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G = C42.425C23  order 128 = 27

286th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.425C23, C4.672- (1+4), C8⋊Q8⋊21C2, C4⋊C4.137D4, C8⋊2Q8⋊16C2, Q8.Q8⋊26C2, D4.Q8⋊26C2, D4⋊Q8⋊29C2, C4.Q16⋊29C2, C2.33(D4○D8), C4⋊C8.77C22, (C4×C8).79C22, C2.33(Q8○D8), C22⋊C4.29D4, C4⋊C4.182C23, (C2×C4).441C24, (C2×C8).169C23, C23.304(C2×D4), C4⋊Q8.125C22, C2.D8.42C22, C4.Q8.44C22, C8⋊C4.34C22, (C2×D4).184C23, (C4×D4).122C22, C4⋊D4.48C22, C22⋊C8.68C22, (C2×Q8).172C23, (C4×Q8).119C22, Q8⋊C4.7C22, C22.D8.4C2, C22⋊Q8.48C22, D4⋊C4.54C22, C23.20D4⋊29C2, C23.48D4⋊24C2, (C22×C4).314C23, C23.19D4.4C2, C4.4D4.43C22, C22.701(C22×D4), C42.C2.28C22, C42.7C22⋊17C2, C42.78C22⋊3C2, C23.41C23⋊9C2, C42.28C22⋊10C2, C42⋊C2.171C22, C22.36C24.3C2, C2.89(C23.38C23), (C2×C4).565(C2×D4), (C2×C4⋊C4).656C22, SmallGroup(128,1975)

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

Derived series
C1 — C2×C4 — C42.425C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C23.41C23 — C42.425C23
Lower central
C1 — C2 — C2×C4 — C42.425C23
Upper central
C1 — C22 — C42⋊C2 — C42.425C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.425C23

Subgroups: 300 in 160 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 [×11], D4 [×3], Q8 [×5], C23, C23, C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×8], C4⋊C4 [×10], C2×C8 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×3], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×6], C2×C4⋊C4, C42⋊C2 [×2], C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×3], C22.D4, C4.4D4, C4.4D4, C42.C2 [×2], C42.C2, C42⋊2C2, C4⋊Q8 [×3], C4⋊Q8, C42.7C22, D4⋊Q8, C4.Q16, D4.Q8, Q8.Q8, C22.D8, C23.19D4, C23.48D4, C23.20D4, C42.78C22, C42.28C22, C8⋊2Q8, C8⋊Q8, C22.36C24, C23.41C23, C42.425C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- (1+4) [×2], C23.38C23, D4○D8, Q8○D8, C42.425C23

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

110900000
011090000
110600000
011060000
00000161111
000010011
000011612
00000111616
,
01000000
160000000
00010000
001600000
00000100
000016000
00000012
0000001616
,
10000000
016000000
00100000
000160000
00001000
000001600
0000001615
00000001
,
33000000
314000000
00330000
003140000
00003310
00003141615
000015006
00001130
,
101500000
010150000
001600000
000160000
000001300
00004000
000010748
00000101313

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

Character table of C42.425C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11114822444448888888444488
ρ111111111111111111111111111    trivial
ρ21111-11111-1-11-11-11-11-1-1-1-1-1-111    linear of order 2
ρ311111111-1-11-1-1-11-1-11-11-11-11-11    linear of order 2
ρ41111-1111-11-1-11-1-1-1111-11-11-1-11    linear of order 2
ρ51111-1111-11-1-111-1-11-1-11-11-111-1    linear of order 2
ρ611111111-1-11-1-111-1-1-11-11-11-11-1    linear of order 2
ρ71111-11111-1-11-1-1-11-1-1111111-1-1    linear of order 2
ρ81111111111111-1111-1-1-1-1-1-1-1-1-1    linear of order 2
ρ911111-111-1-11-1-11-111-11-1-11-11-11    linear of order 2
ρ101111-1-111-11-1-11111-1-1-111-11-1-11    linear of order 2
ρ1111111-11111111-1-1-1-1-1-1-1111111    linear of order 2
ρ121111-1-1111-1-11-1-11-11-111-1-1-1-111    linear of order 2
ρ131111-1-1111-1-11-111-111-1-11111-1-1    linear of order 2
ρ1411111-111111111-1-1-1111-1-1-1-1-1-1    linear of order 2
ρ151111-1-111-11-1-11-111-111-1-11-111-1    linear of order 2
ρ1611111-111-1-11-1-1-1-1111-111-11-11-1    linear of order 2
ρ17222220-2-222-2-2-20000000000000    orthogonal lifted from D4
ρ182222-20-2-2-2222-20000000000000    orthogonal lifted from D4
ρ19222220-2-2-2-2-2220000000000000    orthogonal lifted from D4
ρ202222-20-2-22-22-220000000000000    orthogonal lifted from D4
ρ214-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ224-4-4400000000000000002√202√2000    orthogonal lifted from D4○D8
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
ρ254-44-4004-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ264-44-400-44000000000000000000    symplectic lifted from 2- (1+4), Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{425}C_2^3
 
% in TeX
 
G:=Group("C4^2.425C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1975);
 
// by ID
 
G=gap.SmallGroup(128,1975);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,100,675,1018,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=e^2=1,d^2=a^2*b^2,a*b=b*a,c*a*c=d*a*d^-1=a^-1*b^2,e*a*e=a*b^2,c*b*c=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*b^2*c,d*e=e*d>;
 
// generators/relations
 

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