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G = C42.356C23order 128 = 27

217th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.356C23, (C4×D8)⋊5C2, (C4×C8)⋊8C22, C4⋊D824C2, C4⋊C815C22, C4⋊C4.349D4, D4.Q821C2, C22⋊D816C2, D8⋊C411C2, C8⋊C46C22, C2.19(D4○D8), (C4×D4)⋊10C22, C41D46C22, C4⋊C4.75C23, (C2×C8).49C23, C2.D858C22, C4.Q814C22, D4.12(C4○D4), (C2×C4).320C24, C22⋊C4.150D4, C23.259(C2×D4), C42.C22C22, D4⋊C423C22, (C2×D8).127C22, (C2×D4).410C23, C4⋊D4.29C22, C22⋊C8.33C22, C22.11C2412C2, C23.19D420C2, (C22×C4).293C23, C42.7C225C2, C22.580(C22×D4), C22.34C242C2, (C22×D4).362C22, C42⋊C2.131C22, C2.121(C22.19C24), C4.205(C2×C4○D4), (C2×C4).504(C2×D4), SmallGroup(128,1854)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.356C23
C1C2C4C2×C4C42C4×D4C22.11C24 — C42.356C23
C1C2C2×C4 — C42.356C23
C1C22C42⋊C2 — C42.356C23
C1C2C2C2×C4 — C42.356C23

Generators and relations for C42.356C23
 G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, ac=ca, dad=ab2, ae=ea, cbc=ebe=b-1, bd=db, dcd=a2b2c, ece=bc, de=ed >

Subgroups: 476 in 209 conjugacy classes, 88 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×9], C22, C22 [×21], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×12], D4 [×4], D4 [×14], C23, C23 [×10], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×6], C4⋊C4 [×2], C2×C8 [×4], D8 [×8], C22×C4, C22×C4 [×6], C2×D4 [×4], C2×D4 [×9], C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×6], C4×D4 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22.D4 [×2], C42.C2, C41D4, C2×D8 [×4], C22×D4, C42.7C22, C4×D8 [×2], D8⋊C4 [×2], C22⋊D8 [×2], C4⋊D8 [×2], D4.Q8 [×2], C23.19D4 [×2], C22.11C24, C22.34C24, C42.356C23
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○D8 [×2], C42.356C23

Smallest permutation representation of C42.356C23
On 32 points
Generators in S32
(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)
(1 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 30 15 12)(6 31 16 9)(7 32 13 10)(8 29 14 11)
(1 32)(2 29)(3 30)(4 31)(5 26)(6 27)(7 28)(8 25)(9 19)(10 20)(11 17)(12 18)(13 23)(14 24)(15 21)(16 22)
(2 17)(4 19)(5 13)(6 8)(7 15)(9 11)(10 30)(12 32)(14 16)(22 27)(24 25)(29 31)
(1 18)(2 19)(3 20)(4 17)(5 10)(6 11)(7 12)(8 9)(13 30)(14 31)(15 32)(16 29)(21 23)(22 24)(25 27)(26 28)

G:=sub<Sym(32)| (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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,30,15,12)(6,31,16,9)(7,32,13,10)(8,29,14,11), (1,32)(2,29)(3,30)(4,31)(5,26)(6,27)(7,28)(8,25)(9,19)(10,20)(11,17)(12,18)(13,23)(14,24)(15,21)(16,22), (2,17)(4,19)(5,13)(6,8)(7,15)(9,11)(10,30)(12,32)(14,16)(22,27)(24,25)(29,31), (1,18)(2,19)(3,20)(4,17)(5,10)(6,11)(7,12)(8,9)(13,30)(14,31)(15,32)(16,29)(21,23)(22,24)(25,27)(26,28)>;

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), (1,28,20,23)(2,25,17,24)(3,26,18,21)(4,27,19,22)(5,30,15,12)(6,31,16,9)(7,32,13,10)(8,29,14,11), (1,32)(2,29)(3,30)(4,31)(5,26)(6,27)(7,28)(8,25)(9,19)(10,20)(11,17)(12,18)(13,23)(14,24)(15,21)(16,22), (2,17)(4,19)(5,13)(6,8)(7,15)(9,11)(10,30)(12,32)(14,16)(22,27)(24,25)(29,31), (1,18)(2,19)(3,20)(4,17)(5,10)(6,11)(7,12)(8,9)(13,30)(14,31)(15,32)(16,29)(21,23)(22,24)(25,27)(26,28) );

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)], [(1,28,20,23),(2,25,17,24),(3,26,18,21),(4,27,19,22),(5,30,15,12),(6,31,16,9),(7,32,13,10),(8,29,14,11)], [(1,32),(2,29),(3,30),(4,31),(5,26),(6,27),(7,28),(8,25),(9,19),(10,20),(11,17),(12,18),(13,23),(14,24),(15,21),(16,22)], [(2,17),(4,19),(5,13),(6,8),(7,15),(9,11),(10,30),(12,32),(14,16),(22,27),(24,25),(29,31)], [(1,18),(2,19),(3,20),(4,17),(5,10),(6,11),(7,12),(8,9),(13,30),(14,31),(15,32),(16,29),(21,23),(22,24),(25,27),(26,28)])

32 conjugacy classes

class 1 2A2B2C2D···2H2I2J4A···4F4G···4M4N4O8A8B8C8D8E8F
order12222···2224···44···444888888
size11114···4882···24···488444488

32 irreducible representations

dim11111111112224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○D8
kernelC42.356C23C42.7C22C4×D8D8⋊C4C22⋊D8C4⋊D8D4.Q8C23.19D4C22.11C24C22.34C24C22⋊C4C4⋊C4D4C2
# reps11222222112284

Matrix representation of C42.356C23 in GL6(𝔽17)

1300000
0130000
0040150
0004015
00160130
00016013
,
1600000
0160000
0001600
001000
0000016
000010
,
0150000
800000
00125116
005566
00143512
00331212
,
1600000
010000
0016000
0001600
0013010
0001301
,
1600000
010000
000100
001000
000001
000010

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,16,0,0,0,0,4,0,16,0,0,15,0,13,0,0,0,0,15,0,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[0,8,0,0,0,0,15,0,0,0,0,0,0,0,12,5,14,3,0,0,5,5,3,3,0,0,11,6,5,12,0,0,6,6,12,12],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,13,0,0,0,0,16,0,13,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C42.356C23 in GAP, Magma, Sage, TeX

C_4^2._{356}C_2^3
% in TeX

G:=Group("C4^2.356C2^3");
// GroupNames label

G:=SmallGroup(128,1854);
// by ID

G=gap.SmallGroup(128,1854);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,1018,304,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^2,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d=a^2*b^2*c,e*c*e=b*c,d*e=e*d>;
// generators/relations

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