρ141111-1-1-11111-111-11-1-11-11-11-1-1-1-111    linear of order 2
ρ151111-11-1-111-11-1-1-1-1111-111-1-111-1-11    linear of order 2
ρ16111111111111-111-1111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ172222-2-20-2-2-2-2002202000000000000    orthogonal lifted from D4
ρ182222220-2-2-2-2002-20-2000000000000    orthogonal lifted from D4
ρ192222-2202-2-2200-220-2000000000000    orthogonal lifted from D4
ρ2022222-202-2-2200-2-202000000000000    orthogonal lifted from D4
ρ212-22-200002-202i2i002i02i00000-200200    complex lifted from C4○D4
ρ222-22-200002-202i2i002i02i00000200-200    complex lifted from C4○D4
ρ232-22-200002-202i2i002i02i00000-200200    complex lifted from C4○D4
ρ242-22-200002-202i2i002i02i00000200-200    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-4-440004i004i000000000000000000    complex lifted from D8⋊C22
ρ274-4-440004i004i000000000000000000    complex lifted from D8⋊C22
ρ2844-4-4000000000000000000002√-22√-2000    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002√-22√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{492}C_2^3
 
% in TeX
 
G:=Group("C4^2.492C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2083);
 
// by ID
 
G=gap.SmallGroup(128,2083);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,100,2019,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,e*a*e=a^-1,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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