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G = C42.360C23order 128 = 27

221st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.360C23, C4⋊C4.353D4, Q8⋊D46C2, C4⋊SD168C2, Q8.Q821C2, (C4×SD16)⋊32C2, C4⋊C8.59C22, C4⋊C4.79C23, (C2×C8).53C23, (C2×C4).324C24, (C4×C8).263C22, Q8.14(C4○D4), C22⋊C4.154D4, (C4×D4).86C22, (C2×D4).95C23, C23.263(C2×D4), SD16⋊C418C2, (C4×Q8).82C22, C8⋊C4.16C22, C2.D8.93C22, C2.32(D4○SD16), C4⋊D4.32C22, C41D4.61C22, C22⋊C8.37C22, (C2×Q8).383C23, C4.Q8.156C22, D4⋊C4.39C22, C23.19D421C2, C42.7C229C2, (C22×C4).297C23, Q8⋊C4.39C22, C22.584(C22×D4), C42.C2.14C22, (C2×SD16).147C22, (C22×Q8).295C22, C23.32C2310C2, C42⋊C2.135C22, C22.34C24.1C2, C2.125(C22.19C24), C4.209(C2×C4○D4), (C2×C4).508(C2×D4), SmallGroup(128,1858)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.360C23
C1C2C4C2×C4C42C4×Q8C23.32C23 — C42.360C23
C1C2C2×C4 — C42.360C23
C1C22C42⋊C2 — C42.360C23
C1C2C2C2×C4 — C42.360C23

Generators and relations for C42.360C23
 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=a2b2c, ece-1=bc, de=ed >

Subgroups: 364 in 191 conjugacy classes, 88 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×13], C22, C22 [×9], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], D4 [×8], Q8 [×4], Q8 [×6], C23, C23 [×2], C42 [×2], C42 [×5], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×4], SD16 [×8], C22×C4, C22×C4 [×3], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4 [×2], C4×Q8 [×4], C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C41D4, C2×SD16 [×4], C22×Q8, C42.7C22, C4×SD16 [×2], SD16⋊C4 [×2], Q8⋊D4 [×2], C4⋊SD16 [×2], Q8.Q8 [×2], C23.19D4 [×2], C23.32C23, C22.34C24, C42.360C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D4○SD16 [×2], C42.360C23

Smallest permutation representation of C42.360C23
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 20 26 23)(2 17 27 24)(3 18 28 21)(4 19 25 22)(5 11 62 14)(6 12 63 15)(7 9 64 16)(8 10 61 13)(29 34 41 39)(30 35 42 40)(31 36 43 37)(32 33 44 38)(45 49 57 54)(46 50 58 55)(47 51 59 56)(48 52 60 53)
(1 48)(2 45)(3 46)(4 47)(5 35)(6 36)(7 33)(8 34)(9 32)(10 29)(11 30)(12 31)(13 41)(14 42)(15 43)(16 44)(17 54)(18 55)(19 56)(20 53)(21 50)(22 51)(23 52)(24 49)(25 59)(26 60)(27 57)(28 58)(37 63)(38 64)(39 61)(40 62)
(1 57)(2 46)(3 59)(4 48)(5 29)(6 42)(7 31)(8 44)(9 36)(10 38)(11 34)(12 40)(13 33)(14 39)(15 35)(16 37)(17 50)(18 56)(19 52)(20 54)(21 51)(22 53)(23 49)(24 55)(25 60)(26 45)(27 58)(28 47)(30 63)(32 61)(41 62)(43 64)
(1 32 26 44)(2 29 27 41)(3 30 28 42)(4 31 25 43)(5 58 62 46)(6 59 63 47)(7 60 64 48)(8 57 61 45)(9 52 16 53)(10 49 13 54)(11 50 14 55)(12 51 15 56)(17 39 24 34)(18 40 21 35)(19 37 22 36)(20 38 23 33)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G···4Q4R4S8A8B8C8D8E8F
order12222224···44···444888888
size11114882···24···488444488

32 irreducible representations

dim11111111112224
type++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16
kernelC42.360C23C42.7C22C4×SD16SD16⋊C4Q8⋊D4C4⋊SD16Q8.Q8C23.19D4C23.32C23C22.34C24C22⋊C4C4⋊C4Q8C2
# reps11222222112284

Matrix representation of C42.360C23 in GL6(𝔽17)

1300000
0130000
000010
000001
001000
000100
,
100000
010000
000100
0016000
000001
0000160
,
6160000
1110000
0001360
00130011
0060013
00011130
,
7130000
12100000
0011004
00011130
0001360
004006
,
100000
010000
001313314
001341414
003141313
001414134

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[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],[6,1,0,0,0,0,16,11,0,0,0,0,0,0,0,13,6,0,0,0,13,0,0,11,0,0,6,0,0,13,0,0,0,11,13,0],[7,12,0,0,0,0,13,10,0,0,0,0,0,0,11,0,0,4,0,0,0,11,13,0,0,0,0,13,6,0,0,0,4,0,0,6],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,13,13,3,14,0,0,13,4,14,14,0,0,3,14,13,13,0,0,14,14,13,4] >;

C42.360C23 in GAP, Magma, Sage, TeX

C_4^2._{360}C_2^3
% in TeX

G:=Group("C4^2.360C2^3");
// GroupNames label

G:=SmallGroup(128,1858);
// by ID

G=gap.SmallGroup(128,1858);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,1018,304,4037,1027,124]);
// 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*b^2*c,e*c*e^-1=b*c,d*e=e*d>;
// generators/relations

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