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G = C42.411C23  order 128 = 27

272nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.411C23, C4.1172+ (1+4), C4⋊C4.134D4, Q8⋊D4⋊13C2, C4⋊2Q16⋊29C2, D4⋊D4.4C2, C8.12D4⋊6C2, C8.2D4⋊15C2, C4⋊C8.67C22, C22⋊C4.26D4, C2.31(Q8○D8), D4.D4⋊12C2, D4.2D4⋊28C2, D4.7D4⋊31C2, C4⋊C4.164C23, (C4×C8).114C22, (C2×C4).423C24, (C2×C8).164C23, C22⋊Q16⋊24C2, Q8.D4⋊28C2, C4.SD16⋊17C2, (C2×D8).26C22, C23.295(C2×D4), C4⋊Q8.122C22, C8⋊C4.24C22, C2.48(D4○SD16), (C2×D4).172C23, (C4×D4).110C22, C4⋊D4.46C22, C22⋊C8.58C22, (C2×Q16).28C22, (C4×Q8).107C22, (C2×Q8).160C23, C22⋊Q8.46C22, D4⋊C4.47C22, (C22×C4).311C23, (C2×SD16).39C22, C4.4D4.42C22, C22.683(C22×D4), C42.7C22⋊15C2, C42.28C22⋊6C2, C22.36C24⋊8C2, Q8⋊C4.103C22, (C22×Q8).327C22, C42⋊C2.162C22, C23.38C23⋊17C2, C2.94(C22.29C24), (C2×C4).552(C2×D4), (C2×C4○D4).182C22, SmallGroup(128,1957)

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

Derived series
C1 — C2×C4 — C42.411C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — C23.38C23 — C42.411C23
Lower central
C1 — C2 — C2×C4 — C42.411C23
Upper central
C1 — C22 — C42⋊C2 — C42.411C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.411C23

Subgroups: 388 in 188 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×7], Q8 [×11], C23, C23 [×2], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×7], C2×C8 [×4], D8, SD16 [×4], Q16 [×3], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×3], C22.D4 [×3], C4.4D4 [×3], C4.4D4, C42⋊2C2, C4⋊Q8 [×3], C2×D8, C2×SD16 [×4], C2×Q16 [×3], C22×Q8, C2×C4○D4, C42.7C22, Q8⋊D4, D4⋊D4, C22⋊Q16, D4.7D4, D4.D4, C4⋊2Q16, D4.2D4, Q8.D4, C4.SD16, C42.28C22, C8.12D4, C8.2D4, C23.38C23, C22.36C24, C42.411C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C22.29C24, D4○SD16, Q8○D8, C42.411C23

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

000130000
00400000
013000000
40000000
000050120
000005012
0000120120
0000012012
,
01000000
160000000
00010000
001600000
00000100
000016000
00000001
000000160
,
003140000
0014140000
314000000
1414000000
000014300
00003300
000000143
00000033
,
10000000
016000000
00100000
000160000
00001000
000001600
00000010
000000016
,
00100000
00010000
160000000
016000000
00000010
00000001
000016000
000001600

G:=sub<GL(8,GF(17))| [0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,0,5,0,12,0,0,0,0,0,0,5,0,12,0,0,0,0,12,0,12,0,0,0,0,0,0,12,0,12],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,3,14,0,0,0,0,0,0,14,14,0,0,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,14,3,0,0,0,0,0,0,3,3],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;
 

Character table of C42.411C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114882244444888888444488
ρ111111111111111111111111111    trivial
ρ21111-1-1-111-11-11-111-1-1111-11-1-11    linear of order 2
ρ31111-1-1-1111-11-1-11-111-111111-1-1    linear of order 2
ρ4111111111-1-1-1-111-1-1-1-111-11-11-1    linear of order 2
ρ511111-1111111111-1-111-1-1-1-1-1-1-1    linear of order 2
ρ61111-11-111-11-11-11-11-11-1-11-111-1    linear of order 2
ρ71111-11-1111-11-1-111-11-1-1-1-1-1-111    linear of order 2
ρ811111-1111-1-1-1-11111-1-1-1-11-11-11    linear of order 2
ρ9111111-111-1-1-1-11-1-1-1111-11-11-11    linear of order 2
ρ101111-1-11111-11-1-1-1-11-111-1-1-1-111    linear of order 2
ρ111111-1-1111-11-11-1-11-11-11-11-111-1    linear of order 2
ρ12111111-11111111-111-1-11-1-1-1-1-1-1    linear of order 2
ρ1311111-1-111-1-1-1-11-11111-11-11-11-1    linear of order 2
ρ141111-111111-11-1-1-11-1-11-11111-1-1    linear of order 2
ρ151111-11111-11-11-1-1-111-1-11-11-1-11    linear of order 2
ρ1611111-1-11111111-1-1-1-1-1-1111111    linear of order 2
ρ172222-200-2-2-2-2222000000000000    orthogonal lifted from D4
ρ182222200-2-2-222-2-2000000000000    orthogonal lifted from D4
ρ192222-200-2-222-2-22000000000000    orthogonal lifted from D4
ρ202222200-2-22-2-22-2000000000000    orthogonal lifted from D4
ρ214-44-40004-400000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4000-4400000000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ244-4-4400000000000000002√202√2000    symplectic lifted from Q8○D8, Schur index 2
ρ2544-4-4000000000000000002√-202√-200    complex lifted from D4○SD16
ρ2644-4-4000000000000000002√-202√-200    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{411}C_2^3
 
% in TeX
 
G:=Group("C4^2.411C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1957);
 
// by ID
 
G=gap.SmallGroup(128,1957);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,219,675,1018,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b^2,a*b=b*a,c*a*c=d*a*d=a^-1*b^2,e*a*e^-1=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e^-1=a^2*b^2*c,d*e=e*d>;
 
// generators/relations
 

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