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G = C42.15C23order 128 = 27

15th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.15C23, C4⋊C86C22, D4.8(C2×D4), C4⋊Q86C22, C4⋊C4.336D4, (C2×D4).150D4, D4.D43C2, (C2×C8).17C23, C4.73(C22×D4), D4.2D415C2, C4.34(C4⋊D4), C4⋊C4.383C23, (C2×C4).246C24, C22⋊C4.137D4, (C4×D4).67C22, (C2×D8).55C22, C4.4D46C22, C23.443(C2×D4), (C2×Q8).40C23, D4⋊C475C22, C2.12(D4○SD16), Q8⋊C478C22, (C22×SD16)⋊23C2, (C2×D4).388C23, C23.38D48C2, (C2×SD16).5C22, C22.11C2410C2, C23.24D423C2, C22.81(C4⋊D4), (C22×C4).976C23, (C22×C8).340C22, C42.6C224C2, C22.506(C22×D4), C23.38C238C2, (C22×D4).341C22, (C2×M4(2)).53C22, (C22×Q8).273C22, C42⋊C2.101C22, C4.156(C2×C4○D4), (C2×C4).466(C2×D4), C2.64(C2×C4⋊D4), (C2×C8⋊C22).11C2, (C2×C4).277(C4○D4), (C2×C4○D4).118C22, SmallGroup(128,1774)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.15C23
C1C2C4C2×C4C22×C4C22×D4C22.11C24 — C42.15C23
C1C2C2×C4 — C42.15C23
C1C22C42⋊C2 — C42.15C23
C1C2C2C2×C4 — C42.15C23

Generators and relations for C42.15C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=b2, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2b2c, de=ed >

Subgroups: 508 in 247 conjugacy classes, 100 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×7], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×2], C22 [×17], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×4], D4 [×10], Q8 [×8], C23, C23 [×9], C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C4⋊C4 [×4], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×4], SD16 [×12], C22×C4, C22×C4 [×6], C2×D4, C2×D4 [×6], C2×D4 [×4], C2×Q8, C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×4], C24, D4⋊C4 [×2], Q8⋊C4 [×6], C4⋊C8 [×4], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×4], C4×D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C22×C8, C2×M4(2), C2×D8 [×2], C2×SD16 [×6], C2×SD16 [×4], C8⋊C22 [×4], C22×D4, C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C42.6C22, D4.D4 [×4], D4.2D4 [×4], C22.11C24, C23.38C23, C22×SD16, C2×C8⋊C22, C42.15C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×8], C23 [×15], C2×D4 [×12], C4○D4 [×2], C24, C4⋊D4 [×4], C22×D4 [×2], C2×C4○D4, C2×C4⋊D4, D4○SD16 [×2], C42.15C23

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J4A4B4C4D4E···4L4M4N4O8A8B8C8D8E8F
order1222222222244444···4444888888
size1111224444822224···4888444488

32 irreducible representations

dim111111111122224
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D4D4C4○D4D4○SD16
kernelC42.15C23C23.24D4C23.38D4C42.6C22D4.D4D4.2D4C22.11C24C23.38C23C22×SD16C2×C8⋊C22C22⋊C4C4⋊C4C2×D4C2×C4C2
# reps111144111122444

Matrix representation of C42.15C23 in GL6(𝔽17)

1300000
040000
0000160
0000016
0016000
0001600
,
1600000
0160000
00161600
002100
00001616
000021
,
020000
900000
000500
0010000
0000012
000070
,
100000
0160000
001100
0001600
000011
0000016
,
1300000
040000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,16,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,2,0,0,0,0,16,1,0,0,0,0,0,0,16,2,0,0,0,0,16,1],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,10,0,0,0,0,5,0,0,0,0,0,0,0,0,7,0,0,0,0,12,0],[1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,0,0,0,1,16],[13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.15C23 in GAP, Magma, Sage, TeX

C_4^2._{15}C_2^3
% in TeX

G:=Group("C4^2.15C2^3");
// GroupNames label

G:=SmallGroup(128,1774);
// by ID

G=gap.SmallGroup(128,1774);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,1018,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=d^2=1,c^2=b^2,e^2=a^2*b^2,a*b=b*a,c*a*c^-1=a^-1*b^2,a*d=d*a,e*a*e^-1=a*b^2,c*b*c^-1=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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