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G = C42.355C23  order 128 = 27

216th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.355C23, C4⋊C4.348D4, Q8⋊Q8⋊7C2, (C4×Q16)⋊23C2, (C4×SD16)⋊8C2, C4.Q16⋊24C2, C4⋊C4.74C23, C4⋊C8.56C22, (C2×C8).48C23, Q8⋊D4.2C2, C2.18(Q8○D8), Q16⋊C4⋊10C2, (C4×C8).112C22, (C2×C4).319C24, Q8.11(C4○D4), C22⋊Q16⋊16C2, Q8.D4⋊20C2, C22⋊C4.149D4, (C4×D4).82C22, (C2×D4).94C23, C23.258(C2×D4), C4⋊Q8.106C22, SD16⋊C4⋊15C2, (C4×Q8).79C22, C4.Q8.21C22, C8⋊C4.13C22, C2.29(D4○SD16), C4⋊D4.28C22, C22⋊C8.32C22, (C2×Q8).381C23, (C2×Q16).61C22, C2.D8.175C22, C22⋊Q8.28C22, C23.20D4⋊20C2, (C22×C4).292C23, C42.7C22⋊4C2, Q8⋊C4.37C22, C4.4D4.28C22, C23.19D4.2C2, C22.579(C22×D4), D4⋊C4.163C22, C23.32C23⋊9C2, (C2×SD16).144C22, (C22×Q8).294C22, C42⋊C2.130C22, C22.36C24.1C2, C2.120(C22.19C24), C4.204(C2×C4○D4), (C2×C4).503(C2×D4), SmallGroup(128,1853)

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

Derived series
C1 — C2×C4 — C42.355C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C23.32C23 — C42.355C23
Lower central
C1 — C2 — C2×C4 — C42.355C23
Upper central
C1 — C22 — C42⋊C2 — C42.355C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.355C23

Generators and relations for C42.355C23
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd=a2c, ece-1=bc, de=ed >

Subgroups: 324 in 185 conjugacy classes, 88 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4×C8, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42⋊2C2, C4⋊Q8, C2×SD16, C2×Q16, C22×Q8, C42.7C22, C4×SD16, C4×Q16, SD16⋊C4, Q16⋊C4, Q8⋊D4, C22⋊Q16, Q8.D4, Q8⋊Q8, C4.Q16, C23.19D4, C23.20D4, C23.32C23, C22.36C24, C42.355C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, C22.19C24, D4○SD16, Q8○D8, C42.355C23

Smallest permutation representation of C42.355C23
►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 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 63 10)(6 16 64 11)(7 13 61 12)(8 14 62 9)(29 36 41 38)(30 33 42 39)(31 34 43 40)(32 35 44 37)(45 55 58 50)(46 56 59 51)(47 53 60 52)(48 54 57 49)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G···4Q4R4S4T8A8B8C8D8E8F
order1222224···44···4444888888
size1111482···24···4888444488

32 irreducible representations

dim11111111111111122244
type+++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16Q8○D8
kernelC42.355C23C42.7C22C4×SD16C4×Q16SD16⋊C4Q16⋊C4Q8⋊D4C22⋊Q16Q8.D4Q8⋊Q8C4.Q16C23.19D4C23.20D4C23.32C23C22.36C24C22⋊C4C4⋊C4Q8C2C2
# reps11111111211111122822

Matrix representation of C42.355C23 ►in GL6(𝔽17)

400000
040000
000010
000001
0016000
0001600
,
100000
010000
000100
0016000
000001
0000160
,
100000
4160000
001000
0001600
000010
0000016
,
940000
1480000
000701
00100160
0001010
0016070
,
100000
010000
000033
0000314
00141400
0014300

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

C42.355C23 in GAP, Magma, Sage, TeX 

C_4^2._{355}C_2^3
 
% in TeX
 
G:=Group("C4^2.355C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1853);
 
// by ID
 
G=gap.SmallGroup(128,1853);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,1018,80,4037,1027,124]);
 
// 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,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d=a^2*c,e*c*e^-1=b*c,d*e=e*d>;
 
// generators/relations
 

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