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G = C42.355C23order 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⋊Q87C2, (C4×Q16)⋊23C2, (C4×SD16)⋊8C2, C4.Q1624C2, C4⋊C4.74C23, C4⋊C8.56C22, (C2×C8).48C23, Q8⋊D4.2C2, C2.18(Q8○D8), Q16⋊C410C2, (C4×C8).112C22, (C2×C4).319C24, Q8.11(C4○D4), C22⋊Q1616C2, Q8.D420C2, C22⋊C4.149D4, (C4×D4).82C22, (C2×D4).94C23, C23.258(C2×D4), C4⋊Q8.106C22, SD16⋊C415C2, (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, C23.20D420C2, (C22×C4).292C23, C42.7C224C2, Q8⋊C4.37C22, C4.4D4.28C22, C23.19D4.2C2, C22.579(C22×D4), D4⋊C4.163C22, C23.32C239C2, (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

C1C2×C4 — C42.355C23
C1C2C4C2×C4C42C4×Q8C23.32C23 — C42.355C23
C1C2C2×C4 — C42.355C23
C1C22C42⋊C2 — C42.355C23
C1C2C2C2×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 [×3], C2 [×2], C4 [×2], C4 [×14], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×3], Q8 [×4], Q8 [×9], C23, C23, C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8 [×3], C2×Q8 [×6], C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C42⋊C2 [×2], C42⋊C2 [×2], C4×D4, C4×Q8 [×5], C4×Q8 [×2], C4⋊D4, C22⋊Q8, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C422C2, C4⋊Q8, C2×SD16 [×2], C2×Q16 [×2], C22×Q8, C42.7C22, C4×SD16, C4×Q16, SD16⋊C4, Q16⋊C4, Q8⋊D4, C22⋊Q16, Q8.D4 [×2], Q8⋊Q8, C4.Q16, C23.19D4, C23.20D4, C23.32C23, C22.36C24, C42.355C23
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, 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 27 19)(2 24 28 20)(3 21 25 17)(4 22 26 18)(5 62 14 12)(6 63 15 9)(7 64 16 10)(8 61 13 11)(29 38 36 41)(30 39 33 42)(31 40 34 43)(32 37 35 44)(45 59 49 55)(46 60 50 56)(47 57 51 53)(48 58 52 54)
(5 10)(6 11)(7 12)(8 9)(13 63)(14 64)(15 61)(16 62)(17 21)(18 22)(19 23)(20 24)(29 38)(30 39)(31 40)(32 37)(33 42)(34 43)(35 44)(36 41)(45 47)(46 48)(49 51)(50 52)(53 59)(54 60)(55 57)(56 58)
(1 48)(2 49)(3 46)(4 51)(5 40)(6 44)(7 38)(8 42)(9 35)(10 29)(11 33)(12 31)(13 39)(14 43)(15 37)(16 41)(17 56)(18 57)(19 54)(20 59)(21 60)(22 53)(23 58)(24 55)(25 50)(26 47)(27 52)(28 45)(30 61)(32 63)(34 62)(36 64)
(1 29 27 36)(2 30 28 33)(3 31 25 34)(4 32 26 35)(5 60 14 56)(6 57 15 53)(7 58 16 54)(8 59 13 55)(9 51 63 47)(10 52 64 48)(11 49 61 45)(12 50 62 46)(17 40 21 43)(18 37 22 44)(19 38 23 41)(20 39 24 42)

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

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

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

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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