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G = C42.25C23  order 128 = 27

25th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.25C23, C4.332+ (1+4), C4.142- (1+4), C4⋊C4.140D4, C8⋊8D4.2C2, C8⋊D4.3C2, Q8.Q8⋊31C2, D4.Q8⋊31C2, Q8⋊Q8⋊15C2, C4.Q16⋊32C2, C4⋊2Q16⋊32C2, C8.D4⋊17C2, C4⋊C8.86C22, C22⋊C4.32D4, C2.34(Q8○D8), C23.93(C2×D4), D4.D4⋊15C2, C8.18D4⋊20C2, C4⋊C4.197C23, (C2×C4).456C24, (C2×C8).340C23, Q8.D4⋊31C2, C4⋊Q8.128C22, C2.D8.49C22, C4.Q8.96C22, C2.53(D4○SD16), (C4×D4).135C22, D4⋊C4.8C22, (C2×D4).197C23, C4⋊D4.51C22, (C2×Q16).77C22, (C4×Q8).132C22, (C2×Q8).185C23, C22⋊Q8.51C22, (C22×C8).191C22, Q8⋊C4.60C22, (C2×SD16).91C22, C4.4D4.46C22, C22.716(C22×D4), C42.C2.31C22, C22.35C24⋊7C2, (C22×C4).1111C23, C42.6C22⋊13C2, (C2×M4(2)).94C22, C42⋊C2.174C22, C22.36C24.5C2, C2.75(C22.31C24), (C2×C4).580(C2×D4), SmallGroup(128,1990)

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

Derived series
C1 — C2×C4 — C42.25C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.36C24 — C42.25C23
Lower central
C1 — C2 — C2×C4 — C42.25C23
Upper central
C1 — C22 — C42⋊C2 — C42.25C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.25C23

Subgroups: 308 in 167 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×3], Q8 [×7], C23, C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×6], C4⋊C4 [×11], C2×C8 [×4], C2×C8, M4(2), SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8 [×3], C2×Q8, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×4], C4.Q8 [×2], C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×3], C4⋊D4, C22⋊Q8 [×3], C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C42.C2 [×2], C42⋊2C2 [×3], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16 [×2], C42.6C22, D4.D4, C4⋊2Q16, Q8.D4 [×2], C8⋊8D4, C8.18D4, C8⋊D4, C8.D4, Q8⋊Q8, C4.Q16, D4.Q8, Q8.Q8, C22.35C24, C22.36C24, C42.25C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, D4○SD16, Q8○D8, C42.25C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b2, ab=ba, cac-1=a-1, dad=ab2, ae=ea, cbc-1=ebe=b-1, bd=db, dcd=a2c, ece=bc, ede=a2d >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

115000000
116000000
016010000
1161600000
00000010
000016161615
00001000
00000001
,
160000000
016000000
001600000
000160000
00000100
000016000
000016161615
00001011
,
007100000
50700000
051250000
1251250000
000012500
00005500
0000121277
000050510
,
101500000
001610000
001600000
011600000
00006010
00001651615
0000160110
00001212012
,
10000000
01000000
101600000
100160000
00001000
000001600
00000010
00001601616

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

Character table of C42.25C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114822444448888888444488
ρ111111111111111111111111111    trivial
ρ21111-11111-1-11-1-11-1-11-111-11-1-11    linear of order 2
ρ31111-1111-11-1-111-1-111-1-1-11-11-11    linear of order 2
ρ411111111-1-11-1-1-1-11-111-1-1-1-1-111    linear of order 2
ρ51111-11111-1-11-1-1-111-1-11-11-111-1    linear of order 2
ρ611111111111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ711111111-1-11-1-1-11-11-11-11111-1-1    linear of order 2
ρ81111-1111-11-1-11111-1-1-1-11-11-11-1    linear of order 2
ρ91111-1-1111-1-11-11-111-11-11-11-1-11    linear of order 2
ρ1011111-11111111-1-1-1-1-1-1-1111111    linear of order 2
ρ1111111-111-1-11-1-111-11-1-11-1-1-1-111    linear of order 2
ρ121111-1-111-11-1-11-111-1-111-11-11-11    linear of order 2
ρ1311111-11111111-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ141111-1-1111-1-11-111-1-111-1-11-111-1    linear of order 2
ρ151111-1-111-11-1-11-1-1-111111-11-11-1    linear of order 2
ρ1611111-111-1-11-1-11-11-11-111111-1-1    linear of order 2
ρ17222220-2-22-2-2-220000000000000    orthogonal lifted from D4
ρ182222-20-2-2-2-22220000000000000    orthogonal lifted from D4
ρ19222220-2-2-22-22-20000000000000    orthogonal lifted from D4
ρ202222-20-2-2222-2-20000000000000    orthogonal lifted from D4
ρ214-44-4004-4000000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-400-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2344-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-4000000000000000002√202√200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16
ρ264-4-4400000000000000002√-202√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{25}C_2^3
 
% in TeX
 
G:=Group("C4^2.25C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1990);
 
// by ID
 
G=gap.SmallGroup(128,1990);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,675,1018,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=b^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,d*c*d=a^2*c,e*c*e=b*c,e*d*e=a^2*d>;
 
// generators/relations
 

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